workbook_full_solutions_1 (1).pdf

WORKBOOK FULL SOLUTIONS
NEW SYLLABUS
MATHEMATICS
7th
EDITION
1
with
New Trend
Questions
1

1
1
ANSWERS
Chapter 1 Primes, Highest Common Factor
and Lowest Common Multiple
Basic
1. (a) 101 is an odd number, so it is not divisible by 2.
Since the sum of the digits 1 + 0 + 1 = 2 is not
divisible by 3 (divisibility test for 3), then 101 is
not divisible by 3. The last digit of 101 is neither
0 nor 5, so 101 is not divisible by 5.
A calculator may be used to test whether 101 is
divisible by prime numbers more than 5. Since
101 is not divisible by any prime numbers less than
101, 101 is a prime number.
(b) 357 is an odd number, so it is not divisible by 2.
Since the sum of the digits 3 + 5 + 7 = 15 which is
divisible by 3, therefore 357 is divisible by 3
(divisibility test for 3).
∴357 is a composite number.
(c) 411 is an odd number, so it is not divisible by 2.
Since the sum of the digits 4 + 1 + 1 = 6 which is
divisible by 3, therefore 411 is divisible by 3
(divisibility test for 3).
∴ 411 is a composite number.
(d) 1223 is an odd number, so it is not divisible by 2.
Since the sum of the digits 1 + 2 + 2 + 3 = 8 which
is not divisible by 3, then 1223 is not divisible by
3. The last digit of 1223 is neither 0 nor 5, so 1223
is not divisible by 5.
A calculator may be used to test whether 1223 is
divisible by prime numbers more than 5. Since
1223 is not divisible by any prime numbers less
than 1223, 1223 is a prime number.
(e) 1555 is an odd number, so it is not divisible by
2. Since the sum of the digits 1 + 5 + 5 + 5 = 16
which is not divisible by 3, so 1555 is not divisible
by 3. The last digit of 1555 is 5, so 1555 is divisible
by 5.
(f) 3127 is an odd number, so it is not divisible by
2. Since the sum of the digits 3 + 1 + 2 + 7 = 13,
then 3127 is not divisible by 3. A calculator may
be used to test whether 3127 is divisible by prime
numbers more than 3 and 3127 is divisible by 53,
which is a prime number.
2. The prime numbers less than 30 are 2, 3, 5, 7, 11, 13,
17, 19, 23 and 29.
Sum of prime numbers less than 30
= 2 + 3 + 5 +7 + 11 + 13 + 17 + 19 + 23 + 29
= 129
3. The two prime numbers between 20 and 30 are 23 and
29.
Difference of the two prime numbers = 29 – 23 = 6.
4. (a) Divide 315 by the smallest prime factor and
continue the process until we obtain 1.
3 315
3 105
5 35
7 7
1
315 = 3 × 3 × 5 × 7 = 32
× 5 × 7
(b) 2 8008
2 4004
2 2002
7 1001
11 143
13 13
1
8008 = 2 × 2 × 2 × 7 × 11 × 13
= 23
× 7 × 11 × 13

1 2
(c) 2 61 200
2 30 600
2 15 300
2 7650
3 3825
3 1275
5 425
5 85
17 17
1
61 200 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 × 17
= 24
× 32
× 52
× 17
(d) 2 58 752
2 29 376
2 14 688
2 7344
2 3672
2 1836
2 918
3 459
3 153
3 51
17 17
1
58 752 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 17
= 27
× 33
× 17
(e) 2 117 800
2 58 900
2 29 450
5 14 725
5 2945
19 589
31 31
1
117 800 = 2 × 2 × 2 × 5 × 5 × 19 × 31
= 23
× 52
× 19 × 31
5. (a) 2025 = 3 × 3 × 3 × 3 × 5 × 5
= (3 × 3 × 5) × (3 × 3 × 5)
= (3 × 3 × 5)2
∴ 2025 = 3 × 3 × 5 = 45
(b) 2304 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
= (2 × 2 × 2 × 2 × 3) × (2 × 2 × 2 × 2 × 3)
= (2 × 2 × 2 × 2 × 3)2
∴ 2304 = 2 × 2 × 2 × 2 × 3 = 48
(c) 3969 = 3 × 3 × 3 × 3 × 7 × 7
= (3 × 3 × 7) × (3 × 3 × 7)
= (3 × 3 × 7)2
∴ 3969 = 3 × 3 × 7 = 63
(d) 7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7
= 24
× 32
× 72
= (22
× 3 × 7)2
∴ 7056 = 22
× 3 × 7 = 84
(e) 5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3
= 23
× 36
= (2 × 32
)3
5832
3
= 2 × 32
= 18
(f) 9261 = 3 × 3 × 3 × 7 × 7 × 7
= 33
× 73
= (3 × 7)3
9261
3
= 3 × 7 = 21
(g) 17 576 = 2 × 2 × 2 × 13 × 13 × 13
= (2 × 13) × (2 × 13) × (2 × 13)
= (2 × 13)3
17 576
3
= 2 × 13 = 26
(h) 39 304 = 2 × 2 × 2 × 17 × 17 × 17
= (2 × 17) × (2 × 17) × (2 × 17)
= (2 × 17)3
39 304
3
= 2 × 17 = 34
6. 3136 = 26
× 72
∴ 3136 = ×
2 7
6 2
= 23
× 7
= 56
7. 59 319 = 33
× 133
∴ 59 319
3
= ×
3 13
3 3
3
= 3 × 13
= 39
8. (a) We observe that 48 is close to 49 which is a perfect
square. Thus 48 ≈ 49 = 7.
(b) We observe that 626 is close to 625 which is a
perfect square. Thus 626 ≈ 625 = 25.
(c) 65 is close to 64 which is a perfect cube. Thus
65
3
≈ 64
3
= 4.
(d) 998 is close to 1000 which is a perfect cube. Thus
998
3
≈ 1000
3
= 10.
(e) We observe that 99 is close to 100 which is a perfect
square and 28 is close to 27 which is a perfect cube.
Thus 99 – 28
3
≈ 100 – 27
3
= 7.
(f) We observe that 19 is close to 20 and 10 004 is
close to 10 000 which is a perfect square. Thus
192
× 10 004 ≈ 202
× 10 000 = 400 × 100
= 40 000.

1
3
(g) We observe that 11 is close to 10 and 7999 is
close to 8000 which is a perfect cube. Thus
113
+ 7999
3
≈ 103
+ 8000
3
= 1000 + 20 = 1020.
9. (a) 693
+ 1262
− 71 289 × 912 673
3
= 318 486
(b)
12 167
3
× 572
− 563
153 664
= −257.3699 (to 4 d.p.)
(c)
576 + 961 − 12 167
3
( )
4096
3
= 2
(d)
183
5184
+
162
− 75 357
3
223
−1032
− 753 571
3
= 76.8892 (to 4 d.p.)
10. (a) 16 = 2 × 2 × 2 × 2
24 = 2 × 2 × 2 × 3
2 2 2
HCF of 16 and 24 = 2 × 2 × 2 = 8
(b) 45 = 3 × 3 × 5
63 = 3 × 3 × 7
3 3
HCF of 45 and 63 = 3 × 3 = 9
(c) 56 = 2 × 2 × 2 × 7
70 = 2 × 5 × 7
2 7
HCF of 56 and 70 = 2 × 7 = 14
(d) 90 = 2 × 3 × 3 × 5
126 = 2 × 3 × 3 ×7
2 3 3
HCF of 90 and 126 = 2 × 3 × 3 = 18
(e) 1008 = 2 × 2 × 2 × 2 × 3 × 3 × 7
1960 = 2 × 2 × 2 × 5 × 7 × 7
2 2 2 7
HCF of 1008 and 1960 = 2 × 2 × 2 × 7 = 56
(f) 1080 = 2 × 2 × 2 × 3 × 3 × 3 × 5
1584 = 2 × 2 × 2 × 2 × 3 × 3 × 11
2 2 2 3 3
HCF of 1080 and 1584 = 2 × 2 × 2 × 3 × 3 = 72
(g) 42 = 2 × 3 × 7
66 = 2 × 3 × 11
78 = 2 × 3 × 13
2 3
HCF of 42, 66 and 78 = 2 × 3 = 6
(h) 132 = 2 × 2 × 3 × 11
156 = 2 × 2 × 3 × 13
180 = 2 × 2 × 3 × 3 × 5
2 2 3
HCF of 132, 156 and 180 = 2 × 2 × 3 = 12
(i) 84 = 2 × 2 × 3 × 7
98 = 2 × 7 × 7
112 = 2 × 2 × 2 × 2 × 7
2 7
HCF of 84, 98 and 112 = 2 × 7 = 14
(j) 195 = 3 × 5 × 13
270 = 2 × 3 × 3 × 3 × 5
345 = 3 × 5 × 23
3 5
HCF of 195, 270 and 345 = 3 × 5 = 15
(k) 147 = 3 × 7 × 7
231 = 3 × 7 × 11
273 = 3 × 7 × 13
3 7
HCF of 147, 231 and 273 = 3 × 7 = 21
(l) 225 = 3 × 3 × 5 × 5
495 = 3 × 3 × 5 × 11
810 = 2 × 3 × 3 × 3 × 3 × 5
3 3 5
HCF of 225, 495 and 810 = 3 × 3 × 5 = 45
11. (a) 48 = 2 × 2 × 2 × 2 × 3
72 = 2 × 2 × 2 × 3 × 3
2 2 2 2 3 3
LCM of 48 and 72 = 2 × 2 × 2 × 2 × 3 × 3 = 144
(b) 75 = 3 × 5 × 5
105 = 3 × 5 × 7
3 5 5 7
LCM of 75 and 105 = 3 × 5 × 5 × 7 = 525
(c) 243 = 3 × 3 × 3 × 3 × 3
405 = 3 × 3 × 3 × 3 × 5
3 3 3 3 3 5
LCM of 243 and 405 = 3 × 3 × 3 × 3 × 3 × 5
= 1215

1 4
(d) 261 = 3 × 3 × 29
435 = 3 × 5 × 29
3 3 5 29
LCM of 261 and 435 = 3 × 3 × 5 × 29 = 1305
(e) 144 = 2 × 2 × 2 × 2 × 3 × 3
306 = 2 × 3 × 3 × 17
2 2 2 2 3 3 17
LCM of 144 and 306 = 2 × 2 × 2 × 2 × 3 × 3 × 17
= 2448
(f) 264 = 2 × 2 × 2 × 3 × 11
504 = 2 × 2 × 2 × 3 × 3 × 7
2 2 2 3 3 7 11
LCM of 264 and 504 = 2 × 2 × 2 × 3 × 3 × 7 × 11
= 5544
(g) 1176 = 2 × 2 × 2 ×3 × 7 × 7
1960 = 2 × 2 × 2 × 5 × 7 × 7
2 2 2 3 5 7 7
LCM of 1176 and 1960 = 2 × 2 × 2 × 3 × 5 × 7 × 7
= 5880
(h) 56 = 2 × 2 × 2 × 7
72 = 2 × 2 × 2 × 3 × 3
104 = 2 × 2 × 2 × 13
2 2 2 3 3 7 13
LCM of 56, 72 and 104
= 2 × 2 × 2 × 3 × 3 × 7 × 13 = 6552
(i) 324 = 2 × 2 × 3 × 3 × 3 × 3
756 = 2 × 2 × 3 × 3 × 3 × 7
972 = 2 × 2 × 3 × 3 × 3 × 3 × 3
2 2 3 3 3 3 3 7
LCM of 324, 756 and 972
= 2 × 2 × 3 × 3 × 3 × 3 × 3 × 7
= 6804
(j) 450 = 2 × 3 × 3 × 5 × 5
720 = 2 × 2 × 2 × 2 × 3 × 3 × 5
1170 = 2 × 3 × 3 × 5 × 13
2 2 2 2 3 3 5 5 13
LCM of 450, 720 and 1170
= 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 × 13
= 46 800
12. (a) 84 = 2 × 2 × 3 × 7
189 = 3 × 3 × 3 × 7
3 7
HCF of 84 and 189 = 3 × 7 = 21
84 = 2 × 2 × 3 × 7
189 = 3 × 3 × 3 × 7
2 2 3 3 3 7
LCM of 84 and 189 = 2 × 2 × 3 × 3 × 3 × 7
= 756
(b) 315 = 3 × 3 × 5 × 7
720 = 2 × 2 × 2 × 2 × 3 × 3 × 5
3 3 5
HCF of 315 and 720 = 3 × 3 × 5 = 45
315 = 3 × 3 × 5 × 7
720 = 2 × 2 × 2 × 2 × 3 × 3 × 5
2 2 2 2 3 3 5 7
LCM of 315 and 720
= 2 × 2 × 2 × 2 × 3 × 3 × 5 × 7 = 5040
(c) 392 = 2 × 2 × 2 × 7 × 7
616 = 2 × 2 × 2 × 7 × 11
2 2 2 7
HCF of 392 and 616 = 2 × 2 × 2 × 7 = 56
392 = 2 × 2 × 2 × 7 × 7
616 = 2 × 2 × 2 × 7 × 11
2 2 2 7 7 11
LCM of 392 and 616 = 2 × 2 × 2 × 7 × 7 × 11
= 4312
(d) 1008 = 2 × 2 × 2 × 2 × 3 × 3 × 7
1764 = 2 × 2 × 3 × 3 × 7 × 7
2 2 3 3 7
HCF of 1008 and 1764 = 2 × 2 × 3 × 3 × 7
= 252
1008 = 2 × 2 × 2 × 2 × 3 × 3 × 7
1764 = 2 × 2 × 3 × 3 × 7 × 7
2 2 2 2 3 3 7 7
LCM of 1008 and 1764
= 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7
= 7056

1
5
(e) 140 = 2 × 2 × × 5 × 7
224 = 2 × 2 × 2 × 2 × 2 × 7
560 = 2 × 2 × 2 × 2 × 5 × 7
2 2 7
HCF of 140, 224 and 560 = 2 × 2 × 7 = 28
140 = 2 × 2 × 5 × 7
224 = 2 × 2 × 2 × 2 × 2 × 7
560 = 2 × 2 × 2 × 2 × 5 × 7
2 2 2 2 2 5 7
LCM of 140, 224 and 560
= 2 × 2 × 2 × 2 × 2 × 5 × 7 = 1120
(f) 315 = 3 × 3 × 5 × 7
525 = 3 × 5 × 5 × 7
1400 = 2 × 2 × 2 × 5 × 5 × 7
5 7
HCF of 315, 525 and 1400 = 5 × 7 = 35
315 = 3 × 3 × 5 × 7
525 = 3 × 5 × 5 × 7
1400 = 2 × 2 × 2 × 5 × 5 × 7
2 2 2 3 3 5 5 7
LCM of 315, 525 and 1400
= 2 × 2 × 2 × 3 × 3 × 5 × 5 × 7
= 12 600
(g) 252 = 2 × 2 × 3 × 3 × 7
378 = 2 × 3 × 3 × 3 × 7
567 = 3 × 3 × 3 × 3 × 7
3 3 7
HCF of 252, 378 and 567 = 3 × 3 × 7 = 63
252 = 2 × 2 × 3 × 3 × 7
378 = 2 × 3 × 3 × 3 × 7
567 = 3 × 3 × 3 × 3 × 7
2 2 3 3 3 3 7
LCM of 252, 378 and 567
= 2 × 2 × 3 × 3 × 3 × 3 × 7 = 2268
(h) 330 = 2 × 3 × 5 × 11
792 = 2 × 2 × 2 × 3 × 3 × 11
1188 = 2 × 2 × 3 × 3 × 3 × 11
2 3 11
HCF of 330, 792 and 1188 = 2 × 3 × 11 = 66
330 = 2 × 3 × 5 × 11
792 = 2 × 2 × 2 × 3 × 3 × 11
1188 = 2 × 2 × 3 × 3 × 3 × 11
2 2 2 3 3 3 5 11
LCM of 330, 792 and 1188
= 2 × 2 × 2 × 3 × 3 × 3 × 5 × 11
= 11 880
13. (a) 22
× 32
× 11
24
× 3 × 7
22
3
HCF = 22
× 3
22
× 32
× 11
24
× 3 × 7
24
32
7 11
LCM = 24
× 32
× 7 × 11
(b) 34
× 52
× 7
33
× 73
× 11
33
7
HCF = 33
× 7
34
× 52
× 7
33
× 73
× 11
34
52
73
11
LCM = 34
× 52
× 73
× 11
(c) 22
× 36
× 52
23
× 33
× 56
22
33
52
72 900 =
HCF = 22
× 33
× 52
22
× 36
× 52
23
× 33
× 56
23
36
56
72 900 =
LCM = 23
× 36
× 56

1 6
(d) 33
× 72
× 112
35
× 76
× 11
33
72
11
160 083 =
HCF = 33
× 72
× 11
33
× 72
× 112
35
× 76
× 11
35
76
112
160 083 =
LCM = 35
× 76
× 112
Intermediate
14. (a) The first 7 odd numbers are 1, 3, 5, 7, 9, 11 and
13.
The sum of the first 7 odd numbers
= 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49.
Difference between the square of 7 and the sum of
the first 7 odd numbers = 0.
(b) The length of its edge = 29 791
3
= 31.
The area of one side of box = 312
= 961 cm2
.
(c) 585 = 32
× 5 × 13
In order for 585p to be perfect square, 585p must
be expressed as a product of the square of its prime
factors.
Thus 32
× 52
× 132
is a perfect square and
32
× 5 × 13 × 5 × 13 = 585 × 5 × 13.
Thus the smallest p = 5 × 13 = 65.
15. 720 = 24
× 32
× 5
1575 = 32
× 52
× 7
(i) Largest prime factor of 720 and 1575 = 5
(ii) LCM of 720 and 1575 = 24
× 32
× 52
× 7
= 25 200
16. 374 = 2 × 11 × 17
34 = 2 × 17
So the smallest number that gives LCM of 374 is 11.
Thus m = 11.
17. (i) Divide 1764 by the smallest prime number until
we get 1.
2 1764
2 882
3 441
3 147
7 49
7 7
1
1764 = 22
× 32
× 72
= (2 × 3 × 7)2
1764 = 2 × 3 × 7 = 42
(ii) Divide 3375 by the smallest prime number until
we get 1.
3 3375
3 1125
3 375
5 125
5 25
5 5
1
3375 = 33
× 53
= (3 × 5)3
3375
3
= 3 × 5 = 15
(iii)Find the HCF and LCM of 15 and 42.
15 = 3 × 5
42 = 2 × 3 × 7
3
HCF of 15 and 42 = 3
15 = 3 × 5
42 = 2 × 3 × 7
2 3 5 7
LCM = 2 × 3 × 5 × 7 = 210

1
7
18. (a) (i) Divide 216 000 by the smallest prime number
until we get 1.
2 216 000
2 108 000
2 54 000
2 27 000
2 13 500
2 6750
3 3375
3 1125
3 375
5 125
5 25
5 5
1
216 000 = 26
× 33
× 53
(ii) Divide 518 400 by the smallest prime number
until we get 1.
2 518 400
2 259 200
2 129 600
2 64 800
2 32 400
2 16 200
2 8100
2 4050
3 2025
3 675
3 225
3 75
5 25
5 5
1
518 400 = 28
× 34
× 52
(b) (i) 216 000 = 26
× 33
× 53
= (22
× 3 × 5)3
216 000
3 = 22
× 3 × 5 = 60
(ii) 518 400 = 28
× 34
× 52
= (24
× 32
× 5)2
518 400 = 24
× 32
× 5 = 720
(iii) 26
× 33
× 53
28
× 34
× 52
26
33
52
216 000 =
518 400 =
HCF of 216 000 and 518 400 = 26
× 33
× 52
= 43 200
(iv) 60 = 22
× 3 × 5
720 = 24
× 32
× 5
24
32
5
LCM of 60 and 720 = 24
× 32
× 5 = 720
19. 84 = 2 × 2 × 3 × 7
126 = 2 × 3 × 3 × 7
(i) To find the length of each square is to find the
largest whole number which is a factor of both 84
and 126.
84 = 2 × 2 × 3 × 7
126 = 2 × 3 × 3 × 7
2 3 7
HCF of 84 and 126 = 2 × 3 × 7 = 42
Thus the length of each square is 42 cm.
(ii) Area of the rectangular sheet = 84 × 126
= 10 584 cm2
Area of each square = 42 × 42 = 1764 cm2
Number of squares that she can cut
= 10 584 ÷ 1764 = 6
20. 48 = 2 × 2 × 2 × 2 × 3
72 = 2 × 2 × 2 × 3 × 3
96 = 2 × 2 × 2 × 2 × 2 × 3
2 2 2 3
(i) Greatest number of discussion topics
= HCF of 48, 72 and 96
= 2 × 2 × 2 × 3
= 24
(ii) Number of participants from China in each
discussion group
= 96 ÷ 24
= 4

1 8
21. 8 = 2 × 2 × 2
10 = 2 × 5
12 = 2 × 2 × 3
2 2 2 3 5
∴LCM of 8, 10 and 12 = 2 × 2 × 2 × 3 × 5 = 120
∴ The three canteens will serve noodle soup again after
120 days.
22. (a) Volume of paper box = 8 × 12 × 16 = 1536 m3
Volume of each small cube = 2 × 2 × 2 = 8 m3
Number of small cubes that he is able to pack
= 1536 ÷ 8
= 192
(b) The length of each cube is the largest whole number
which is a factor of 8, 12 and 16.
8 = 2 × 2 × 2
12 = 2 × 2 × 3
16 = 2 × 2 × 2 × 2
2 2
HCF of 8, 12 and 16 = 4
Thus the length of each cube is 4 m.
23. 160 = 2 × 2 × 2 × 2 × 2 × 5
192 = 2 × 2 × 2 × 2 × 2 × 2 × 3
240 = 2 × 2 × 2 × 2 × 3 × 5
2 2 2 2
(i) Largest possible length of each piece of ribbon
= 2 × 2 × 2 × 2
= 16 cm
(ii) Total number of ribbons
= (160 ÷ 16) + (192 ÷ 16) + (240 ÷ 16)
= 37
24. To find the time when they next meet again is to find
the LCM of 126, 154 and 198 seconds.
126 = 2 × 3 × 3 × 7
154 = 2 × 7 × 11
198 = 2 × 3 × 3 × 11
2 3 3 7 11
LCM of 126, 154 and 198 = 2 × 3 × 3 × 7 × 11
= 1386
Time when they next meet again
= 4 pm + 23 min 6 s
= 4.23 pm
25. (i) To find the greatest number of hampers that can
be packed is to find the HCF of the boxes of
chocolates, the bottles of water and the tins
of biscuits.
420 = 2 × 2 × 3 × 5 × 7
630 = 2 × 3 × 3 × 5 × 7
1260 = 2 × 2 × 3 × 3 × 5 × 7
2 3 5 7
HCF of 420, 630 and 1260 = 2 × 3 × 5 × 7 = 210
(ii) Number of boxes of chocolate = 1260 ÷ 210
= 6
Number of bottles of water = 420 ÷ 210 =
2 Number of tins of biscuits = 630 ÷ 210 =
3
26. (i) Divide 13 824 by the smallest prime number until
we get 1.
2 13 824
2 6912
2 3456
2 1728
2 864
2 432
2 216
2 108
2 54
3 27
3 9
3 3
1
13 824 = 29
× 33
5 42 875
5 8575
5 1715
7 343
7 49
7 7
1
42 875 = 53
× 73
13 824 × 42 875 = 29
× 33
× 53
× 73
(ii) 13 824 × 42 875 = 29
× 33
× 53
× 73
= (23
× 3 × 5 × 7)3
×
13 824 42 875
3
= 23
× 3 × 5 × 7 = 840

1
9
Advanced
27. (a) True
If a and b are two prime numbers, a < b and
(a + b) is another prime number, the only possible
set of a and b is 2 and another prime number. The
only possible set of a and b are 2 and other prime
numbers. When 2 is added to the number, the sum
will turn out to be an odd number. As such, some
of the numbers will turn out to be prime numbers.
When an odd number (prime number) is added to
another prime number, the sum is an even number,
which will not be a prime number.
(b) False
Consider 1 × 2 = 2. 2 is a prime number.
(c) False
a + b = 2
(d) False
The digits of c × d = 56 as 383
= 54 872.
(e) False
When x = 62, the sum of the digits = 6 + 2 = 8. But
62 is not divisible by 8.
(f) True
One example to verify this statement is 12 × 32
= 4 × 96 = 384.
(g) False
2 × 24 ≠ 6 × 8 × 12
28. (i) 15 = 3 × 5
20 = 2 × 2 × 5
27 = 3 × 3 × 3
LCM of 15, 20 and 27 = 2 × 2 × 3 × 3 × 3 × 5
= 540
The next event will happen 540 seconds or
9 minutes later, i.e. at 12.09 am.
(ii) Since it happens after every 9 minutes and there are
60 minutes between midnight and 1 am, it will
happen for another 6 times.
29. 24 = 2 × 2 × 2 × 3
42 = 2 × 3 × 7
60 = 2 × 2 × 3 × 5
LCM of 24, 42 and 60 = 2 × 2 × 2 × 3 × 5 × 7 = 840
Shortest possible length = 840 cm
30. 36 = 2 × 2 × 3 × 3
56 = 2 × 2 × 2 × 7
1512 = 2 × 2 × 2 × 3 × 3 × 3 × 7
Smallest value of n
= 3 × 3 × 3
= 27
31. A = 22
× 34
× 52
× 74
× 133
B = 24
× 36
× 52
× 75
× 1116
C = 37
× 52
× 7 × 172
(i) (a) HCF of A, B and C = 34
× 52
× 7
(b) LCM of A, B and C
= 24
× 37
× 52
× 75
× 1116
× 133
× 172
(ii) For B × D to be a perfect square, the powers of
B × D must be even. Hence D = 7 so that
B × D = 24
× 36
× 52
× 75
× 1116
× 7
= (22
× 33
× 5 × 73
× 118
)2
.
(iii)A × C = 22
× 34
× 52
× 74
× 133
× 37
× 52
× 7 × 172
= 22
× 311
× 54
× 75
× 133
× 172
In order for A × C × E to be a perfect cube, the
powers of A × C × E must be multiples of 3.
Thus E = 2 × 3 × 52
× 7 × 17.
32. Consider multiples of 4 and they are 8, 12, 16 and 20.
We can find the corresponding numbers which give
HCF = 4 and LCM = 120.
Case 1
4 = 2 × 2
LCM = 2 × 2 × 30 = 120. Thus the next number is
2 × 2 × 30 = 120.
The first set of numbers is 4 and 120.
Case 2
8 = 2 × 2 × 2
LCM = 2 × 2 × 2 × 15 = 120. Thus the next number is
2 × 2 × 15 = 60.
The second set of numbers is 8 and 60.
Case 3
12 = 2 × 2 × 3
2 × 2 × 10 = 40.
The third set of numbers is 12 and 40.
Case 4
20 = 2 × 2 × 5
2 × 2 × 6 = 24.
The last set of numbers is 20 and 24.
33. By observation, 19 × 11 = 209 where 19 + 11 = 30 but
209 does not contain all prime numbers.
So, we can try 19 × 2 × 3 × 5. But 19 + 2 + 3 + 5 ≠ 30
and 19 × 2 × 3 × 5 = 570 and 0 is not a prime number.
Therefore we can try 19 × 2 × 2 × 7.
19 + 2 + 2 + 7 = 30
and 19 × 2 × 2 × 7 = 532 and 5, 3 and 2 are prime
numbers.
So, the 3-digit number that satisfies all the conditions
is 532.

1 10
New Trend
34. (a) 504 = 23
× 32
× 7
(b) HCF: 2 × 3
LCM: 23
× 32
× 7
First number = 2 × 3 × 7 = 42
Second number = 23
× 32
= 72
35. (a) Total surface area = 2(10 × 12 + 10 × 8 + 12 × 8)
= 592 cm2
(b) 455 = 5 × 7 × 13
Length of side of each cube = 2 cm
∴ Dimensions of the cuboid are 10 cm by 14 cm
by 26 cm.
(c) Number of cubes required to form the largest cube
= 73
= 343
Number of cubes left = 455 − 343
= 112
36. (a) 2 3234
3 1617
7 537
7 77
11 11
1
3234 = 2 × 3 × 7 × 7 × 11 = 2 × 3 × 72
× 11
(b) 4 = 2 × 2
30 = 2 × 3 × 5
LCM of 4 and 30 = 2 × 2 × 3 × 5
= 60
Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20,
30, 60
37. (a) Divide 1200 by the smallest prime number until
we get 1.
2 1200
2 600
2 300
2 150
3 75
5 25
5 5
1
1200 = 24
× 3 × 52
= 2a
× 3b
× 5c
∴ a = 4, b = 1, c = 2
(b) (i) Divide 3240 by the smallest prime number until
we get 1.
2 3240
2 1620
2 810
3 405
3 135
3 45
3 15
5 5
1
3240 = 23
× 34
× 5
Divide 4212 by the smallest prime number until
we get 1.
2 4212
2 2106
3 1053
3 351
3 117
3 39
13 13
1
4212 = 22
× 34
× 13
(ii) The largest whole number which is a factor of
both 3240 and 4212 is 22
× 34
= 324.
(c) 3240 = 23
× 34
× 5
In order for
3240
k
to be a square number, the
powers of
3240
k
must be even.
Thus, k = 2 × 5 = 10.
38. A = 22
× 34
× 52
× 72
B = 24
× 36
× 52
× 1116
C = 37
× 52
× 7
(i) LCM of A, B and C = 24
× 37
× 52
× 72
× 1116
(ii) HCF of A, B and C = 34
× 52
= 2025
∴ The greatest number that will divide A, B and C
exactly is 2025
(iii)A × B = 26
× 310
× 54
× 72
× 1116
= (23
× 35
× 52
× 7 × 118
)2
×
A B = 23
× 35
× 52
× 7 × 118
(iv) In order for Ck to be a perfect cube, the powers of
Ck have to be multiples of 3.
Thus Ck has to be 39
× 53
× 73
which means
37
× 52
× 7 × 32
× 5 × 72
= C × 32
× 5 × 72
.
Thus k = 32
× 5 × 72
= 2205.

1
11
Chapter 2 Integers, Rational Numbers and
Real Numbers
Basic
1. (a) If –15 represents 15 m below sea level, then +20
represents 20 m above sea level.
(b) If –10 represents the distance of 10 km of a car
travelling south, then +10 represents the distance
of 10 km of a car travelling north.
(c) If +100 represents a profit of $100 on the sale of
a mobile phone, then –91 represents a loss of $91
on the sale.
(d) If +90° represents a clockwise rotation of 90°, then
–90° represents rotating 90° anticlockwise.
(e) If –5 represents 5 flights down the stairs, then 14
flights up the stairs is represented by +14.
(f) If +600 represents a deposit of $600 in the bank,
then a withdrawal of $60 is represented by –60.
2. (a)
10
5
5 10 15 20 25
20
15 25
(b)
1
1
0 2
2
4
4
7
7
10
10
3
3
6
6
9
9
12
5
5
8
8
11
11
(c) –5 –3 –1 1
–4 –2 0 2 3
–5 –3 –1 1
–4 –2 0 2 3
3. (a) 6 > –6
(b) 0 > –
4
5
(c) (–1)3
= –1
Since –1 < 3,
Therefore, (–1)3
> 3
(d) –12 > –16
(e) –
12
2
= –6
Since –6 < –5,
Therefore, –
12
2
> –5
(f) –5.6 > –3.4
(g) –64
3
= –4
– 16 = –4
Therefore, –64
3
= – 16
(h) – 25 = –5
– 15 = –3.873
Therefore, – 25 > – 15
4. (a)
–10
–15
–15 –1
0
–5 5 10
8
2 3
–15, –1, 2, 3, 8
(b)
–80 –40
–60 –20
–100
–100 –50 –8
–2
0
0
–100, –50, –8, –2, 0
(c) –
7
4
= –1.75
–
5
3
≈ –1.67
–
3
2
= –1.5
–
4
3
≈ –1.33
–3 –1
–2 0
–4
–3.6 1.5
–2
1 2
0
–
5
3
–
3
2
–
4
3
–
7
4
–3.6, –2, –
7
4
, –
5
3
, –
3
2
, –
4
3
, 0, 1.5
5. (a) –5 + 13 = 13 – 5
= 8
(b) –25 + 12 = 12 – 25
= –13
(c) 5 + (–4) = 5 – 4
= 1
(d) 19 + (–26) = 19 – 26
= –7
(e) –2 + (–2) = –2 – 2
= –4
(f) –5 + (–3) = –5 – 3
= –8
(g) –11 + (–10) = –11 – 10
= –21
(h) –25 + (–65) = –25 – 65
= –90
6. (a) 14 – 18 = –4
(b) –5 – 3 = –8
(c) –12 – 13 = –25
(d) –(–13) = 13

1 12
(e) 6 – (–11) = 6 + 11
= 17
(f) –8 – (–11) = –8 + 11
= 11 – 8
= 3
(g) (–17) – (–35) = –17 + 35
= 35 – 17
= 18
(h) (–25) – (–10) = –25 + 10
= 10 – 25
= –15
7. (a) 5 × (–4) = 5 × (–1 × 4)
= 5 × (–1) × 4
= (–1) × 20
= –20
(b) –3 × 8 = (–1 × 3) × 8
= (–1) × 3 × 8
= (–1) × 24
= –24
(c) (–4) × (–12) = (–1 × 4) × (–12)
= (–1 × 4) × (–1 × 12)
= (–1) × 4 × (–1) × 12
= (–1) × (–1) × 4 × 12
= 1 × 48
= 48
(d) –5(–16) = (–1 × 5) × (–16)
= (–1 × 5) × (–1 × 16)
= (–1) × 5 × (–1) × 16
= (–1) × (–1) × 5 × 16
= 1 × 80
= 80
(e) –10(–20) = (–1 × 10) × (–20)
= (–1 × 10) × (–1 × 20)
= (–1) × 10 × (–1) × 20
= (–1) × (–1) × 10 × 20
= 1 × 200
= 200
(f) 0 × (–18) = 0 × (–1) × 18
= (–1) × 0 × 18
= (–1) × 0
= 0
(g) 56 ÷ (–7) =
56
–7
= 56 ×
1
–7
= 56 × –
1
7






= –8
(h) 0 ÷ (–12) =
0
–12
= 0 ×
1
–12
= 0
(i) –100 ÷ (–4) =
–100
–4
= –100 ×
1
–4
= –100 × –
1
4






= 25
(j) (–75) ÷ (–25) =
–75
–25
= –75 ×
1
–25
= –75 × –
1
25






= 3
(k)
70
–14
= 70 ×
1
–14
= 70 × –
1
14






= –5
(l)
–90
–15
= –90 ×
1
–15
= –90 × –
1
15






= 6
8. (a) (–2) × (–3) × (–4) × (–5) = 6 × (–4) × (–5)
= –(6 × 4) × (–5)
= (–24) × (–5)
= 120
(b) (–8) × (–3) × 5 × (–6) = 24 × 5 × (–6)
= 120 × (–6)
= –(120 × 6)
= –720
(c) (–2) × 5 × (–9) × (–7) = –(2 × 5) × (–9) × (–7)
= –10 × (–9) × (–7)
= 90 × (–7)
= –630
(d) 4 × (–4) × (–5) × (–16)
= –(4 × 4) × (–5) × (–16)
= –16 × (–5) × (–16)
= 80 × (–16)
= –1280
(e) 5 × 6 × (–1) × (–12) = 30 × (–1) × (–12)
= (–30 × 1) × (–12)
= –30 × (–12)
= 360

1
13
(f) (–1) × (–8) × 3 × 5 = 8 × 3 × 5
= 24 × 5
= 120
(g) 140 ÷ (–7) ÷ 4 =
140
–7





 ÷ 4
= 140 ×
–1
7





 ÷ 4
= (–20) ÷ 4
=
–20
4
= –20 ×
1
4






= –5
(h) (–264) ÷ 11 ÷ 8 =
–264
11





 ÷ 8
= –264 ×
1
11





 ÷ 8
= (–24) ÷ 8
= –3
(i) (–390) ÷ (–13) ÷ (–5) =
–390
–13





 ÷ (–5)
= –390 ×
1
–13





 ÷ (–5)
= (30) ÷ (–5)
= –(30 ÷ 5)
= –6
(j) (–9) × (–4) ÷ (–12) = (36) ÷ (–12)
=
36
–12






= (36) × –
1
12






= –3
(k) (–56) ÷ (–8) × 15 =
–56
–8





 × 15
= –56 ×
1
–8





 × 15
= 7 × 15
= 105
(l) –288 ÷ (–2) × (–3)2
=
–288
–2







 × (–3)2

= –288 ×
1
–2













 × (–3)2
= 144
( ) × (–3)2
= 12 × (–3)2
= 12 × 9
= 108
9. (a) (–2) × (–3) × (–4) × (–5) = 120
(b) (–8) × (–3) × 5 × (–6) = –720
(c) (–2) × 5 × (–9) × (–7) = –630
(d) 4 × (–4) × (–5) × (–16) = –1280
(e) 5 × 6 × (–1) × (–12) = 360
(f) (–1) × (–8) × 3 × 5 = 120
(g) 140 ÷ (–7) ÷ 4 = –5
(h) (–264) ÷ 11 ÷ 8 = –3
(i) (–390) ÷ (–13) ÷ (–5) = –6
(j) (–9) × (–4) ÷ (–12) = –3
(k) (–56) ÷ (–8) × 15 = 105
(l) –288 ÷ (–2) × (−3)2
= 108
10. (a) [(–3) + (–4)] ÷ 7 = [(–3) – 4] ÷ 7
= (–7) ÷ 7
=
–7
7
= –7 ×
1
7
= –1
(b) (–56) ÷ [7 + (–14)] = (–56) ÷ [7 – 14]
= (–56) ÷ (–7)
=
–56
–7
= –56 × –
1
7






= 8
(c) (–72) ÷ [–14 – (–23)] = (–72) ÷ (–14 + 23)
= (–72) ÷ (9)
=
–72
9
= –72 ×
1
9
= –8
(d) 32 + (–16) ÷ (–2)2
= 32 + (–16) ÷ 4
= 32 +
–16
4






= 32 + –16 ×
1
4






= 32 + (–4)
= 32 – 4
= 28
(e) 5 × (–4)2
– (–3)3
= 5 × (16) – (–27)
= 80 – (–27)
= 80 + 27
= 107

1 14
(f) (47 + 19 – 36) ÷ (–5) = (66 – 36) ÷ (–5)
= 30 ÷ (–5)
=
30
–5
= 30 × –
1
5
= –6
(g) 6 – (–3)2
+ 6 ÷ (–3) = 6 – 9 + 6 ÷ (–3)
= 6 – 9 +
6
–3






= 6 – 9 + 6 × –
1
3






= 6 – 9 + (–2)
= –3 – 2
= –5
(h) (–2)3
× (–2)2
– 8 ÷ (–2)3
= (–8) × 4 – 8 ÷ (–8)
= –32 – 8 ÷ (–8)
= –32 +
–8
–8






= –32 + –8 × –
1
8






= –32 + 1
= –31
11. (a) [(–3) + (–4)] ÷ 7 = –1
(b) (–56) ÷ [7 + (–14)] = 8
(c) (–72) ÷ [–14 – (–23)] = –8
(d) 32 + (–16) ÷ (–2)2
= 28
(e) 5 × (–4)2
– (–3)3
= 107
(f) (47 + 19 – 36) ÷ (–5) = –6
(g) 6 – (–3)2
+ 6 ÷ (–3) = –5
(h) (–2)3
× (–2)2
– 8 ÷ (–2)3
= –31
12. (a) 2
5
9
– 3
1
4
=
23
9
–
13
4
=
×
×
23 4
9 4
–
×
×
13 9
4 9
=
92
36
–
117
36
=
92 – 117
36
= –
25
36
(b) 2
1
4
+ –1
3
5





 = 2
1
4
– 1
3
5
=
9
4
–
8
5
=
×
×
9 5
4 5
–
×
×
8 4
5 4
=
45 – 32
20
=
13
20
(c) 9
1
4
+ –7
3
5





 = 9
1
4
– 7
3
5
=
37
4
–
38
5
=
×
×
37 5
4 5
–
×
×
38 4
5 4
=
185 – 152
20
=
33
20
= 1
13
20
(d)
2
5
– –
1
6





 =
2
5
+
1
6
=
×
×
2 6
5 6
+
×
×
1 5
6 5
=
+
12 5
30
=
17
30
(e) –1
1
5
+ –1
1
3





 = –
6
5
– 1
1
3
= –
6
5
–
4
3
=
×
×
–6 3
5 3
–
×
×
4 5
3 5
=
–18 – 20
15
=
–38
15
= –2
8
15
(f) –2
1
3
– –1
1
2





 = –2
1
3
+ 1
1
2
= –
7
3
+
3
2
=
×
×
–7 2
3 2
+
×
×
3 3
2 3
=
–14 9
6
+
= –
5
6
(g) –4
2
9
– –1
1
6





 = –4
2
9
+ 1
1
6
= –
38
9
+
7
6
=
×
×
–38 6
9 6
+
×
×
7 9
6 9
=
+
–228 63
54
= –
165
54
= –
55
18

1
15
(h) – –
7
8





 – 1
3
4
=
7
8
– 1
3
4
=
7
8
–
7
4
=
×
×
7 4
8 4
–
×
×
7 8
4 8
=
28 – 56
32
= –
28
32
= –
7
8
13. (a) 2
5
9
– 3
1
4
= –
25
36
(b) 2
1
4
+ –1
3
5





 =
13
20
(c) 9
1
4
+ –7
3
5





 = 1
13
20
(d)
2
5
– –
1
6





 =
17
30
(e) –1
1
5
+ –1
1
3





 = –2
8
15
(f) –2
1
3
– –1
1
2





 = –
5
6
(g) –4
2
9
– –1
1
6





 = –
55
18
(h) – –
7
8





 – 1
3
4
= –
7
8
14. (a) 5 × –2
2
5





 = 5
1
×
–12
5 1








= –12
(b) –
4
5





 ÷ (–16) = –
1
4
5








×
–1
16 4








=
1
20
(c) 16
3
10
× –
5
8





 =
163
2 10







 × –
5
1
8








= –
163
16
= –10
3
16
(d) –
4
9
×
3
14
= –
4
9
2
3
×
3
14
1
7
= –
2
21
(e) –3
1
2





 × 2
3
5
= –
7
2
×
13
5
= –
91
10
= –9
1
10
(f) –7
1
3





 ÷ 1
5
6
= –
22
3
÷
11
6
= –
22
3
2
1
×
6
11
2
1
= – 4
(g) –
7
18
1
2
× –
9
1
14 2








= –
1
2
× –
1
2






= – –
1
4






=
1
4
(h) –
5
6





 ÷ –1
3
4





 = –
5
6
÷ –
7
4






= –
5
6
3
× –
4
2
7








=
10
21
15. (a) 5 × –2
2
5





 = –12
(b) –
4
5





 ÷ (–16) =
1
20
(c) 16
3
10
× –
5
8





 = –10
3
16
(d) –
4
9
×
3
14
= –
2
21
(e) –3
1
2





 × 2
3
5
= –9
1
10
(f) –7
1
3





 ÷ 1
5
6
= – 4
(g) –
7
18
1
2
× –
9
1
14 2








=
1
4
(h) –
5
6





 ÷ –1
3
4





 =
10
21
16. (a) 1 4.8
× 6.2
2 9 6
+ 8 8 8
9 1.7 6
14.8 × 6.2 = 91.76
(b) 1 4 4.7 3 5
× 0.1 5
7 2 3 6 7 5
+ 1 4 4 7 3 5
2 1.7 1 0 2 5
144.735 × 0.15 = 21.710 25

1 16
(c) 0.3 5
× 0.0 9 6
2 1 0
+ 3 1 5
0.0 3 3 6 0
0.35 × 0.096 = 0.0336
(d) 1.8 4
× 0.0 9 2
3 6 8
+ 1 6 5 6
0.1 6 9 2 8
1.84 × 0.092 = 0.169 28
(e) 1.45 ÷ 0.16
=
1.45
0.16


=
145
16
9 . 0 6 2 5
16) 1 4 5 . 0 0 0 0
– 1 4 4
1 0 0
– 9 6
4 0
– 3 2
8 0
– 8 0
0
1.45 ÷ 0.16 = 9.0625
(f) 4.86 ÷ 1.20
=


4.86
1.20
=
486
120
4. 0 5
120) 4 8 6
–4 8 0
6 0
– 0
6 0 0
– 6 0 0
0
486 ÷ 120 = 4.05
(g) 1.921 68 ÷ 62.8
=


1.92168
62.8
=
19.2168
628
0.0 3 06
628) 1 9.2 1 68
–0
1 9 2
– 0
1 9 2 1
–1 8 8 4
3 7 6
– 0
3 7 68
– 3 7 68
0
1.92168 ÷ 62.8 = 0.0306
(h) 0.003 48 ÷ 0.048
=


0.00348
0.048
=
3.48
48
0. 0 7 2 5
48) 3. 4 8
–0
3 4
– 0
3 4 8
–3 3 6
1 2 0
– 9 6
2 4 0
– 2 4 0
0
0.003 48 ÷ 0.048 = 0.0725
17. (a) 5.3 – (–4.9)
= 5.3 + 4.9
= 10.2
(b) 3.3 + (–2.7)
= 3.3 – 2.7
= 0.6
(c) –15.4 + 8.9
= –(15.4 – 8.9)
= –6.5
(d) –17.3 – 6.25
= –(17.3 + 6.25)
= –23.55

1
17
Intermediate
18. (a)
45 46 48 51 54
47
47
50 53
53
49 52 55
(b) Factors of 64 and 80 are 1, 2, 4, 5, 8 and 16.
Composite numbers that are factors of both 64
and 80 are 4, 8, 16.
4
4
7 10 13
6 9 12 15 16
16
5 8
8
11 14
(c) Natural numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9,
10……
1
1
2
2
0 4
4
7
7
10
10
3
3
6
6
9
9
5
5
8
8
19. (a)
2
11
= 0.1818
0.325 = 0.1803
(0.428)2
= 0.1830
2
11 (0.428)2
0.1800 0.1840
0.1820 0.1860 0.1870
0.1810 0.1850
0.1830 0.1880 0.1890
.
0.18
√0.325
0.18
.
, (0.428)2
,
2
11
, 0.325
(b)
10
11
= 0.9090
0.955
3
= 0.984
3
√0.955
10
11
0.9000 0.9800
0.9400
0.9200 1.000
0.9600
.
0.9
.
0.909
0.9
.
, 0.955
3
, 0.909
.
,
10
11
(c) π
3
= 1.047 20
1
1
9
= 1.1111
12
11
= 1.0909
5
2
= 1.118 03
1.00 1.04 1.08
1.02 1.06 1.10
1.01 1.05 1.09
1.03 1.07 1.11 1.12
. .
1.01 1
1
9
p
3
12
11
√5
2

5
2
, 1
1
9
,
12
11
, π
3
, 1.01
.
.
20. (a) 4 + (–15) – 21
= 4 – 15 – 21
= –11 – 21
= –(11 + 21)
= –32
(b) –4 + (–12) + 10
= –4 – 12 + 10
= –16 + 10
= –(16 – 10)
= –6
(c) –5 + (–7) – (–13)
= –5 – 7 + 13
= –12 + 13
= –(12 – 13)
= –(–1)
= 1
(d) 20 + (–9) – (–16)
= 20 – 9 – (–16)
= 20 – 9 + 16
= 11 + 16
= 27
(e) 3 – (–7) – 4 + (–4)
= 3 + 7 – 4 – 4
= 10 – 4 – 4
= 6 – 4
= 2
(f) –27 – (–35) – 5 + (–9)
= –27 + 35 – 5 – 9
= –(27 – 35) – 5 – 9
= –(–8) – 5 – 9
= 8 – 5 – 9
= 3 – 9
= –6
(g) 35 – (–5) + (–12) – (–8)
= 35 + 5 – 12 + 8
= 40 – 12 + 8
= 28 + 8
= 36
(h) 23 + (–3) – (–7) + (–22)
= 23 – 3 + 7 – 22
= 20 + 7 – 22
= 27 – 22
= 5
(i) –14 – [–6 + (–15)]
= –14 – (–6 – 15)
= –14 – (–21)
= –14 + 21
= 7

1 18
(j) [–4 + (–14)] + [–8 – (–26)]
= (–4 – 14) + (–8 + 26)
= (–18) + (26 – 8)
= –18 + 18
= 0
21. [–2 + (–14) – 10] – [(–6)2
+ (–17) – (–9)]
= (–2 –14 – 10) – [36 + (–17) – (–9)]
= (–26) – (36 – 17 + 9)
= –26 – 28
= –(26 + 28)
= –54
22. (a)
(–2) (–5) (–20)
(–10)
× +
=
10 (–20)
–10
+
=
10 – 20
–10
=
–10
–10
= 1
(b)
(–123) [19 (–19)]
38
× +
=
(–123) (19 – 19)
38
×
=
–123 0
38
×
=
0
38
= 0
(c) (–11) × [–52 + (–17) – (–39)]
= (–11) × (–52 – 17 + 39)
= (–11) × (–69 + 39)
= (–11) × (–30)
= –(–330)
= 330
(d) [109 – (–19)] ÷ (–2)3
× (–5)
= (109 + 19) ÷ (–2)3
× (–5)
= 128 ÷ (–8) × (–5)
=
128
–8





 × (–5)
= –16 × (–5)
= –(–80)
= 80
(e) (13 – 9)2
– 52
– (28 – 31)3
= 42
– 52
– (–3)3
= 16 – 25 – (–27)
= 16 – 25 + 27
= 18
(f) 16 + (–21) ÷ 7 × {9 + [56 ÷ (–8)]}
= 16 + (–21) ÷ 7 × [9 + (–7)]
= 16 + (–21) ÷ 7 × (9 – 7)
= 16 + (–21) ÷ 7 × 2
= 16 +
–21
7
× 2






= 16 + (–3 × 2)
= 16 + (–6)
= 16 – 6
= 10
(g) 8 ÷ [3 + (–15)] ÷ [(–2) × 4 × (–3)]
= 8 ÷ (3 – 15) ÷ [(–2) × (–12)]
= 8 ÷ (–12) ÷ (24)
=
8
–12





 ÷ 24
= –
2
3





 ×
1
24
= –
1
36
(h) [(–5) × (–8)2
– (–2)3
× 7] ÷ (–11)
= [(–5) × 64 – (–8) × 7] ÷ (–11)
= [–(5 × 64) – [–(8 × 7)]] ÷ (–11)
= [–320 – (–56)] ÷ (–11)
= (–320 + 56) ÷ (–11)
= (–264) ÷ (–11)
= 24
(i) {[(–23) – (–11)] ÷ 6 – 7 ÷ (–7)} × 1997
= [(–23 + 11) ÷ 6 – 7 ÷ (–7) × 1997
= [(–12) ÷ 6 – 7 ÷ (–7)] × 1997
=
–12
6





 –
7
–7











 × 1997
= [(–2) – (–1)] × 1997
= (–2 + 1) × 1997
= (–1) × 1997
= –1997
(j) (–7)3
+ (–2)3
– [(–21) + 35 – 125
3
× (–8)]
= –343 + (–8) – [(–21) + 35 – 5 × (–8)]
= –343 + (–8) – [(–21) + 35 – (–40)]
= –343 + (–8) – [(–21) + 35 + 40]
= –343 + (–8) – (14 + 40)
= –343 – 8 – 54
= –(343 + 8 + 54)
= –405

1
19
23. (a)
(–2) (–5) (–20)
(–10)
× +
= 1
(b)
(–123) [19 (–19)]
38
× +
= 0
(c) (–11) × [–52 + (–17) – (–39)] = 330
(d) [109 – (–19)] ÷ (–2)3
× (–5) = 80
(e) (13 – 9)2
– 52
– (28 – 31)3
= 18
(f) 16 + (–21) ÷ 7 × {9 + [56 ÷ (–8)]} = 10
(g) 8 ÷ [3 + (–15)] ÷ [(–2) × 4 × (–3)] = –
1
36
(h) [(–5) × (–8)2
– (–2)3
× 7] ÷ (–11) = 24
(i) {[(–23) – (–11)] ÷ 6 – 7 ÷ (–7)} × 1997 = –1997
(j) (–7)3
+ (–2)3
– [(–21) + 35 – 125
3
× (–8)] = –405
24. (a) –5
2
9
– 3
1
4
– 3
5
9
= –5
8
36
– 3
9
36
– 3
20
36
= (–5 – 3 – 3) –
8
36
–
9
36
–
20
36
= –11 –
(8 9 20)
36
+ +
= –11 –
37
36
= –12
1
36
(b) –3
4
5
– 1
3
10
– –2
3
4






= –3
16
20
– 1
6
20
– –2
15
20






= –3
16
20
– 1
6
20
+ 2
15
20






= (–3 – 1 + 2) –
16
20
–
6
20
+
15
20
= –2 +
(–16 – 6 15)
20
+
= –2 +
–7
20






= –2 –
7
20
= –2
7
20
(c) –2
3
4
+ –1
1
2





 – –1
2
3






= –2
3
4
– 1
1
2
+ 1
2
3
= –2
9
12
– 1
6
12
+ 1
8
12
= (–2 – 1 + 1) –
9
12
–
6
12
+
8
12
= –2 –
7
12
= –2
7
12
(d) – –3
5
7





 + 1
3
5
– –
3
7






= 3
5
7
+ 1
3
5
+
3
7
= 3
25
35
+ 1
21
35
+
15
35
= (3 + 1) +
25
35
+
21
35
+
15
35
= 4 +
61
35
= 4 + 1
26
35
= 5
26
35
(e) –
1
5
+
1
3





 +
1
10
+ –
1
5











 + –
1
25






= –
3
15
+
5
15





 +
1
10
+ –
2
10











 + –
1
25






=
2
15
+ –
1
10





 –
1
25
=
2
15
–
1
10
–
1
25
=
20
150
–
15
150
–
6
150
= –
1
150
25. (a) –5
2
9
– 3
1
4
– 3
5
9
= –12
1
36
(b) –3
4
5
– 1
3
10
– –2
3
4





 = –2
7
20
(c) –2
3
4
+ –1
1
2





 – –1
2
3





 = –2
7
12
(d) – –3
5
7





 + 1
3
5
– –
3
7





 = 5
26
35
(e) –
1
5
+
1
3





 +
1
10
+ –
1
5











 + –
1
25





 = –
1
150
26. (a) (–4) ÷ –
1
4





 × (–4)
= (–4) × (–4) × (–4)
= 16 × (–4)
= –64
(b) –2
2
5





 ×
5
6





 ÷ (–13)
= –
2
12
1 5







 ×
51
61





 ÷ (–13)
= –2 ÷ (–13)
=
–2
–13
=
2
13

1 20
(c) 1
7
15





 ÷ –17
2
7





 × 3
3
14






=
22
15





 ÷
–121
7





 ×
45
14






=
22
1 15







 × –
7
1
121








×
45
3
14 2








= –
66
242
= –
3
11
(d) –2
5
7





 ÷ 1
1
3
×
3
4






= –2
5
7





 ÷
4
3
×
3
4






= –2
5
7





 ÷ 1
= –2
5
7
(e) 3
3
5





 × (–6) ÷ –4
4
5






=
18
5





 × (–6) ÷ –
24
5






=
18
1 5







 × (–6)
1
× –
5
1
24 4








=
18
4
= 4
2
4
= 4
1
2
(f)
1
4
+ –
3
4





 × –1
1
4






=
1
4
+ –
3
4





 × –
5
4






=
1
4
+
15
16






=
4
16
+
15
16






=
19
16
= 1
3
16
(g) –9
1
4
– –7
3
5

















 ÷ 2
3
4
= –9
1
4
+ 7
3
5











 ÷ 2
3
4
= –9
5
20
+ 7
12
20











 ÷ 2
3
4
= (–9 + 7) –
5
20
+
12
20





 ÷ 2
3
4
= (–2) +
7
20





 ÷ 2
3
4
= –
33
20





 ÷ 2
3
4
= –
33
20





 ÷
11
4
= –
33
20





 ×
4
11
= –
3
5
(h) –1
1
4





 + 1
2
5





 ÷ (–6) –
4
7
× –2
3
4












= –1
5
20





 + 1
8
20





 ÷ (–6) –
4
7
× –
11
4












= (–1 + 1) –
5
20
+
8
20





 ÷ (–6) –
4
7
× –
11
4












=
3
20





 ÷ (–6) – –
11
7












=
3
20





 ÷ –
42
7





 +
11
7












=
3
20
÷ –
31
7






=
3
20
× –
7
31






= –
21
620
(i) –
3
4





 × 1
1
2
+ –
3
4





 × –2
1
2






= –
3
4





 ×
3
2
+ –
3
4





 × –
5
2






= –
9
8





 +
15
8
=
6
8
=
3
4

1
21
27. (a) (–4) ÷ –
1
4





 × (–4) = –64
(b) –2
2
5





 ×
5
6





 ÷ (–13) =
2
13
(c) 1
7
15





 ÷ –17
2
7





 × 3
3
14





 = –
3
11
(d) –2
5
7





 ÷ 1
1
3
×
3
4





 = –2
5
7
(e) 3
3
5





 × (–6) ÷ –4
4
5





 = 4
1
2
(f)
1
4
+ –
3
4





 × –1
1
4





 = 1
3
16
(g) –9
1
4
– –7
3
5

















 ÷ 2
3
4
= –
3
5
(h) –1
1
4





 + 1
2
5





 ÷ (–6) –
4
7
× –2
3
4











 = –
21
620
(i) –
3
4





 × 1
1
2
+ –
3
4





 × –2
1
2





 =
3
4
28. (a)
0.25
0.05
×


–
0.18
1.3






=
25
5
×
–1.8
13






= 5 ×
–1.8
13






= –
9
13
(b)
0.0064
0.04
×


–
1.8
0.16






=
0.64
4
× –
180
16






= 0.16 × –
45
4






= –1.8
(c) (–0.3)2
×
–1.4
0.07





 – 0.78
= –
3
10






2
×
–140
7





 – 0.78
=
9
100





 × (–20) – 0.78
= –1.8 – 0.78
= –2.58
(d) (–0.4)3
×
–3.3
0.11





 + 0.123
= –
4
10






3
×
–3.3
0.11





 + 0.123
= –
64
1000





 ×
–330
11





 + 0.123
= –
64
1000





 × (–30) + 0.123
= 1.92 + 0.123
= 2.043
29. (a)
1
8
13
×
13
42
+5
1
5
÷
7
45
7
9
+
7
18





÷
1
18
×
1
7
= 11
13
42
(b)
13
3
– 7
48 – 101
3 = –0.130 (to 3 d.p.)
(c)
42.7863
3
× 41.567
( )
2
94 536.721
= 0.064 (to 3 d.p.)
(d)
9206× 29.5
( )
3
11.86
( )
2
3 = 118.884 (to 3 d.p.)
(e)
46.32
+85.92
– 70.72
2× 46.3×85.9
= 0.754 (to 3 d.p.)
(f)
18× 4.359
( )
2
+10× 3.465
( )
2
4.359
( )
3
+ 3× 3.465
( )
3
= 1.492 (to 3 d.p.)
30. Altitude at which the plane is flying now
= 650 – 150 + 830
= 500 + 830
= 1330 m
31. Temperature of Singapore after rain stops
= 24°C + 8°C – 12°C + 6°C
= 32°C – 12°C + 6°C
= 20°C + 6°C
= 26°C
32. Let x be the number of boys.
Number of sweets each boy will have = 6 – 1
= 5
Since Raj took the last sweet,
total number of sweets = 41
5x + 1 = 41
5x = 40
x =
40
5
x = 8
8 boys were seated around the table.
33. (a)
Packet 1 2 3 4 5
Mass
above
or
below
the
standard
mass (g)
–28 –13 +10 –19 +5
Actual
mass
(g)
1000 – 28
= 972 g
1000 – 13
= 987 g
1000 + 10
= 1010 g
1000 – 19
= 981 g
1000 + 5
= 1005 g
Packet 5
(b) (i) Difference = 1005 – 972
= 33 g

1 22
(ii) Difference = 1005 – 981
= 24 g
(iii)Difference = 1005 – 987
= 18 g
Packet 5 and packet 1 have the largest difference.
(c) Mass of rice in packet 6
=
972 + 1010
2
=
1982
2
= 991 g
34.
Reservoir A B C D
Water level
–2 + 6 + 8
= 12
+1 + 3 – 7
= –3
–3 – 1 – 2
= –6
–5 + 9 – 1
= 3
(a) Reservoir A caught the most rain.
(b) Reservoir C caught the least rain.
(c) Reservoir D because 3 > –3.
35. (i) Cost of ride = $5 – $3.36
= $1.64
(ii) Total value of card = $20 + $3.36
= $23.36
Total cost in a day = $1.83 × 2
= $3.66
Therefore, number of days before he needs
to top up his card
=
23.36
3.66
= 6.38 (to 3 s.f.)
Hence, he will need to top up his card next
Sunday.
36. Let the length of shorter piece of rope be x m.
Therefore, length of the longer piece of rope =
5
4
x m.
x +
5
4
x = 6.3
4
4
x +
5
4
x = 6.3
9
4
x = 6.3
x = 6.3 ×
4
9
x = 2.8
Length of the shorter piece of rope is 2.8 m
37. Number of cups of flour Priya used
= 2
1
2
× 9





 + 2
3
4
× 3






=
5
2
× 9





 +
11
4
× 3






=
45
2





 +
33
4






=
90
4
+
33
4
=
123
4
= 30
3
4
38. Number of students in class A
=
4
19
× 247
= 52
Number of students in class that travel to school by bus
=
8
13
× 52





 + 7
= 32 + 7
= 39
Therefore, number of students in class A who do not
travel by bus
= 52 – 39
= 13
39. (i) Fraction of cost price of refrigerator that Huixian
pays
= 1 –
3
10
–
9
20
=
20
20
–
6
20
–
9
20
=
5
20
=
1
4
(ii) Cost of refrigerator
= $525 ÷
1
4
= $525 × 4
= $2100

1
23
40. Fraction of students who failed the test
= 1 –
1
7
–
1
3
–
1
2
=
42
42
–
6
42
–
14
42
–
21
42
=
1
42
Fraction of students who scored A and B
=
1
7
+
1
3
=
3 7
21
+
=
10
21
=
20
42
Therefore, number of students who failed the test
=
100
20
× 1
= 5
41. Let the money that Junwei has be $x.
His wife will receive $
3
7
x.
Rui Feng will receive x –
3
7
x





 ×
1
2
=
4
7
x ×
1
2
= $
2
7
x
Fraction of money distributed to each child
= x –
3
7
x +
2
7
x











 ÷ 3
= x –
5
7
x





 ÷ 3
=
2
7
x ÷ 3
=
2
7
x ×
1
3
=
x
2
21
Therefore,
x
2
21
= 400
2x = 8400
x = 4200
Hence, his wife will receive
=
3
7
× 4200
= $1800
Advanced
42. –4 (–5.5) – [–2 (–3) 8(–2) – 8 2] 12 – (–4)
2 3
4
× × + × +
= –4 (–5.5) – [6 (–16) – 16] 12 – (–4)
2 3
4
× + +
= –4 (–5.5) – (6 – 16 – 16) 144 – (–64)
4 × +
= –4 (–5.5) – (–26) 144 64
4 × + +
= 22 26 144 64
4
+ + +
= 256
4
= 4
43. Fraction of land used for phase 1
=
11
18
+
3
7





 1 –
11
18






=
11
18
+
3
7






7
18






=
11
18
+
1
6






=
11
18
+
3
18
=
14
18
=
7
9
Fraction of land used for phase 2
=
1
4
× 1 –
7
9






=
1
4
×
2
9
=
1
18
Fraction of land used for shopping malls and medical
facilities
= 1 –
7
9
–
1
18
=
18
18
–
14
18
–
1
18
=
3
18
=
1
6
New Trend
44. Arranging in ascending order,
0.85
3
2
,
π
4
, 0.64 , 0.801

1
Chapter 3 Approximation and Estimation
Basic
1. (a) 789 500 ( to the nearest 100)
(b) 790 000 (to the nearest 1000)
(c) 790 000 (to the nearest 10 000)
2. (a) 2.5 (to 1 d.p.)
(b) 18.5 (to 1 d.p.)
(c) 36.1 (to 1 d.p.)
(d) 138.1 (to 1 d.p.)
3. (a) 4.70 ( to 2 d.p.)
(b) 14.94 (to 2 d.p.)
(c) 26.80 (to 2 d.p.)
(d) 0.05 (to 2 d.p.)
4. (a) 4.826 (to 3 d.p.)
(b) 6.828 (to 3 d.p.)
(c) 7.450 (to 3 d.p.)
(d) 8.445 (to 3 d.p.)
(e) 11.639 (to 3 d.p.)
(f) 13.451 (to 3 d.p.)
(g) 32.929 (to 3 d.p.)
(h) 0.038 (to 3 d.p.)
5. (a) 36.3 (to 1 d.p.)
(b) 36 (to the nearest whole number)
(c) 36.260 (to 3 d.p.)
6. (a) All zeros between non-zero digits are significant.
5 significant figures
(b) In a decimal, all zeros before a non-zero digit are not
significant.
4 significant figures
(c) 5 significant figures
(d) 9 or 10 significant figures.
7. (a) 3.9 (to 2 s.f.)
(b) 20 (to 2 s.f.)
(c) 38 (to 2 s.f.)
(d) 4.07 (to 3 s.f.)
(e) 18.1 (to 3 s.f.)
(f) 0.0326 (to 3 s.f.)
(g) 0.0770 (to 3 s.f.)
(h) 0.008 17 (to 3 s.f.)
(i) 18.14 (to 4 s.f.)
(j) 240.0 (to 4 s.f.)
(k) 5004 (to 4 s.f.)
(l) 0.054 45 (to 4 s.f.)
8. (a) 20 (to 1 s.f.)
(b) 19.1 (to 1 d.p.)
(c) 19 (to 2 s.f.)
9. (a) 0.007 (to 1 s.f.)
(b) 0.007 (to 3 d.p.)
(c) 0.007 20 (to 3 s.f.)
10. (a) 984.61 (to 2 d.p.)
(b) 984.6 (to 4 s.f.)
(c) 984.608 (to 3 d.p.)
(d) 984.61 (to the nearest hundredth)
11. (a) 0.000 143 (to 3 s.f.)
(b) 5.1 (to 1 d.p.)
(c) 1000 (to 2 s.f.)
12. (a) 0.3403 (to 4 s.f.)
(b) 10.255 (to 5 s.f.)
(c) 64 704 800 (to 6 s.f.)
13. (a) 428.2 (to 4 s.f.)
The number of decimal places in the answer is 1.
(b) 0.000 90 (to 5 d.p.)
The number of significant figures is 1 or 2, depending
on whether the last zero is included or otherwise.
14. (a) 4 cm (to the nearest cm)
(b) 24 cm (to the nearest cm)
(c) 107 cm (to the nearest cm)
(d) 655 cm (to the nearest cm)
15. (a) 14.0 kg (to the nearest 0.1 kg)
(b) 57.5 kg (to the nearest 0.1 kg)
(c) 108.4 kg (to the nearest 0.1 kg)
(d) 763.2 kg (to the nearest 0.1 kg)
16. (a) 7.0 cm2
(to the nearest
1
10
cm2
)
(b) 40.1 cm2
(to the nearest
1
10
cm2
)
(c) 148.3 cm2
(to the nearest
1
10
cm2
)
(d) 168.4 cm2
(to the nearest
1
10
cm2
)
17. (a) 5620 km (to the nearest 10 km)
(b) 900 cm (to the nearest 100 cm)
(c) 2.45 g (to the nearest
1
100
g)
(d) $50 000 (to the nearest $10 000)
18. (a) 61.994 06 – 29.980 78
= 32.013 28
=
30 (to 1 s.f.)
(b) 64.967 02 – 36.230 87
= 28.736 15
=
30 (to 1 s.f.)
(c) 4987 × 91.2
= 454 814.4
=
500 000 (to 1 s.f.)
24

1
(d) 30.9 × 98.6
= 3046.74
=
3000 (to 1 s.f.)
(e) 0.0079 × 21.7
= 0.171 43
=
0.2 (to 1 s.f.)
(f) 1793 × 0.000 97
= 1.739 21
=
2 (to 1 s.f.)
(g) 9801 × 0.0613
= 600.8013
=
600 (to 1 s.f.)
(h) (8.907)2
= 79.334 649
=
80 (to 1 s.f.)
(i) (398)2
× 0.062
= 9821.048
=
10 000 (to 1 s.f.)
(j) 81.09 ÷ 1.592
= 50.935…
=
50 (to 1 s.f.)
(k)
49.82
9.784
= 5.091 98…
=
5 (to 1 s.f.)
(l)
163.4
0.0818
= 1997.555 012…
=
2000 (to 1 s.f.)
(m) 15.002 ÷ 0.019 99 – 68.12
= 682.355 237 6…
=
700 (to 1 s.f.)
(n)
×
59.26 5.109
3.817
=
302.759 34
3.817
= 79.318 663 87…
=
80 (to 1 s.f.)
(o)
×
4.18 0.0309
0.0212
=
0.129 162
0.0212
= 6.092 547 17
=
6 (to 1 s.f.)
(p) 16.02 0.0341
0.079 21
×
=
0.546 282
0.079 21
= 6.896 629 213…
=
7 (to 1 s.f.)
(q)
35.807
101.09
= 0.354 209 12
= 0.595 154 703…
=
0.6 (to 1 s.f.)
(r) ×
18.01 36.01
1.989
=
648.5401
1.989
= 326.063 398 7
= 18.057 225 66…
=
20 (to 1 s.f.)
19. 340 ÷ 21
≈ 340 ÷ 20
= 34 ÷ 2
= 17
Rui Feng’s answer is wrong.
Using a calculator, the actual answer is 16.190 476 19.
Hence, his estimated value 15 is close to actual value
16.190 476 19.
He has underestimated the value by using the estimation
300 ÷ 20.
20. (i) (a) 45.3125 = 45 (to 2 s.f.)
(b) 3.9568 = 4.0 (to 2 s.f.)
(ii) 45.3125 ÷ 3.9568
≈ 45 ÷ 4.0
= 11.25
(iii) Using a calculator, the actual value is 11.451 804 49.
The estimated value is close to the actual value. The
estimated value is approximately 0.20 less than the
actual value.
21. (a) 0.052 639 81 = 0.052 640 (to 5 s.f.)
(b) 1793 × 0.000 979
= 1.755 347
= 1.8 (to 1 d.p.)
(c)
×
31.205 4.97
1.925
=
155.088 85
1.925
= 80.565 636 36…
=
80 (to 1 s.f.)
22. The calculation is 297 ÷ 19.91.
297 ÷ 19.91
≈ 300 ÷ 20
=
15 (to 2 s.f.)
15 litres of petrol is used to travel 1 km.
25

1
23. Total cost of set meals = $6.90 × 9
= $7 × 9
= $63
Ethan should pay less than $63 for the set meals.
Therefore, he has paid the wrong amount.
Intermediate
24. (a) (16.245 – 5.001)3
× 122.05
= 15 704.76…
=
20 000 (to 1 s.f.)
(b)
×
6.01 0.0312
0.0622
= 3.014 66…
=
3 (to 1 s.f.)
(c)
×
29.12 5.167
1.895
= 79.400…
=
80 (to 1 s.f.)
(d) 41.41
10.02 0.018 65
×
= 221.594 344 8
=
200 (to 1 s.f.)
(e)
π(8.52
− 7.52
) × 26
169.8
= 7.6967…
=
8 (to 1 s.f.)
(f)
×
24.997 28.0349
19.897
= 7.044 58…
=
7 (to 1 s.f.)
(g) 2905 (0.512)
0.004 987
3
×
= 78 183.77…
=
80 000 (to 1 s.f.)
(h)
+
59.701 41.098
998.07
3
= 10.086 393 09…
=
10 (to 1 s.f.)
(i)
−
×
4.311 2.9016
981 0.0231
3
= 6.140 437 069…
=
6 (1 s.f.)
(j)
−
−
(20.315) 82.0548
85.002 21.997
3
3
= 2104.695 751…
=
2000 (to 1 s.f.)
25. (i)
×
12.01 4.8
2.99
≈
×
12 4.8
3.0
= 19.2
=
20 (to 1 s.f.)
(ii)
×
12.01 0.048
0.299
≈ 12 × 4.8 ÷ 100
3.0 ÷ 10
= 20 ÷ 10
= 2
26. (a) (i) 24.988 = 25 (to 2 s.f.)
(ii) 39.6817 = 40 (to 2 s.f.)
(iii) 198.97 = 200 (to 2 s.f.)
(b)
×
24.988 39.6817
198.97
≈
×
25 40
200
=
×
5 40
200
=
1 (to 1 s.f.)
27. (a)
×
−
17.47 6.87
5.61 3.52
= 57.425 311
=
57.425 (to 5 s.f.)
(b)
× ×
1.743 5.3 2.9454
(11.71)2
= 0.198 428 362…
=
0.198 43 (to 5 s.f.)
(c) 7.593 − 6.219 ×
1.47
(1.4987)3
= 4.877 225 103…
=
4.8772 (to 5 s.f.)
(d) 119.73 − 13.27 × 4.711
88.77 ÷ 66.158
= 42.640 891 68…
=
42.641 (to 5 s.f.)
(e)
32.41 − 10.479
7.218





 ×
4.7103 × 21.483
8.4691






= 36.303 441 14…
=
36.303 (to 5 s.f.)
(f)
−
(0.629) 7.318
2.873
2
=
−0.803 877 207…
=
−0.803 88 (to 5 s.f.)
(g)
×
11.84 0.871
0.9542
3
= 2.210 939 278…
=
2.2109 (to 5 s.f.)
26

1
(h)
−
7.295 7.295
(7.295)2 +
−
(6.98) 6.98
6.98
3
3
= 0.086 327 152 + 174.290 757 4
= 174.377 084 6…
=
174.38 (to 5 s.f.)
28. (a) (i) 271.569 = 270 (to 2 s.f.)
(ii) 9.9068 = 10 (to the nearest whole number)
(iii) 3.0198 = 3.0 (to 1 d.p.)
(b)
×
271.569 (9.9068)
(3.0198)
2
3
≈
×
270 (10)
(3.0)
2
3
=
×
270 100
27
=
1000 (to 1 s.f.)
(c)
×
271.569 (9.9068)
(3.0198)
2
3
= 967.859 777 4…
=
970 (to 2 s.f.)
(d) No, the answers are close but not the same.
The estimated value is 30 more than the actual value.
29. (a) Perimeter of the rectangular sheet of metal
= 2(9.96 + 5.08)
= 2(15.04)
= 30.08
=
30 m (to 1 s.f.)
(b) Area of rectangular sheet of metal
= 9.96 × 5.08
= 50.5968
= 50.6 m2
30. (a) Smallest possible number of customers = 250
(b) Largest possible number of customers = 349
31. Total number of students that the school can accommodate
= 33 × 37
= 1221
=
1200 (to 2 s.f.)
The school can accommodate approximately 1200
students.
32. Number of pens bought
= 815 ÷ 85
= 9.588…
=
9 (to 1 s.f.)
The greatest number of pens that he can buy is 9.
33. (i) Thickness of each piece of paper
=
60 ÷ 10
500
=
6
500
= 0.012
=
0.01 cm (to 1 d.p.)
(ii) Thickness of a piece of paper
=
0.012 cm
= 0.000 12 m
=
0.0001 m (to 1 s.f.)
34. (i) Length of the carpet
=
11.9089
4.04
= 2.947 747 525…
=
2.95 m (to 3 s.f.)
(ii) Perimeter of the carpet
≈ 2(2.9477 + 4.04)
= 2(6.9877)
= 13.9754
=
13.98 m (to 4 s.f.)
35. (i) 18 905 = 19 000 (to 2 s.f.)
(ii) Cost of each ticket
=
7000000
19 000
=
7000
19
≈ 368.421 052 6
= $368 (to the nearest dollar)
36. (a) (i) Radius
= 497
=
500 mm (to 2 s.f)
Circumference of circle
= 2p(500)
= 1000p
= 3141.59…
=
3000 mm (to 1 s.f.)
(ii) Radius
= 5.12
=
5.1 m (to 2 s.f.)
Circumference of circle
= 2p(5.1)
= 10.2p
= 32.044…
=
30 m (to 1 s.f.)
(b) (i) Radius
= 10.09
=
10 m (to 2 s.f.)
Area of circle
= p(10)2
= 100p
= 314.159…
= 300 m2
(to 1 s.f.)
27

1
(ii) Radius
= 98.4
=
98 mm (to 2 s.f.)
Area of circle
= p(98)2
= 9604p
= 30 171.855 …
= 30 000 mm2
(to 1 s.f.)
37. Total value of 20-cent coins
= 31 × 0.2
= $6.20
Total value of 5-cent coins
=
$7.35 − $6.20
= $1.15
Number of 5-cent coins
=
1.15
0.05
= 1.2
0.05


(to 2 s.f.)
=
120
5
= 24
There are about 24 5-cent coins in the box.
38. Total amount that Lixin has to pay
= 18 × (0.99 ÷ 3) + 1.2 × 1.5 + 2 × 0.81 + 2.2 × 3.4
= 18 × 0.33 + 1.2 × 1.5 + 2 × 0.8 + 2.2 × 3.4
= 5.94 + 1.8 + 1.6 + 7.48
= $16.84
The total amount she has to pay, to the nearest dollar, is
$17.
39. KRW 900 ≈ S$1
Price of a shirt in KRW = KRW 27 800
� KRW 27 900
Price of shirt in S$ =
27900
900
=
279
9
= S$31
40. For Airline A,
cost = 0.8 × $88.020
= 0.8 × $90
= $72
For Airline B,
cost = $93 – $35
= $58
For Airline C,
cost = 0.9 × $75
= $67.50
Airline B’s offer is the best.
41. For option A,
700 ml costs about $4.00.
For option B,
1400 ml costs $8.90.
Thus 700 ml will cost about (8.90 ÷ 2) = $4.45
For option C,
950 ml costs $9.90.
Thus 700 ml will cost about (9.90 ÷ 950) × 700
≈$7.00
Option A is better value for money.
Advanced
42. (a) 406 A45 when correct to 3 significant figures is
406 000, so A < 5.
The maximum prime value of A is 3.
(b) 398 200 is the estimated value for
398 150 to 398 199, if corrected to 4 significant figures;
398 195 to 398 204, if corrected to 5 significant figures;
398 200.1 to 398 200.4, if corrected to 6 significant
figures.
m = 4, 5 or 6
43. 2000 is the estimated value for 1999 to 2004, if corrected
to 1, 2 and 3 significant figures.
The smallest number is 1999 and the largest number is
2004.
44. Rp 7872.5300 = S$1
Rp 8000 ≈ S$1
Price of cup noodle in Rp
= Rp 27 800
≈ Rp 28 000
Price of cup noodle in S$ = S$
28000
8000
= S$3.50
The cup noodle costs S$3.50.
28

1
45.
16 500.07 × 39.59 − 119 999.999 +
485 200.023
(2.6)2






1.02 × (13.5874 + 19.0007)2
− 99.998
3
≈
17000 × 40 − 120000 +
490000
(2.6)2






1.0 × (14 + 20)2
− 100
3
=
17000 × 40 − 120000 +
490000
6.76






989
3
≈
680000 − 120000 +
490000
7






1000
3
(Note: 6.76 and 989 are estimated so that the division and
cube root can be carried out, without the use of calculator)
=
−
680000 190000
10
=
490 000
10
=
700
10
=
70 (to 1 s.f.)
New Trend
46. (a)
16.85
3(7.1) – 1.55
≈ 67 760
(b) 67 760 = 67 800 (to 3 s.f.)
47. (a) (0.984 52) 2525
102.016
3
×
≈
×
(1.0) 2500
100
3
=
0.5 (to 1 s.f.)
(b) (0.984 52) 2525
102.016
3
×
= 0.470 041 311
= 0.47 (to 2 s.f.)
48.
−
(1.92)
(4.3) 4.788
2
3
3
= 0.362 609 371
=
0.362 61 (to 5 s.f.)
49. (a) 8.5 kg
(b) Greatest possible mass of 1 m3
of wood
=
9.5
2.5
= 3.8 kg
29

1
Chapter 4 Basic Algebra and Algebraic
Manipulation
Basic
1. (a) (2x + 5y) – 4 = 2x + 5y – 4
(b) (3x)(7y) + 9z = 21xy + 9z
(c) (7x)(11y) × 2z = 77xy × 2z
= 154xyz
(d) (3z + 7s) ÷ 5a =
3z + 7s
5a
(e) r3
– (p ÷ 3q) = r3
–
p
q
3
(f) 3w ÷ (3x + 7y) =
+
w
x y
3
3 7
(g) (k ÷ 2y) − 9(x)(3h) =
k
y
2
– 27xh
2. (a) 7b – 3c + 4a
= 7(2) – (3)(−1) + 4(3)
= 14 + 3 + 12
= 29
(b) 3a3
= 3(3)3
= 3(27)
= 81
(c) (5b)2
= (5 × 2)2
= (10)2
= 100
(d) (2a + b + c)(5b – 3a)
= (2 × 3 + 2 + (−1))(5 × 2 – 3 × 3)
= (7)(1)
= 7
(e) (a – b)2
– (b – c)2
= (3 – 2)2
– (2 – (−1))2
= 12
– (3)2
= −8
(f) 2a2
– 3b2
+ 3abc
= 2(3)2
– 3(2)2
+ 3(3)(2)(−1)
= 18 – 12 – 18
= –12
(g) (a + 3b)3
= (3 + 3(2))3
= 93
= 729
(h) ab
– ca
+ bc
= (3)2
– (−1)3
+ (2)(−1)
= 9 + 1 +
1
2
= 10
1
2
(i)
a
b
–
b
c
=
3
2
–
2
–1
= 1
1
2
+ 2
= 3
1
2
(j)
b a
c
8 – (3 )2
=
8(2) – (3 3)
(–1)
2
×
=
16 – 9
–1
2
=
16 – 81
–1
= 65
(k)
b c
a
+
+
a bc
b
+
=
2 (–1)
3
+
+
3 (2)(–1)
2
+
=
1
3
+
1
2
=
5
6
(l)
a b
c
–
2 2
2
–
a c
c b
–
– 3
3
=
3 – 2
(–1)
2 2
2
–
3 – (–1)
(–1) – 3(2)
3
=
5
1
–
28
–7
= 5 + 4
= 9
3. (a) 3x + 9y + (−11y)
= 3x + 9y – 11y
= 3x – 2y
(b) −a – 3b + 7a – 10b
= 7a – a – 3b – 10b
= 6a – 13b
(c) 13d + 5c + (−13c + 5d)
= 13d + 5c – 13c + 5d
= 13d + 5d + 5c – 13c
= 18d – 8c
= −8c + 18d
(d) 7pq – 11hk + (−3pq − 21kh)
= 7pq – 11hk – 3pq – 21kh
= 4pq – 32hk
4. (a) 5x + 7y – 2x – 4y
= 5x – 2x + 7y – 4y
= 3x + 3y
30

1
(b) −3a – 7b + 11a + 11b
= −3a + 11a – 7b + 11b
= 11a – 3a + 11b – 7b
= 8a + 4b
(c) 5u – 7v – 7u – 9v
= 5u – 7u – 7v – 9v
= −2u – 16v
(d) 5p + 4q – 7r – 5q + 4p
= 5p + 4p + 4q – 5q – 7r
= 9p – q – 7r
(e) 5pq – 7qp + 21 – 7
= −2pq + 14
(f) 15x + 9y + 5x – 3y – 13
= 15x + 5x + 9y – 3y – 13
= 20x + 6y – 13
(g) 8ab – 5bc + 21ba – 7cb
= 8ab + 21ab – 5bc – 7cb
= 29ab – 12bc
(h) −7 + mn + 9mn – 3mn – 25
= mn + 9mn – 3mn – 25 – 7
= 7mn – 32
(i) 3h – 4gh +
2
3
h –
1
3
gh
= 3h +
2
3
h – 4gh –
1
3
gh
= 3
2
3
h – 4
1
3
gh
(j)
3
5
x –
2
3
xy +
1
4
x –
1
5
xy
=
3
5
x +
1
4
x –
2
3
xy –
1
5
xy
=
17
20
x –
13
15
xy
(k) 2.5p – 3.6q + 1.1p – 6.3q
= 2.5p + 1.1p – 3.6q – 6.3q
= 3.6p – 9.9q
(l) −0.5a – 0.65b + 0.375a – 0.258b
= −0.5a + 0.375a – 0.65b – 0.258b
= −0.125a – 0.908b
5. (a) 3(3x – 5)
= 9x –15
(b) 7(5 – 7x)
= 35 – 49x
(c) 11(4x + 5y)
= 44x + 55y
(d) −3(9k – 2)
= −27k + 6
(e) −7(−3h – 5)
= 21h + 35
(f) 4(3a – 2b + c)
= 12a – 8b + 4c
(g) –5
1
4
p –
2
5
q +
1
2
r






= –
5
4
p + 2q –
5
2
r
(h) –
1
4
(8a – 5b + 3c)
= –2a +
5
4
b –
3
4
c
6. (a) 5a – 3(2p + 3)
= 5a – 6p – 9
(b) 3x – 5(x – y)
= 3x – 5x + 5y
= −2x + 5y
(c) 5(a + 4) + 7(b – 2)
= 5a + 20 + 7b – 14
= 5a + 7b + 20 – 14
= 5a + 7b + 6
(d) 3(2p – 3q) – 5(3p – 5q)
= 6p – 9q – 15p + 25q
= 6p – 15p + 25q – 9q
= −9p + 16q
(e) r(3x – y) – 3r(x – 7y)
= 3xr – ry – 3xr + 21ry
= 3xr – 3xr + 21ry – ry
= 20ry
(f) 3(x + y + z) + 5y – 4z
= 3x + 3y + 3z + 5y – 4z
= 3x + 3y + 5y + 3z – 4z
= 3x + 8y – z
7. In 4 years’ time,
Rui Feng will be (x + 4) years old.
∴ His brother will be 3(x + 4) years old.
8. Let the largest odd integer be x.
Then the previous odd integer will be (x − 2).
The smallest odd integer is (x − 2) − 2 = x − 4.
Sum of three consecutive odd integers
= x + (x − 2) + (x − 4)
= x + x − 2 + x − 4
= x + x + x − 2 − 4
= 3x − 6
9. (a)
1
3
x +
1
5
y –
1
9
x –
1
15
y
=
1
3
x –
1
9
x +
1
5
y –
1
15
y
=
3
9
x –
1
9
x +
3
15
y –
1
15
y
=
2
9
y +
2
15
y
31

1
(b)
3
4
a –
1
5
b + 3a –
4
7
b
=
3
4
a + 3a –
4
7
b –
1
5
b
= 3
3
4
a –
20
35
b –
7
35
b
= 3
3
4
a –
27
35
b
(c)
5
6
c +
8
7
d –
2
9
c –
5
3
d
=
5
6
c –
2
9
c +
8
7
d –
5
3
d
=
15
18
c –
4
18
c +
24
21
d –
35
21
d
=
11
18
c –
11
21
d
(d) 5f –
5
7
h +
7
8
k –
4
3
f –
4
5
h +
12
11
k
= 5f –
4
3
f –
5
7
h –
4
5
h +
12
11
k +
7
8
k
= 3
2
3
f –
25
35
h –
28
35
h +
96
88
k +
77
88
k
= 3
2
3
f – 1
18
35
h + 1
85
88
k
10. Amount of money spent on buying apples
= 10 × $
x
4
= $
5
2
x
Amount of money spent on buying bananas
= $1.25 × m
= $1.25m
Amount of money spent on buying oranges
= $
3
4
× (3n + 1)
= $
+
n
3(3 1)
4
Total money spent = $
5
2
x + $1.25m + $
+
n
3(3 1)
4
= $
5x
2


 + 1.25m +
3(3n + 1)
4



11. (a) 5a + 3b – 2c + 3
1
2
a + 2
1
2
b – 3
1
2
c






= 5a + 3b – 2c + 3
1
2
a + 2
1
2
b – 3
1
2
c
= 5a + 3
1
2
a + 3b + 2
1
2
b – 2c – 3
1
2
c
= 8
1
2
a + 5
1
2
b – 5
1
2
c
(b)
1
2
[5y – 2(x – 3y)]
=
5
2
y – (x – 3y)
=
5
2
y – x + 3y
=
5
2
y + 3y – x
=
11
2
y – x
(c)
3
4
[8q – 7p – 3(p – 2q)]
=
3
4
[8q – 7p – 3p + 6q)]
=
3
4
[–7p – 3p + 6q + 8q]
=
3
4
[–10p + 14q]
= –
30
4
p +
42
4
q
=
q p
21 – 15
2
(d)
3
10
[3(5a – b) – 7(2a – 5b)]
=
3
10
[15a – 3b – 14a + 35b]
=
3
10
[15a – 14a + 35b – 3b]
=
3
10
[a + 32b]
=
3
10
a +
96
10
b
=
3
10
(a + 32b)
12. (a)
x
2(5 – 1)
3
–
x – 3
5
=
x
2(5 – 1) 5
3 5
×
×
–
x
( – 3) 3
5 3
×
×
=
x
50 – 10
15
–
x
(3 – 9)
15
=
x x
50 – 10 – 3 9
15
+
=
x x
50 – 3 9 – 10
15
+
=
x
47 – 1
15
32

1
(b)
x
2
+
x – 3
5
–
x – 4
4
=
x
5
10
+
x
2( – 3)
10
–
x – 4
4
=
x x
5 2 – 6
10
+
–
x
( – 4)
4
=
x
7 – 6
10
–
x
( – 4)
4
=
x
(7 – 6) 2
10 2
×
×
–
x
( – 4) 5
4 5
×
×
=
x
2(7 – 6)
20
–
x
5( – 4)
20
=
x x
14 – 12 – 5 20
20
+
=
x
9 8
20
+
(c)
x 5
3
+
–
x
2 – 7
6
+
x
2
=
x
2( 5)
6
+
–
x
(2 – 7)
6
+
x
2
=
x x
2 10 – 2 7
6
+ +
+
x
2
=
17
6
+
x
3
6
=
x
3 17
6
+
(d)
x
3 – 7
2
–
x 4
5
+
–
3
4
=
x
5(3 – 7)
10
–
x
2( 4)
10
+
–
3
4
=
x x
15 – 35 – 2 – 8
10
–
3
4
=
x x
15 – 2 – 8 – 35
10
–
3
4
=
x
13 – 43
10
–
3
4
=
x
2(13 – 43)
20
–
15
20
=
x
2(13 – 43) – 15
20
=
x
26 – 86 – 15
20
=
x
26 – 101
20
(e)
x y
3
+
–
2
5
–
x y
3 – 2
6
=
x y
5( )
15
+
–
6
15
–
x y
3 – 2
6
=
x y
5 5 – 6
15
+
–
x y
3 – 2
6
=
x y
2(5 5 – 6)
30
+
–
x y
5(3 – 2 )
30
=
x y x y
10 10 – 12 – 15 10
30
+ +
=
x x y y
10 – 15 10 10 – 12
30
+ +
=
x y
–5 20 – 12
30
+
13. (a) 15x – 3 = 3(5x – 1)
(b) −21y – 48 = −3(7y + 16)
(c) 64b – 27bc = b(64 – 27c)
(d) 18ax + 6a – 36az = 6a(3x + 1 – 6z)
(e) 14p – 56pq – 42pr
= 7p(2 – 8q – 6r)
= 14p(1 – 4q – 3r)
Intermediate
14. (a) k + 8 Add 8 to a number k
5(k + 8) Multiply the sum by 5
5(k + 8) − (2k – 1) Subtract (2k – 1) from the result
= 5k + 40 – 2k + 1
= 5k – 2k + 40 + 1
= 3k + 41
(b) Cost of 7 pencils
= 7 × p
= 7p cents
Change after buying the pencils
= $4.20
= 420 cents
Amount Kate had before buying the pencils
= (420 + 7p) cents
(c) Cost price of the apples
= x(y + 3) cents
Selling price of the apples
= x(2y – 5) cents
Profit
= selling price – cost price
= x(2y – 5) – x(y + 3)
= 2xy – 5x – xy – 3x
= 2xy – xy – 5x – 3x
= (xy – 8x) cents
33

1
(d) Cost price of the microchips
= (n)(2x)
= $2nx
Selling price of the microchips
= (n)(n – x)
= $n(n – x)
Loss
= cost price – selling price
= 2nx – n(n – x)
= 2nx – n2
+ nx
= $(3nx – n2
)
15. (a) When a = 4, m = −2 and n = −1,
4(−2)2
– 3(4) – 5(−1)
= 16 – 12 + 5
= 4 + 5
= 9
(b) When a = 4, m = −2 and n = −1,
7(–1) + 3
3
4
(4) – (–2 – 4)
= −7 +15 – (−6)
= 8 + 6
= 14
16. (a) When a = 2, c = −1, d = 5 and e = −4,
(2) − (−1)(5 – (−4))
= 2 + (5 + 4)
= 2 + 5 + 4
= 11
(b) When a = 2, c = −1, d = 5 and e = −4,
2(–4) – 2
(–1) – 5(–4)
2
=
+
–8 – 2
1 20
=
10
21
17. When a = 2, b = −1, c = 0 and d =
1
2
(a) (2a – b)2
= (2 × 2 – (−1))2
= (4 + 1)2
= 52
= 25
(b) (3a – b)(2c + d)
= [3(2) – (–1)] 2 (0) +
1
2






= (6 + 1)
1
2






=
7
2
= 3
1
2
(c) (5a – b)(2c + d) – b(ab + bc – 4cd)
[5(2) – (–1)] 2 (0) +
1
2






– (–1) (2)(–1) + (–1)(0) – 4(0)
1
2












= (10 + 1)
1
2





 + (–2)
= 3
1
2
18. When x = −3,
(2x – 1)(2x + 1)(2x + 3)
= (2(−3) – 1)(2(−3) + 1)(2(−3) + 3)
= (−6 – 1)(−6 + 1)(−6 + 3)
= (−7)(−5)(−3)
= −105
19. When x = −2,
(–2) 1
(–2) – 1
+
+
2(–2) – 1
2(–2) 1
+
=
–1
–3
+
–5
–3






=
1
3
+
5
3
= 2
20. When x = −2,
(–2) – 5
(–2) 7
+
– 3(–2)2
=
–7
5
– 12
= −13
2
5
21. When x = 2 and y = −1,
(2)3
+ 2(2)(−1)2
+ (−1)3
= 8 + 4 – 1
= 11
22. When a = −2, b = 3 and c = −5,
3(–2) (3)(–5)
2(3) – 3(–5)
2
–
(3)(–5)
(–2)
=
3(4)(3)(–5)
6 – (–15)
–
–15
–2






=
–180
21
–
15
2
= −16
1
14
34

1
23. When y = −3 and z = −1
1
2
,
5x = (–3)2
–
(–3)3
–1
1
2






5x = 9 – –27 ÷ –
3
2












5x = 9 – –27 × –
2
3












5x = 9 – 18
5x = –9
∴ x =
–9
5
= –1
4
5
24. When y = −3,
x
x
5(–3)
5 – 7(–3)
+
=
1
4
+
x
x
– 15
5 21
=
1
4
4(x – 15) = 5x + 21
4x – 60 = 5x + 21
5x – 4x = −60 − 21
x = −81
25. (a) a + b + c + (2b – c) + (3c + a)
= a + b + c + 2b – c + 3c + a
= a + a + b + 2b + c – c + 3c
= 2a + 3b + 3c
(b) 2ab + 3bc + (5ac – 5ba) + (2cb + 5ab)
= 2ab + 3bc + 5ac – 5ba + 2cb + 5ab
= 2ab + 5ab – 5ba + 3bc + 2cb + 5ac
= 2ab + 5ab – 5ab + 3bc + 2bc + 5ac
= 2ab + 5bc + 5ac
(c)
1
2
xy +
1
3
xz –
1
4
yx
⎛
⎝
⎜
⎞
⎠
⎟ +
1
6
xz + xy
⎛
⎝
⎜
⎞
⎠
⎟
=
1
2
xy +
1
3
xz –
1
4
yx +
1
6
xz + xy
= 1
1
4
xy +
1
2
xz
(d) a + b – c + (2c – b + a) + (5a + 7c)
= a + b – c + 2c – b + a + 5a + 7c
= a + a + 5a + b – b – c + 2c + 7c
= 7a + 8c
(e) 5abc – 7cb + 4ac + (4cba – 4bc + 3ca)
= 5abc – 7cb + 4ac + 4cba – 4bc + 3ca
= 5abc + 4cba + 4ac + 3ca – 7cb – 4bc
= 5abc + 4abc + 4ac + 3ac – 7bc – 4bc
= 9abc + 7ac – 11bc
26. (a) 5(2x – 7y) – 4(y – 3x)
= 10x – 35y – 4y + 12x
= 10x + 12x – 35y – 4y
= 22x – 39y
(b) 3a + 5ac – 2c – 4c – 6a – 8ca
= 3a – 6a + 5ac – 8ca – 2c – 4c
= −3a – 3ac – 6c
(c) 5p + 3q – 4r – (6q − 3p + r)
= 5p + 3q – 4r – 6q + 3p – r
= 5p + 3p + 3q – 6q – 4r – r
= 8p – 3q – 5r
(d) 3b + 5a – 2(a – 2b)
= 3b + 5a – 2a + 4b
= 3a + 7b
(e) 2(z – 5x) – 7(y + z – 1)
= 2z – 10x – 7y – 7z + 7
= −10x – 7y + 2z – 7z + 7
= −10x – 7y – 5z + 7
(f) 7m – 2[6m – (3m – 4p)]
= 7m – 2[6m – 3m + 4p]
= 7m – 12m + 6m – 8p
= m – 8p
(g) 7x – {3x – [4x – 2(x + 3y)]}
= 7x – {3x – [4x – 2x – 6y]}
= 7x – {3x – [2x – 6y]}
= 7x – {3x – 2x + 6y}
= 7x – {x + 6y}
= 7x – x – 6y
= 6x – 6y
(h) 8a – {2a – [3c – 6(a – 2c)]}
= 8a – {2a – [3c – 6a + 12c]}
= 8a – {2a – [3c + 12c – 6a]}
= 8a – {2a – [15c – 6a]}
= 8a – {2a – 15c + 6a}
= 8a – {2a + 6a – 15c}
= 8a – {8a – 15c}
= 8a – 8a + 15c
= 15c
(i) 12a – 3{a – 4[c – 5(a – c)]}
= 12a – 3{a – 4[c – 5a + 5c]}
= 12a – 3{a – 4[c + 5c – 5a]}
= 12a – 3{a – 4[6c – 5a]}
= 12a – 3{a – 24c + 20a}
= 12a – 3{a + 20a – 24c}
= 12a – 3{21a – 24c}
= 12a – 63a + 72c
= 72c – 51a
35

1
(j) 7m – 4n – 5(m – 3n) + 4(n − 5)
= 7m – 4n – 5m + 15n + 4n – 20
= 7m – 5m – 4n + 15n + 4n − 20
= 2m + 15n – 20
(k) 2a – 5(3ab – 4b) – 2(a – 2ba)
= 2a – 15ab + 20b – 2a + 4ab
= 2a – 2a – 15ab + 4ab + 20b
= −11ab + 20b
= 20b – 11ab
(l) 4(x – 5y) – 5(2y – 3x) – (2x – 5y)
= 4x – 20y – 10y + 15x – 2x + 5y
= 4x + 15x – 2x – 20y – 10y + 5y
= 17x – 25y
(m) 2(3x + y) – 5[3(x – 3y) – 4(2x – y)]
= 2(3x + y) – 5[3x – 9y – 8x + 4y]
= 2(3x + y) – 5[3x – 8x – 9y + 4y]
= 2(3x + y) – 5[– 5x – 5y]
= 6x + 2y + 25x + 25y
= 6x + 25x + 2y + 25y
= 31x + 27y
(n)
1
2
14x –
2
3
(9x – 21y) − 2(x + y)






=
1
2
[14x – 6x + 14y – 2x – 2y]
=
1
2
[14x – 6x – 2x + 14y – 2y]
=
1
2
[6x + 12y]
= 3x + 6y
27. (a) 3a – 2b – 11 – (10a + 5b – 7)
= 3a – 2b – 11 – 10a – 5b + 7
= 3a – 10a – 2b – 5b – 11 + 7
= −7a – 7b – 4
(b) 4x – 2z + 7 – (x – 3y – 5z + 5)
= 4x – 2z + 7 – x + 3y + 5z – 5
= 4x – x + 3y – 2z + 5z + 7 – 5
= 3x + 3y + 3z + 2
(c) 4p + 2q – 5r – 1 – (7p – q + 3r + 3)
= 4p + 2q – 5r – 1 – 7p + q – 3r – 3
= 4p – 7p + 2q + q – 5r – 3r – 1 – 3
= −3p + 3q – 8r – 4
(d) 6(2 + 3n + 5m) – 4m(n + 5) – [2(3m – 5n) + 5mn]
= 12 + 18n + 30m – 4mn – 20m – (6m – 10n + 5mn)
= 12 + 18n + 30m – 20m – 4mn – 6m + 10n – 5mn
= 12 + 18n + 10n + 30m – 20m – 6m – 4mn – 5mn
= 12 + 28n + 4m – 9mn
28. (i) Let the second number be n.
Then the first number is n – 2.
Then the third number is n + 2.
Lastly, the fourth number is (n + 2) + 2 = n + 4.
(ii) Sum of the four numbers
= n – 2 + n + n + 2 + n + 4
= n + n + n + n – 2 + 2 + 4
= 4n + 4
29. Perimeter of Figure 1
= 7y + 3x + 7y + 3x
= 7y + 7y + 3x + 3x
= 14y + 6x
= (6x + 14y) cm
Perimeter of Figure 2
= 5x + (x + 5y) + 7y
= 5x + x + 5y + 7y
= (6x + 12y) cm
∴ Since (6x + 14y) > (6x + 12y), Figure 1 has a larger
perimeter.
30. (i) Amount of money spent on the fruits
= (120h + 180k) cents
(ii) Number of bags in which each bag contains
2 apples and 3 oranges
= 120 ÷ 2
= 60
Total amount of money for which he sold all
bags of fruits
= [60(3h + 4k)]
= (180h + 240k) cents
(iii)Amount earned from selling the fruits
= [60(3h + 4k)] − (120h + 180k)
= 180h + 240k – 120h – 180k
= 180h – 120h + 240k – 180k
= (60h + 60k) cents or $(0.6h + 0.6k)
31. (i) Number of 50-cent coins Shirley has
= n – x – 3x
= n – 4x
(ii) Since the number of 10-cent coins is x, then
the number of 50-cent coins is
1
4
x.
Total value of all the coins
= 10x + 20(3x) + 50 Q
1
4
xR
= 10x + 60x +
50
4
x
= 82
1
2
x cents
36

1
(iii)Ratio of number of 20-cent coins to 50-cent coins
= 5 : 3
5 parts is 3x.
1 part is
x
3
5
.
3 parts is
x
3
5
× 3 =
x
9
5
.
Total value of all the coins
= 10x + 20(3x) + 50
9x
5






= 10x + 60x + 90x
= 160x cents
32. (a)
x
3( – 2)
3
+
x
2( 3)
4
+
=
x
12( – 2)
12
+
x
6( 3)
12
+
=
x x
12( – 2) 6( 3)
12
+ +
=
x x
12 – 24 6 18
12
+ +
=
x
18 – 6
12
=
x
3 – 1
2
(b)
x
5(3 1)
4
+
–
x
7(5 – 3)
12
=
x
15(3 1)
12
+
–
x
7(5 – 3)
12
=
x x
45 15 – 35 21
12
+ +
=
x x
45 – 35 15 21
12
+ +
=
x
10 36
12
+
=
x
5 18
6
+
(c) 1 +
x
2 1
3
+
+
x
4( – 3)
6
=
3
3
+
x
2 1
3
+
+
x
2( – 3)
3
=
+ + +
x x
3 2 1 2( – 3)
3
=
x x
3 2 1 2 – 6
3
+ + +
=
x x
2 2 1 3 – 6
3
+ + +
=
x
4 – 2
3
(d)
x y
3 – 4
6
+
x y
– 2
4
–
x y
5
+
=
x y
2(3 – 4 )
12
+
x y
3( – 2 )
12
–
x y
5
+
=
x y x y
6 – 8 3 – 6
12
+
–
x y
5
+
=
x y
5(9 – 14 )
60
–
x y
12( )
60
+
=
x y x y
5(9 – 14 ) – 12( )
60
+
=
x y x y
45 – 70 – 12 – 12
60
=
x y
33 – 82
60
(e)
x
2 – 5
3
–
x 4
6
+
+
x
3(5 – )
9
=
x
2(2 – 5)
6
–
x 4
6
+
+
x
3(5 – )
9
=
+
x x
2(2 – 5) – ( 4)
6
+
x
3(5 – )
9
=
x x
4 – 10 – – 4
6
+
x
3(5 – )
9
=
x x
4 – – 10 – 4
6
+
x
3(5 – )
9
=
x
3 – 14
6
+
x
3(5 – )
9
=
x
3(3 – 14)
18
+
x
6(5 – )
18
=
x x
3(3 – 14) 6(5 – )
18
+
=
x x
9 – 42 30 – 6
18
+
=
x x
9 – 6 – 42 30
18
+
=
x
3 – 12
18
=
x – 4
6
37

1
(f)
x
4(3 4)
10
+
–
x 7
15
+
–
x
2 – 1
5
=
x
12(3 4)
30
+
–
x
2( 7)
30
+
–
x
2 – 1
5
=
x x
12(3 4) – 2( 7)
30
+ +
–
x
2 – 1
5
=
x x
36 48 – 2 – 14
30
+
–
x
2 – 1
5
=
x x
36 – 2 48 – 14
30
+
–
x
6(2 – 1)
30
=
x
34 34
30
+
–
x
6(2 – 1)
30
=
x x
34 34 – 12 6
30
+ +
=
x x
34 – 12 34 6
30
+ +
=
x
22 40
30
+
=
x
11 20
15
+
(g) −1 −
x
3( 7)
7
+
–
x
4(2 – 1)
5
=
–7
7
–
x
3( 7)
7
+
–
x
4(2 – 1)
5
=
x
–7 – 3( 7)
7
+
–
x
4(2 – 1)
5
=
x
–7 – 3 – 21
7
–
x
4(2 – 1)
5
=
x
–3 – 28
7
–
x
4(2 – 1)
5
=
x
5(–3 – 28)
35
–
x
28(2 – 1)
35
=
x x
5(–3 – 28) – 28(2 – 1)
35
=
x x
–15 – 140 – 56 28
35
+
=
x x
–15 – 56 – 140 28
35
+
=
x
–71 – 112
35
(h)
x
3 – 7
4
– (x – 5) –
x – 1
3
=
x
3 – 7
4
–
x – 1
3
– (x – 5)
=
x
3(3 – 7)
12
–
x
4( – 1)
12
– (x – 5)
=
x x
3(3 – 7) – 4( – 1)
12
– (x – 5)
=
x x
9 – 21 – 4 4
12
+
– (x – 5)
=
x x
9 – 4 – 21 4
12
+
– (x – 5)
=
x
5 – 17
12
–
x
12( – 5)
12
=
x x
5 – 17 – 12 60
12
+
=
x x
5 – 12 – 17 60
12
+
=
x
–7 43
12
+
(i)
x
2(3 – 1)
5
– (x – 3) –
x
2 1
3
+
=
x
2(3 – 1)
5
–
x
2 1
3
+
– (x – 3)
=
x x
6(3 – 1) – 5(2 1)
15
+
– (x – 3)
=
x x
18 – 6 – 10 – 5
15
– (x – 3)
=
x x
18 – 10 – 6 – 5
15
– (x – 3)
=
x
8 – 11
15
–
x
15( – 3)
15
=
x x
8 – 11 – 15( – 3)
15
=
x x
8 – 11 – 15 45
15
+
=
x x
8 – 15 – 11 45
15
+
=
x
–7 34
15
+
38

1
Advanced
33. (a) a(5b – 3) – b(4a – 1) + a(1 – 2b)
= 5ab – 3a – 4ab + b + a – 2ab
= 5ab – 4ab – 2ab – 3a + a + b
= −ab – 2a + b
(b) 3x – {2x – 4(x – 3y) – [(3x – 4y) – (y – 2x)]}
= 3x – {2x – 4(x – 3y) – [3x – 4y – y + 2x]}
= 3x – {2x – 4(x – 3y) – [3x + 2x – 4y – y]}
= 3x – {2x – 4x + 12y – [5x – 5y]}
= 3x – {– 2x + 12y – 5x + 5y]}
= 3x – {– 2x – 5x + 12y + 5y]}
= 3x – {– 7x + 17y}
= 3x + 7x – 17y
= 10x – 17y
34. Let Raj’s present age be p years.
Then Ethan’s present age is 5p years.
In 5 years’ time,
Raj is (p + 5) years old and Ethan is (5p + 5) years old.
p + 5 + (5p + 5) = x
p + 5 + 5p + 5 = x
p + 5p + 5 + 5 = x
6p = x – 5 – 5
6p = x – 10
p =
x – 10
6
Raj’s present age is
x – 10
6
years old.
35. Total age of the girls
= (n + 5)q years
Total age of the group of boys and girls
= (m + 2 + n + 5)p
= (m + n + 7)p years
Total age of the boys
= p(m + n + 7) – q(n + 5) years
Average age of the boys
=
+ + +
+
p m n q n
m
( 7) – ( 5)
( 2)
years old
36. (a) 3ac – ad + 2ba – 15a
= a(3c – d + 2b – 15)
(b) 2x + 4xy − 7xyz + 2xz
= x(2 + 4y – 7yz + 2z)
(c) 4ba + 5bca + 9dab
= ab(4 + 5c + 9d)
(d) 5m + 20pmn − 10mn + 35pm
= 5m(1 + 4pn – 2n + 7p)
(e) 6pqr – 3(p + q – 2r)
= 6pqr – 3p – 3q + 6r
= 3(2pqr – p – q + 2r)
37. (a)
x y
2( – 3 )
4
–
y x
4 –
12
–
x y
4( – 5 )
3
=
x y
6( – 3 )
12
–
y x
4 –
12
–
x y
4( – 5 )
3
=
x y y x
6( – 3 ) – (4 – )
12
–
x y
4( – 5 )
3
=
x y y x
6 – 18 – 4
12
+
–
x y
4( – 5 )
3
=
x y
7 – 22
12
–
x y
16( – 5 )
12
=
x y x y
7 – 22 – 16( – 5 )
12
=
x y x y
7 – 22 – 16 80
12
+
=
+
x x y y
7 – 16 – 22 80
12
=
x
–9 58
12
+
(b)
p q
–4( – 3 )
5
–
2(q – p)
20
–
3( p – 5q)
4






=
p q
–4( – 3 )
5
–
2(q – p)
20
–
15( p – 5q)
20






=
p q
–4( – 3 )
5
–
2(q – p) – 15( p – 5q)
20






=
p q
–4( – 3 )
5
–
2q – 2 p – 15 p + 75q
20






=
p q
–16( – 3 )
20
–
–17 p + 77q
20






=
p q p q
–16( – 3 ) – (–17 77 )
20
+
=
+ +
p q p q
–16 48 17 – 77
20
=
p p q q
–16 17 48 – 77
20
+ +
=
p q
– 29
20
39

1
(c) −3 +
f h
2( – 3 )
21
–
h f
5( – )
7
+
f h
2(–2 – 3 )
3
= –3 +
f h
2( – 3 )
21
–
h f
15( – )
21
+
f h
2(–2 – 3 )
3
= –3 +
f h h f
2( – 3 ) – 15( – )
21
+
f h
2(–2 – 3 )
3
= –3 +
f h h f
2 – 6 – 15 15
21
+
+
f h
2(–2 – 3 )
3
= –3 +
f h
17 – 21
21
+
f h
2(–2 – 3 )
3
= –3 +
f h
17 – 21
21
+
f h
14(–2 – 3 )
21
= –3 +
f h f h
17 – 21 14(–2 – 3 )
21
+
= –3 +
f h f h
17 – 21 – 28 – 42
21
= –3 +
f f h h
17 – 28 – 42 – 21
21
=
–63
21
+
f h
–11 – 63
21
=
f h
–11 – 63 – 63
21
(d)
x
5
–
x
4
3
=
x
x
3
15
2
–
x
20
15
=
x
x
3 – 20
15
2
(e)
x
1
+
x
1
2
+
x
1
3
=
x
6
6
+
x
3
6
+
x
2
6
=
x
11
6
(f)
x
5
2
–
x
3
3
+
x
7
=
x
15
6
–
x
6
6
+
x
42
6
=
x
15 – 6 42
6
+
=
x
51
6
=
x
17
2
(g)
x
y
2 – 3
5
–
x
y
5 – 2
10
+
x
y
=
x x x
y
2(2 – 3) – (5 – 2 ) 10
10
+
=
x x x
y
4 – 6 – 5 2 10
10
+ +
=
x x x
y
4 2 10 – 6 – 5
10
+ +
=
x
y
16 – 11
10
New Trend
38.
a
5
–
a c
2(3 – 5 )
6
=
a 6
5 6
×
×
–
a c
2(3 – 5 ) 5
6 5
×
×
=
a
6
30
–
a c
10(3 – 5 )
30
=
a a c
6 – 10(3 – 5 )
30
=
a a c
6 – 30 50
30
+
=
a c
24 50
30
− +
=
a c
2(–12 25 )
30
+
=
c a
25 – 12
15
39. (a) BC = 23x − 2 − (3x − 2) − (5x + 1) − (6x −7)
= 23x − 3x − 5x − 6x − 2 + 2 − 1 + 7
= (9x + 6) cm
(b) Since BC = 2AD,
9x + 6 = 2(5x + 1)
9x + 6 = 10x + 2
x = 4
Perimeter of trapezium = 23x − 2
= 23(4) − 2
= 90 cm
40. (a) 2(3x – 5) – 3(7 – 4x)
= 6x – 10 – 21 + 12x
= 6x + 12x – 10 – 21
= 18x – 31
(b) 4(2x + 3y) − 7(x − 2y) = 8x + 12y − 7x + 14y
= x + 26y
40

1
41.
x
3 4
10
+
–
x 7
15
+
–
x
2 – 1
5
=
x
3(3 4)
30
+
–
x
2( 7)
30
+
–
x
2 – 1
5
=
x x
9 12 – 2 – 14
30
+
–
x
2 – 1
5
=
x x
9 – 2 12 – 14
30
+
–
x
2 – 1
5
=
x
7 – 2
30
–
x
6(2 – 1)
30
=
x x
7 – 2 – 6(2 – 1)
30
=
x x
7 – 2 – 12 6
30
+
=
x x
7 – 12 – 2 6
30
+
=
x
–5 4
30
+
42. x ¢ → 1 gram
$y = (100 × y) ¢
= 100y ¢
100y ¢ →
1
x
× 100y
=
100y
x
grams
41

1
Revision Test A1
1. (a) 36 = 22
× 32
54 = 2 × 33
63 = 32
× 7
32
HCF of 36, 54 and 63 = 32
= 9
(b)

63 = 32
× 7
105 = 3 × 5 × 7
420 = 22
× 3 × 5 × 7
22
32
5 7
LCM of 63, 105 and 420 = 22
× 32
× 5 × 7 = 1260
2. (i) (a) 2 576
2 288
2 144
2 72
2 36
2 18
3 9
3 3
1
576 = 26
× 32
(b)
2 5832
2 2916
2 1458
3 729
3 243
3 81
3 27
3 9
3 3
1
5832 = 23
× 36
(ii) 576 = ×
2 3
6 2
= ×
(2 3)
3 2
= 23
× 3
= 24
5832
3
= ×
2 3
3 6
3
= ×
(2 3 )
2 3
3
= 2 × 32
= 18
3. (a)
(–24) + (–30)
–6
=
–24 – 30
–6
=
–54
–6
= 9

(b)
1
5
+
1
4





 ÷ –
1
20






–1
2
5





 × –1
1
4






=
9
20





 ÷ –
1
20






–1
2
5





 × –1
1
4






=
9
1 20







 × –
1
20
1








–
7
1 5





 × –
1
5
4







=
–9
7
4






= –9 ×
4
7
= –5
1
7
4. (a) 79.122
+ 56.192
– 2 × 79.12 × 56.19 × 0.8716
= 79.122
+ 56.192
– 7749.836 281
= 1667.45 (to 2 d.p.)
(b)
× + ×
×
245 269.78 966 294.81
4 (54.783)
3
3
=
1583.0793 + 16 586.250 94
657 653.9379
= 0.03 (to 2 d.p.)
5. (a) Each book costs $4 approximately.
Number of books that can be bought with $10
=
10
4
= 2.5
The number of books that can be purchased is 2.
(b) Number of litres of petrol
=
600
12.1
=
600
12
= 50 l
50 l of petrol are consumed for a 600-km
journey.
42

1
(c) 7.95
3
× 25.04
= 8
3
× 25
= 2 × 25
= 50
6. (a) 3a – (2 – 5a) – 7
= 3a – 2 + 5a – 7
= 3a + 5a – 2 – 7
= 8a – 9
(b) 8x – 3(x – y)
= 8x – 3x + 3y
= 5x + 3y
(c) 4(3x – 7) – 2(6x – 7)
= 12x – 28 – 12x + 14
= 12x – 12x – 28 + 14
= –14
(d)
x
2
5
–
x
3(2 – 5)
3
=
x
2
5
–
x
5(2 – 5)
5
=
x x
2 – 5(2 – 5)
5
=
+
x x
2 – 10 25
5
=
+
x
–8 25
5
7. (a) 5x + 15y = 5(x + 3y)
(b) 4cx – 8dx + 2cdx – 2x
= 2x(2c – 4d + cd – 1)
8. (a) The breadth of the rectangle is x cm.
Then the length of the rectangle is (x + 7) cm.
Perimeter of rectangle
= 2[(x + 7) + x]
= 2[2x + 7]
= (4x + 14) cm
Area of rectangle
= (x)(x + 7)
= (x2
+ 7x) cm2
(b) Let the smaller number be y.
Then the larger number is 4y.
y + 4y = p
5y = p
y =
p
5
The smaller number is
p
5
and the larger number
is
p
4
5
.
43

1
Revision Test A2
1. (a) 24
× 33
× 5
22
× 34
× 53
22
33
5
HCF of the two numbers = 22
× 33
× 5
= 540
(b) 32
× 5
22
× 3 × 52
32
22
52
LCM of the two numbers = 22
× 32
× 52
= 900
(c)
3 11 025
3 3675
5 1225
5 245
7 49
7 7
1
11 025 = 32
× 52
× 72
= (3 × 5 × 7)2
11 025 = × ×
(3 5 7)2
= 3 × 5 × 7
= 105
2. (a) (–2)3
× (–5) – 4 × (–5)2
– (–7)2
–8 × (–5) – 4 × 25 – 49
= 40 – 100 – 49
=
−109
(b) –2
1
3
÷ –
1
5
–
2
3





 ÷
1
15





 – –
1
4






= –2
1
3
÷ –
13
15





 ÷
1
15





 – –
1
4






= –2
1
3
÷ –
13
1 15
×
15
1
1








– –
1
4






= –2
1
3
÷ (–13) – –
1
4






= –
7
3
× –
1
13





 +
1
4
=
7
39
+
1
4
=
67
156

(c)
75 × –
1
2





 × (–13.4)
(0.5) × 7.5
=
502.5
3.75
= 134
3. (a) +
29.76 (8.567 – 0.914)
3 2
= 26 415.738 59
= 29.78 (to 2 d.p.)
(b)
121.56 78.94 – 99.18
2 121.56 78.94
2 2 2
+
× ×
=
11 171.6848
19 191.8926
= 0.76 (to 2 d.p.)
4. (a) 8.4454 = 8.45 (to 2 d.p.)
(b) 0.070 49 = 0.070 (to 2 s.f.)
(c) 25 958 = 26 000 (to the nearest 100)
(d) 15 997 = 16 000 (to the nearest 10)
5. (a) 2a + 5b – 3c – (4b – 3a + 6c)
= 2a + 5b – 3c – 4b + 3a – 6c
= 2a + 3a + 5b – 4b – 3c – 6c
= 5a + b – 9c
(b) [2a – b(a + 3)] + b(3 + 2a)
= [2a – ab – 3b] + 3b + 2ab
= 2a – ab + 2ab – 3b + 3b
= 2a + ab
(c) (2x + 1)(x – 3) – (3 – x)(1 – 5x)
= (2x + 1)(x – 3) + (x – 3)(1 – 5x)
= (x – 3)(2x + 1 + 1 – 5x)
= (x – 3)(−3x + 2)
(d)
x
2( 3)
3
+
– (x – 2) – 4 –
x
3( – 4)
6
=
x
4( 3)
6
+
–
x
3( – 4)
6
– (x – 2) – 4
=
x x
4( 3) – 3( – 4)
6
+
– (x – 2) – 4
=
x x
4 12 – 3 12
6
+ +
– (x – 2) – 4
=
x x
4 – 3 12 12
6
+ +
– (x – 2) – 4
=
x 24
6
+
–
x
6( – 2)
6
–
24
6
=
x x
24 – 6( – 2) – 24
6
+
=
x x
24 – 6 12 – 24
6
+ +
=
x x
– 6 24 12 – 24
6
+ +
=
x
–5 12
6
+
44

1
6. (a) When x = −2, y = −1, z = 0,
(x – y)z – x
=
(−2 – (–1))0 – (−2)
= (–2 + 1)2
=
(−1)2
= 1
(b) When a = 3, b = −2 and c = 5,
(i) a + b + c
= 3 + (−2) + 5
= 3 – 2 + 5
= 1 + 5
= 6
(ii) abc
=
(3)(−2)(5)
=
−30
(iii)
a
1
+
b
1
+
c
1
=
1
3
+
1
–2
+
1
5
=
10
30
–
15
30
+
6
30
=
1
30
7. (a) 7q + 5p – 4r – 5 – (2p + 5q – 4r + 3)
= 7q + 5p – 4r – 5 – 2p – 5q + 4r – 3
= 5p – 2p + 7q – 5q – 4r + 4r – 5 – 3
= 3p + 2q – 8
(b) (i) xy + 2x – 5zxy – 10xz
= x(y + 2 – 5zy – 10z)
(ii) 6ap – 6bp – 3pc + 24pab
= 3p(2a – 2b – c + 8ab)
(c) Total height of the boys = (mp) cm
Total height of the girls = (nq) cm
Total height of the students
= (mp + nq) cm
Average height of the students
=
+
+






mp nq
m n
cm
45

1
Chapter 5 Linear Equations and Simple
Inequalities
Basic
1. (a) 5x + 2 = 7
5x + 2 – 2 = 7 – 2
5x = 5
x
5
5
=
5
5
x = 1
(b) 2x – 7 = 3
2x – 7 + 7 = 3 + 7
2x = 10
x
2
2
=
10
2
x = 5
(c) 15 – 2x = 9
15 – 2x + 2x = 9 + 2x
15 = 9 + 2x
9 + 2x – 9 = 15 – 9
2x = 6
x
2
2
=
6
2
x = 3
(d) 17 + 3x = −3
17 + 3x – 17= – 3 – 17
3x = –20
x
3
3
=
–20
3
x = –6
2
3
(e) –4x + 7 = −15
–4x + 7 – 7 = −15 – 7
–4x = –22
x
–4
–4
=
–22
–4
x = 5
1
2
(f) 2x – 3 = x + 5
2x – 3 + 3 = x + 5 + 3
2x = x + 8
2x – x = x + 8 – x
x = 8
(g) 9x + 4 = 3x – 9
9x + 4 – 4 = 3x – 9 – 4
9x = 3x – 13
9x – 3x = 3x –13 – 3x
6x = –13
x
6
6
=
–13
6
x = –2
1
6
(h) 7x – 14 = 18 – 4x
7x – 14 + 14 = 18 – 4x + 14
7x = 32 – 4x
7x + 4x = 32 – 4x + 4x
11x = 32
x
11
11
=
32
11
x = 2
10
11
2. (a) 3(x – 4) = 7
3x – 12 = 7
3x – 12 + 12 = 7 + 12
3x = 19
x
3
3
=
19
3
x = 6
1
3
(b) 5(2x + 3) = 35
10x + 15 = 35
10x + 15 – 15 = 35 – 15
10x = 20
x
10
10
=
20
10
x = 2
(c) 4(3 – x) = −15
12 – 4x = −15
12 – 4x – 12 = −15 – 12
–4x = –27
x
–4
–4
=
–27
–4
x = 6
3
4
(d) 2(7 – 2x) = 11
14 – 4x = 11
14 – 4x – 14 = 11 – 14
–4x = −3
x
–4
–4
=
–3
–4
x =
3
4
46

1
(e) 2(x – 5) = 5x + 7
2x – 10 = 5x + 7
2x – 10 – 7 = 5x + 7 – 7
2x – 17 = 5x
2x – 17 – 2x = 5x – 2x
−17 = 3x
3x = −17
x
3
3
=
–17
3
x = –5
2
3
(f) 6 – 4x = 5(x – 6)
6 – 4x = 5x – 30
6 – 4x + 4x = 5x – 30 + 4x
6 = 9x – 30
6 + 30 = 9x – 30 + 30
36 = 9x
9x = 36
x
9
9
=
36
9
x = 4
(g) 2x – 3(5 – x) = 35
2x – 15 + 3x = 35
2x + 3x – 15 = 35
5x – 15 = 35
5x – 15 + 15 = 35 + 15
5x = 50
x
5
5
=
50
5
x = 10
(h) 7(x + 4) = 2(x – 4)
7x + 28 = 2x – 8
7x + 28 – 2x = 2x – 8 – 2x
5x + 28 = –8
5x + 28 – 28 = –8 – 28
5x = −36
x
5
5
=
–36
5
x = –7
1
5
(i) 2(5 – 2x) = 4(2 – 3x)
10 – 4x = 8 – 12x
10 – 4x + 12x = 8 – 12x + 12x
8x + 10 = 8
8x + 10 – 10 = 8 – 10
8x = −2
x
8
8
=
–2
8
x = −
1
4
(j) (5x + 3) – (4x – 9)= 0
5x + 3 – 4x + 9 = 0
5x – 4x + 3 + 9 = 0
x + 12 = 0
x + 12 – 12 = 0 – 12
x = −12
(k) 7(3 – 4x) – 5(2x + 8) = 0
21 – 28x – 10x – 40 = 0
21 – 40 – 28x – 10x = 0
–19 – 38x = 0
–19 – 38x + 19 = 0 + 19
–38x = 19
x
–38
–38
=
19
–38
x = −
1
2
(l) 5(2x – 3) – 3(x – 2) = 0
10x – 15 – 3x + 6 = 0
10x – 3x – 15 + 6 = 0
7x – 9 = 0
7x – 9 + 9 = 0 + 9
7x = 9
x
7
7
=
9
7
x = 1
2
7
3. (a)
3
4
x = 15
3
4
x × 4 = 15 × 4
3x = 60
x
3
3
=
60
3
x = 20
(b)
2
5
x – 1 = 4
2
5
x – 1 + 1 = 4 + 1
2
5
x = 5
2
5
x × 5 = 5 × 5
2x = 25
x
2
2
=
25
2
x = 12
1
2
47

1
(c) 5 –
3
4
x = –1
5 –
3
4
x +
3
4
x = –1 +
3
4
x
5 = –1 +
3
4
x
5 + 1 = –1 +
3
4
x + 1
3
4
x = 6
3
4
x × 4 = 6 × 4
3x = 24
x
3
3
=
24
3
x = 8
(d) 3 +
4
7
x = 1
1
3
3 +
4
7
x – 3 = 1
1
3
– 3
4
7
x = –1
2
3
4
7
x × 7 = –1
2
3
× 7
4x = –11
2
3
x
4
4
=
–11
2
3
4
x = –2
11
12
(e) 2x = 0.4x + 12.8
2x – 0.4x = 0.4x + 12.8 – 0.4x
1.6x = 12.8
x
1.6
1.6
=
12.8
1.6
x = 8
(f) 0.3x + 1.2 = 0.25 – 0.2x
0.3x + 1.2 + 0.2x = 0.25 – 0.2x + 0.2x
0.5x + 1.2 = 0.25
0.5x + 1.2 – 1.2 = 0.25 – 1.2
0.5x = −0.95
x
0.5
0.5
=
–0.95
0.5
x = −1.9
(g)
2
3
x + 15 = 4x
2
3
x + 15 –
2
3
x = 4x –
2
3
x
15 = 3
1
3
x
3
1
3
x = 15
x
3
1
3
3
1
3
=
15
3
1
3
x = 4
1
2
(h) 1.3x – 3.6 =
4
5
x + 2
1.3x – 3.6 –
4
5
x =
4
5
x + 2 –
4
5
x
0.5x – 3.6 = 2
0.5x – 3.6 + 3.6 = 2 + 3.6
0.5x = 5.6
x
0.5
0.5
=
5.6
0.5
x = 11.2
(i) 1.5 –
7
8
x = 2.6x +
1
5
1.5 –
7
8
x +
7
8
x = 2.6x +
1
5
+
7
8
x
1.5 = 3.475x +
1
5
1.5 –
1
5
= 3.475x +
1
5
–
1
5
3.475x = 1.3
x
3.475
3.475
=
1.3
3.475
x =
52
139
4. (a)
x
2 – 3
5
= 7
x
2 – 3
5
× 5= 7 × 5
2x – 3 = 35
2x – 3 + 3 = 35 + 3
2x = 38
x
2
2
=
38
2
x = 19
48

1
(b)
x
3 – 4
5
– 7 = 0
x
3 – 4
5
= 7
x
3 – 4
5
× 5 = 7 × 5
3x – 4 = 35
3x – 4 + 4 = 35 + 4
3x = 39
x
3
3
=
39
3
x = 13
(c)
+
x 1
3
=
x
3
5
15 ×
+
x 1
3
= 15 ×
x
3
5
5(x + 1) = 3(3x)
5x + 5 = 9x
5x + 5 – 5x = 9x – 5x
5 = 4x
4x = 5
x
4
4
=
5
4
x = 1
1
4
(d)
x
2 – 1
3
= 1 – x
x
2 – 1
3
× 3 = (1 – x) × 3
2x – 1 = 3(1 – x)
2x – 1 = 3 – 3x
2x – 1 + 3x = 3 – 3x + 3x
5x – 1 = 3
5x – 1 + 1 = 3 + 1
5x = 4
x
5
5
=
4
5
x = 0.8
(e)
2
3
(5x – 7) =
4
5
15 ×
2
3
(5x – 7) = 15 ×
4
5
10(5x – 7) = 12
50x – 70 = 12
50x – 70 + 70 = 12 + 70
50x = 82
x
50
50
=
82
50
x = 1
16
25
(f)
2
3
(6x + 5) = 7(x – 4.5)
4x + 3
1
3
= 7x – 31.5
4x + 3
1
3
– 4x = 7x – 31.5 – 4x
3
1
3
= 3x – 31.5
3
1
3
+ 31.5 = 3x – 31.5 + 31.5
209
6
= 3x
3x =
209
6
x
3
3
=






209
6
3
x = 11
11
18
(g)
1
4
(3x + 5) =
1
3
(5x – 4)
12 ×
1
4
(3x + 5) = 12 ×
1
3
(5x – 4)
3(3x + 5) = 4(5x – 4)
9x + 15 = 20x – 16
9x + 15 – 15 = 20x – 16 – 15
9x = 20x – 31
9x – 20x = 20x – 31 – 20x
–11x = –31
11x = 31
x
11
11
=
31
11
x = 2
9
11
(h)
1
5
(4 – 3x) =
1
7
(3x – 4)
35 ×
1
5
(4 – 3x) = 35 ×
1
7
(3x – 4)
7(4 – 3x) = 5(3x – 4)
28 – 21x = 15x – 20
28 – 21x + 21x = 15x – 20 + 21x
28 = 36x – 20
28 + 20 = 36x – 20 + 20
48 = 36x
36x = 48
x
36
36
=
48
36
x = 1
1
3
49

1
(i)
x
4 – 3
5
=
x
2 – 7
8
8(4x – 3) = 5(2x – 7)
32x – 24 = 10x – 35
32x – 10x = –35 + 24
22x = −11
x
22
22
=
–11
22
x = –
1
2
5. (a) y = a(4a – 5)
When a = 3, y = 3(4 × 3 – 5)
= 3(7) = 21
(b) y = (x + p)(3x – p – 4)
When x = 3, p = 4,
y = (3 + 4)(3 × 3 – 4 – 4)
= (7)(1) = 7
(c) y =
x
2 – 1
3
When x = 5, y =
2(5) – 1
3
=
9
3
= 3
(d) y =
+
r
r
2 5
7 – 9
When r = 6,
y =
+
2(6) 5
7(6) – 9
=
17
33
6. xy – 3y2
= 15
When y = 2,
x(2) – 3(2)2
= 15
2x – 12 = 15
2x = 15 + 12
2x = 27
x = 13
1
2
7. y =
2
3
(24 – x) + 5xy
When x = –3
1
3
,
y =
2
3
24 – –3
1
3











 + 5 –3
1
3





 y
y =
2
3
27
1
3





 – 16
2
3
y
y = 18
2
9
– 16
2
3
y
y + 16
2
3
y = 18
2
9
17
2
3
y= 18
2
9
y = 1
5
159
8. p – 5q = 4qr
When q = 4, r = −1,
p – 5(4) = 4(4)(−1)
p – 20 = −16
p = −16 + 20 = 4
9. (a) D = a2
– b2
(b) The three consecutive numbers are d, d + 2
and d + 4.
S = d + (d + 2) + (d + 4) = 3d + 6 = 3(d + 2)
(c) Perimeter of square = m + m + m + m = 4m
Perimeter of rectangle = 2(n + s)
Perimeter of figure,
P = 4m + 4(n + s)
10. (a) Let the smallest odd number be n.
The next odd number is n + 2.
The largest odd number is (n + 2) + 2 = n + 4.
S = n + n + 2 + n + 4 = 3n + 6
3n + 6 = 243
3n = 243 – 6 = 237
n = 79
The largest odd number is 79 + 4 = 83.
(b) Let the smallest even number be n.
The next even number is n + 2.
The next even number is (n + 2) + 2 = n + 4.
The next even number is (n + 4) + 2 = n + 6.
The largest even number is (n + 6) + 2 = n + 8.
S = n + n + 2 + n + 4 + n + 6 + n + 8
= 5n + 20
5n + 20 = 220
5n = 220 – 20 = 200
n = 40
The smallest of the five numbers is 40.
(c) Let the smaller odd number be n.
The next odd number is n + 2.
3(n + 2) – n = 56
3n + 6 – n = 56
2n = 56 – 6
2n = 50
n = 25
The two numbers are 25 and 27.
(d) Let the smaller even number be n.
The next even number is n + 2.
n + 2 + 3n = 42
4n = 40
n = 10
50

1
11. (a) Let the age of Raj be x years old.
Then Rui Feng is 2x years old.
Khairul is (2x – 7) years old.
x + 2x + (2x – 7) = 38
5x = 38 + 7
5x = 45
x = 9
Raj is 9 years old.
Rui Feng is 2 × 9 = 18 years old.
Khairul is (2 × 9 – 7) = 11 years old.
(b) Let the number of years ago in which Kate’s
father is three times as old as her be n.
50 – n = 3(24 – n)
50 – n = 72 – 3n
2n = 72 – 50
2n = 22
n = 11
Kate’s father was three times as old as Kate
11 years ago.
(c) Let the age of Farhan be x years old.
Then Farhan’s brother’s age is 3x years old.
In 12 years’ time,
Farhan will be (x + 12) years old and his brother
will be (3x + 12) years old.
(x + 12) + (3x + 12) = 10x
4x + 24 = 10x
6x = 24
x = 4
Farhan’s present age is 4 years old and his
brother is 12 years old.
12. (a) Let the first number be x.
Then the second number is 120 – x.
120 – x = 4x
5x = 120
x = 24
The smaller number is 24.
(b) Let the number be x.
12 –
x
4
=
1
6
x
12 =
1
6
x +
x
4
12 =
5
12
x
144 = 5x
x = 28
4
5
The number is 28
4
5
.
13. (a) The cost of 12 pears is equal to the cost of
36 apples.
A pear costs 3 times an apple.
Let the cost of an apple be $x.
Then the cost of a pear is $3x.
The amount of money Michael has is $36x.
Cost of 1 apple and 1 pear
= $3x + $x
= $4x
No. of each fruit Michael can buy
=
x
x
36
4
= 9
(b) Amount of money spent on pencils
= 15 ×
x
2
100
= $
x
3
10
Amount of money spent on pens
= 24 ×
y
4
100
= $
y
24
25
Total amount spent on pencils and pens
=
x
3
10
+
y
24
25
= $
+
x y
15 48
50
14. (a) 3x > 33
x >
33
3
x > 11
(b) 11x ø 25
x ø
25
11
x ø 2
3
11
(c)
1
2
x > 3
2 ×
1
2
x > 3 × 2
x > 6
(d)
x
3
4
ø
3
8
4 ×
x
3
4
ø 4 ×
3
8
3x ø
3
2
3x ÷ 3 ø
3
2
÷ 3
x ø
1
2
51

1
(e)
4
5
x ø 1
1
2
5 ×
4
5
x ø 5 × 1
1
2
4x ø 7
1
2
4x ÷ 4 ø 7
1
2
÷ 4
x ø 1
7
8
(f) 0.4x < 3.2
0.4x ÷ 4 < 3.2 ÷ 0.4
x < 8
Intermediate
15. (a) 5(3x – 2) – 7(x – 1) = 12
15x – 10 – 7x + 7 = 12
15x – 7x – 10 + 7 = 12
8x – 3 = 12
8x = 12 + 3
8x = 15
x =
15
8
= 1
7
8
(b) 4(3 – x) + 3(4x + 5) = –45
12 – 4x + 12x + 15 = –45
–4x + 12x + 12 + 15 = –45
8x + 27 = –45
8x = –45 –27
8x = –72
x = –9
(c) 0.3(4x – 1) = 0.8 + x
1.2x – 0.3 = 0.8 + x
1.2x – x = 0.8 + 0.3
0.2x = 1.1
x
0.2
0.2
=
1.1
0.2
x = 5.5
(d) 3(5x + 2) – 7(3 – x) = (19 + 5x) + (20 – x)
15x + 6 – 21 + 7x = 19 + 20 + 5x – x
15x + 7x – 15 = 39 + 4x
22x – 15 = 39 + 4x
22x – 4x = 39 + 15
18x = 54
x = 3
(e) 2x – [3 + 5(x – 5)] = 10
2x – [3 + 5x – 25] = 10
2x – [5x – 22] = 10
2x – 5x + 22 = 10
−3x = 10 – 22
−3x = –12
x = 4
(f) 3x – [3 – 2(3x – 7)] = 37
3x – [3 – 6x + 14] = 37
3x – [17 – 6x] = 37
3x – 17 + 6x = 37
3x + 6x = 37 + 17
9x = 54
x = 6
16. (a)
x
2( – 1)
3
+
x
3
4
= 0
12 ×
2(x – 1)
3
+
3x
4
= 12 × 0
8(x – 1) + 9x = 0
8x – 8 + 9x = 0
8x + 9x = 8
17x = 8
x =
8
17
(b)
+
x
6 1
7
–
x
2 – 7
3
= 4
21 ×
6x + 1
7
–
2x – 7
3





 = 21 × 4
3(6x + 1) – 7(2x – 7) = 84
18x + 3 – 14x + 49 = 84
4x + 52 = 84
4x = 84 – 52
4x = 32
x = 8
(c) 2x –
x
4
+
x
3
5
= 14 +
x
7
3
2x –
x
4
+
x
3
5
–
x
7
3
= 14
x
60
= 14
60 ×
x
60
= 60 × 14
x = 840
(d) 5x – 1
3
4
= 6 + 1
2
3
x –
5
6
5x – 1
3
4
x = 5
1
6
+ 1
2
3
x
5x – 1
2
3
x = 5
1
6
+ 1
3
4
3
1
3
x = 6
11
12
x = 2
3
40
52

1
(e)
x
4
=
+
x 12
10
+ 0.6
x
4
=
x
10
+
12
10
+ 0.6
x
4
–
x
10
= 1.2 + 0.6
x
3
20
= 1.8
x = 12
(f)
x
3 – 4
6
–
+
x
2 3
8
=
x
2 – 7
24
24 ×
3x – 4
6
–
2x + 3
8





 = 24 ×
x
2 – 7
24
4(3x – 4) – 3(2x + 3) = 2x – 7
12x – 16 – 6x – 9 = 2x – 7
6x – 25 = 2x – 7
6x – 2x = –7 + 25
4x = 18
x = 4
1
2
(g)
x
5 – 1
8
–
x
5 – 7
2
=
x
3(6 – )
6
24 ×
5x – 1
8
–
5 – 7x
2





 = 24 ×
x
3(6 – )
6
3(5x – 1) – 12(5 – 7x) = 12(6 – x)
15x – 3 – 60 + 84x = 72 – 12x
99x – 64 = 72 – 12x
99x + 12x = 72 + 63
111x = 135
x = 1
8
37
(h)
+
x
5 2
7
=
x – 3
5
+ x + 1.5
35 ×
+
x
5 2
7
= 35 ×
x – 3
5
+ x + 1.5






5(5x + 2) = 7(x – 3) + 35x + 52.5
25x + 10 = 7x – 21 + 35x + 52.5
25x + 10 = 42x + 31.5x
25x – 42x = 31.5 – 10
–17x = 21.5
17x = −21.5
x = −1
9
34
(i)
x
3
–
x
7( – 2)
9
= 4 –
x
2 – 5
6
18 ×
x
3
–
7(x – 2)
9





 = 18 × 4 –
2x – 5
6






6(x) – 14(x – 2) = 72 – 3(2x – 5)
6x – 14x + 28 = 72 – 6x + 15
–8x + 28 = 87 – 6x
–8x + 6x = 87 – 28
−2x = 59
x = −29.5
(j) 0.5x + 2 =
1
4
+
x – 1
2
+
x
4
–
1
6
0.5x + 2 =
1
4
+
x
2
–
1
2
+
x
4
–
1
6
0.5x –
x
2
–
x
4
=
1
4
–
1
2
–
1
6
– 2
–
x
4
= –2
5
12
x = 9
2
3
(k) 4x + 1 –
1
2
(3x – 2) –
1
3
(4x – 1) = 0
6 × 4x + 1 –
1
2
(3x – 1) –
1
3
(4x – 1)





 = 6 ×0
24x + 6 – 3(3x – 2) – 2(4x – 1) = 0
24x + 6 – 9x + 6 – 8x + 2 = 0
24x – 9x – 8x + 6 + 6 + 2 = 0
7x + 14 = 0
7x = −14
x = −2
(l)
1
2
2x –
1
2





 =
1
3
3x –
1
4





 +
1
4
(4x – 3)
x –
1
4
= x –
1
12
+ x –
3
4
x – x – x = –
1
12
–
3
4
+
1
4
–x = –
7
12
x =
7
12
17. (a)
x
3
+
x
4
= 5
x
7
= 5
x ×
x
7
= x ×5
7 = 5x
x =
7
5
= 1
2
5
(b)
x
5
2
–
x
7
5
=
2
3
10x ×
5
2x
–
7
5x





 = 10x ×
2
3
25 – 14 = 6
2
3
x
11 = 6
2
3
x
x = 1
13
20
53

1
(c)
x
7
2
+
x
5
3
= 1
5
6
6x ×
7
2x
+
5
3x





 = 6x × 1
5
6
21 + 10 = 11x
31 = 11x
x = 2
9
11
(d)
+
x
5
2
–
+
x
4
2 4
= 6
+
x
5
2
–
+
x
4
2( 2)
= 6
+
x
10
2( 2)
–
+
x
4
2( 2)
= 6
+
x
6
2( 2)
= 6
12(x + 2) = 6
12x + 24 = 6
12x = 6 – 24
12x = −18
x = −1
1
2
(e) 1 –
+
+
x
x
1
3 5
=
1
2
+
+
x
x
1
3 5
= 1 –
1
2
+
+
x
x
1
3 5
=
1
2
2(x + 1) = 3x + 5
2x + 2 = 3x + 5
3x – 2x = 2 – 5
x = −3
18. 5(2x – 3) – 3(x – 2) = 0
10x – 15 – 3x + 6 = 0
10x – 3x – 15 + 6 = 0
7x – 9 = 0
7x – 9 + 11 = 0 + 11
7x + 2 = 11
19. When x = 4,
LHS
= −2 –
×
2 4
5
+
×
3 4
2
= –2 –
8
5
+
12
2
= 2
2
5
≠ 4
3
5
(RHS)
No, x = 4 is not a solution of the equation.
20. When y = 2, p = 5 and q = 6,
x – 2 =
x(2)
5 – 6
x – 2 =
x
2
–1
x – 2 = −2x
x + 2x = 2
3x = 2
x =
2
3
21. When y = 8 and z = 2,
+
x – 1
8 3
–
x
8
=
1
2
x – 1
11
–
x
8
=
1
2
x
11
–
1
11
–
x
8
=
1
2
x
11
–
x
8
=
1
2
+
1
11
–
3
88
x =
13
22
x = –17
1
3
22. v2
= u2
+ 2as
When u = 15, a = 9.81, s = 14.45,
v2
= (15)2
+ 2(9.81)(14.45)
= 508.509
v = ± 508.509
v = ±22.6 (to 3 s.f.)
23. When a = 3
1
2
, h = 10 and k = 15,
x
1
= 3
1
2
– 2






1
10
+
1
15






= 1
1
2






1
6






=
1
4
x = 4
54

1
24. When y = 6 and z = –
1
2
,
3x + 2(6) – 5 –
1
2






6 – 4 –
1
2






=
x
3(6)
+ +
x
3 12 2
1
2
8
=
x
18
+
x
3 14
1
2
8
=
x
18
8 ×
+
x
3 14
1
2
8
= 8 ×
x
18
3x + 14
1
2
=
x
4
9
3x –
x
4
9
= –14
1
2
2
5
9
x = –14
1
2
x = –5
31
46
25. When p = 3, q = −2,
r
5(3) – 3(–2)
=
+
3(–2) – 5(3)
3 (–2)
+
r
15 6
=
–6 – 15
1
r
21
=
–21
1
21 = –21r
r = −1
26. A = P +
PRT
100
(a) When P = 5000, R = 5 and T = 3,
A = 5000 +
(5000)(5)(3)
100
= 5750
(b) When A = 6500, R = 5 and T = 1
2
3
,
6500 = P +
P(5) 1
2
3






100
6500 = P +
1
12
P
6500 = 1
1
12
P
1
1
12
P = 6500
P = 6500 ÷ 1
1
12
= 6000
27.
f
1
=
u
1
+
v
1
(a) When u = 5 and v = 7,
f
1
=
1
5
+
1
7
f
1
=
12
35
12f = 35
f =
35
12
= 2
11
12
(b) When f = 4 and v = 5,
1
4
=
u
1
+
1
5
u
1
=
1
4
–
1
5
u
1
=
1
20
u = 20
28. (a) (i) Let the first number be x.
Then the second number is mx.
Then the third number is mx – n.
Sum of the three numbers S = x + mx + mx – n
= x + 2mx – n
(ii) When S = 109, m = 4, n = 8,
109 = x + 2(4)x – 8
109 = x + 8x – 8
9x = 109 + 8
9x = 117
x = 13
The three numbers are 13, 4(13) = 52
and 52 – 8 = 44.
(b) (i) The cost of the pair of shoes is $C.
Amount of money Nora has after buying the
pair of shoes = $(p – C)
Amount of money Priya has after buying
the pair of shoes = $(q – C)
p – C = 2(q – C)
p – C = 2q – 2C
2C – C = 2q – p
C = 2q – p
(ii) When p = 42, q = 30,
cost of the pair of shoes = 2 × 30 – 42
= $18
55

1
29. (i) Let the number Lixin is thinking of be x.
2x + 14 = 4x – 8
(ii) 2x + 14 = 4x – 8
14 + 8 = 4x – 2x
22 = 2x
x = 11
(iii)The result is 2x + 14 = 2(11) + 14 = 36.
30. Let the denominator of the fraction be x.
Then the numerator is x – 1.
+
+
x
x
– 1 1
2
=
3
4
+
x
x 2
=
3
4
4x = 3(x + 2)
4x = 3x + 6
4x – 3x = 6
x = 6
Then the numerator is 6 – 1 = 5.
The original fraction is
5
6
.
31. (i) The woman’s present age is 8x years old
(ii) Michael’s age two years ago was
(x – 2) years old.
(iii)The woman’s age two years ago was
= (8x – 2) years old
8x – 2 = 15(x – 2)
8x – 2 = 15x – 30
8x – 15x = –30 + 2
–7x = –28
7x = 28
x = 4
(iv) The woman’s present age = 8 × 4 = 32 years old.
The woman’s age in 5 years’ time
= 32 + 5
= 37 years old
32. (i) Amount of time spent cycling =
x
9
hours
(ii) Amount of time spent taking the train
=
28
60
–
x
9
–
3
60
–
1
2
6
=
7
15
–
x
9
–
3
60
–
1
12
=
1
3
–
x
9





 hours
Distance travelled by Ethan on the MRT train
= 60
1
3
–
x
9






= 20 – 6
2
3
x





 km
(iii)x + 20 – 6
2
3
x +
1
2
= 12
6
2
3
x – x = 20 +
1
2
– 12
5
2
3
x = 8
1
2
x = 1
1
2
33. Let the number of apples bought be x.
Then the number of oranges bought is 2x.
Then the number of pears bought is (x – 5).
(i) Amount spent on the fruits = $77
x(0.40) + 2x(0.30) + (x – 5)(0.80) = 77
0.4x + 0.6x + 0.8x – 4 = 77
1.8x – 4 = 77
1.8x = 77 + 4
1.8x = 81
x = 45
(ii) Amount of money spent on buying the pears
= (x – 5)(0.80)
= (45 – 5)(0.80)
= (40)(0.80)
= $32
He spent $32 on buying the pears.
34. Let the number of ducks bought be x.
Then the number of chicken bought is 3x.
The number of geese bought is 0.5x.
Total cost = $607.20
x(7.5) + 3x(3.8) + 0.5x(12.8) = 607.2
7.5x + 11.4x + 6.4x = 607.2
25.3x = 607.2
x = 24
The number of geese bought is 0.5 × 24 = 12.
35. (i) Amount of money the salesman earned in a week
= 90 +
12(580)
100
= 90 + 69.60
= $159.60
(ii) To find the number of articles sold, make n the
subject.
A = 90 +
n
12
100
A – 90 =
n
12
100
12n = 100(A – 90)
n =
A
100( – 90)
12
=
100(190.80 – 90)
12
=
100(100.80)
12
= 840
56

1
(iii)A = 80 +
n
16
100
(iv) For the same amount of money earned before
and after
90 +
n
12
100
= 80 +
n
16
100
90 – 80 =
n
16
100
–
n
12
100
n
25
= 10
n = 250
The number of articles the salesman must sell
in a week to earn the same amount of money before
and after the adjustments is 250.
36. (i) Rui Feng’s brother’s age is 0.5 × 4x = 2x years old.
Sum of their present ages = 4x + 2x = 6x years old
(ii) In 8 years’ time,
Rui Feng is (4x + 8) years old and his brother
is (2x + 8) years old.
Sum of their ages in 8 years’ time
= (4x + 8) + (2x + 8)
= 4x + 2x + 8 + 8
= (6x + 16) years old
37. Let the second number be x.
Then the first number is (x + 5).
Then the third number is 0.5x.
The fourth number is 3[(x + 5) + x] = 3(2x + 5).
The total of the four numbers is 56 × 4 = 224.
(x + 5) + x + 0.5x + 3(2x + 5) = 224
x + 5 + x + 0.5x + 6x + 15 = 224
x + x + 0.5x + 6x + 5 + 15 = 224
8.5x + 20 = 224
8.5x = 224 – 20
8.5x = 204
x = 24
The numbers are 24 + 5 = 29, 24, 0.5(24) = 12 and
3(2 × 24 + 5) = 159.
38. 2x ø 7
x ø 3
1
2
The largest rational number is 3
1
2 .
39. Let $x be the amount of money each student will get.
32x ø 4385
x
32
32
ø
4385
32
x ø 137.031 25
Each student will get a maximum amount of $135 (to
the nearest $5).
40. Let the number of concert tickets be x.
25x ø 115
x ø 4
3
5
The maximum number of tickets that can be
purchased is 4.
41. Let the number of cakes be x.
4x ø 39
x ø 9
3
4
The maximum number of cakes that can be bought
is 9.
42. Let the age of the woman be x years old.
Then her husband is (x + 3) years old.
x + (x + 3) ø 55
2x + 3 ø 55
2x ø 52
x ø 26
The maximum possible age of the woman is 26.
43. Let the first number be x.
Then the second number is x + 1 and the third number
is x + 2.
x + x + 1 + x + 2 < 80
3x + 3 < 80
3x < 77
x < 25
2
3
The largest possible value of the largest integer is
27.
Advanced
44. x3
+ 6x2
= x(x2
+ 5x) + x2
= x(5) + x2
= x2
+ 5x
= 5
45.
x xy
y z
– 3
– 2
2
2
=
y
5
3
When y = 2 and z = –5;
x x
– 3 (2)
(2) – 2(–5)
2
2
=
5(2)
3
x x
– 6
14
2
=
10
3
14 ×
x x
– 6
14
2
= 14 ×
10
3
x2
– 6x = 46
2
3
By trial and error, x is approximately 10.45.
x = 10.5 (to 3 s.f.)
57

1
46. (a) 5(x – 2)2
= 35
5(x – 2)2
÷ 5 = 35 ÷ 5
(x – 2)2
= 7
x – 2 = ± 7
x = 2 ± 7
x = 4.65 (to 2 d.p.) or x = −0.65 (to 2 d.p.)
(b)
x
x
2 – 3
4
– 2
= 3
1
2
x
x
2 – 3
4
– 2
=
7
2
2
2x – 3
4
– 2





 = 7x
x
2 – 3
2
– 4 = 7x
x
2 – 3
2
= 7x + 4
2 ×
x
2 – 3
2
= 2 × (7x + 4)
2x – 3 = 14x + 8
2x – 14x = 8 + 3
–12x = 11
12x = −11
x = –
11
12
Let the second number be 84 – x.
1
2
x –
1
3
(84 – x) = 2
1
2
x +
1
3
x – 28 = 2
5
6
x = 2 + 28
5
6
x = 30
5x = 180
x = 36
48. In 1 hour, Raj can complete
1
3
of the task.
In 1 hour, Farhan can complete
1
2
of the task.
In 1 hour, when they work together, they can complete
1
2
+
1
3
=
5
6
of the task
It takes them
6
5
hours = 1 hour and 12 minutes to
complete the task.
Then the second number is x + 2.
x + x + 2 < 15
2x < 13
x < 6
1
2
The largest possible value of the smaller integer is 5.
New Trend
50.
x
2 – 1
3
–
x
3 – 4
5
=
4
7
15 ×
2x – 1
3
–
3x – 4
5





 = 15 ×
4
7
5(2x – 1) – 3(3x – 4) = 8
4
7
10x – 5 – 9x + 12 = 8
4
7
x + 7 = 8
4
7
x = 8
4
7
– 7
x = 1
4
7
51.
+
x
3
2 5
=
x
4
1 – 3
3(1 – 3x) = 4(2x + 5)
3 – 9x = 8x + 20
9x + 8x = 3 – 20
17x = −17
x = −1
52. 5(2 – 3x) – (1 + 7x) = 5(3 – 6x)
10 – 15x – 1 – 7x = 15 – 30x
9 – 22x = 15 – 30x
–22x + 30x = 15 – 9
8x = 6
x =
6
8
=
3
4
53.
+
x
3 2
4
=
x
2 – 1
3
12 ×
+
x
3 2
4
= 12 ×
x
2 – 1
3
3(3x + 2) = 4(2x – 1)
9x + 6 = 8x – 4
9x – 8x = –4 – 6
x = –4 – 6
= –10
58

1
Chapter 6 Functions and Linear Graphs
Basic
1. We can find the coordinates from the graph. Each
ordered pair determines the points A to R.
A(1, 2) B(7, 1) C(−2, −3)
D(−4, 5) E(6, 6) F(3, −2)
G(−6, −2) H(5, 0) I(0, −5)
J(−7, 4) L(−3, 0) M(0, 3)
N(−5, 2) O(0, 0) P(6, −4)
Q(−3, −6) R(4, −6)
2.
–2
–3
–5
–6
–7
–8 –4 1
–1 4
2 3 6 8
5 7
0
1
7
8
2
3
4
5
6
y
–8
–7
–6
–5
–4
–3
–2
–1
x
N
R
C
L
B
Q
A
K
G
O
D
H
F
I
M
P
E
J
3. (i) When x = 2, y = 5 × 2 + 7 = 17.
(ii) When x = −3, y = 5 × (−3) + 7 = −8.
4. (i) When y = −8,
−8 = 10 – 9x
9x = 10 + 8 = 18
x = 2
(ii) When y = −26,
−26 = 10 – 9x
9x = 10 + 26 = 36
x = 4
59

1
5. (a) y = 2x
x –3 0 3
y = 2x –6 0 6
–2
–3 1
–1 2 3
0
1
2
3
4
5
6
y
–6
–5
–4
–3
–2
–1
x
y = 2x
(b) y = 3x + 2
x –3 0 3
y = 3x + 2 –7 2 11
–2
–3 1
–1 2 3
0
1
7
10
8
11
2
3
4
5
6
9
y
–7
–6
–5
–4
–3
–2
–1
x
y = 3x + 2
(c) y = –2x
x –3 0 3
y = –2x 6 0 –6
–2
–3 1
–1 2 3
0
1
2
3
y
–6
–5
–4
–3
–2
–1
x
y = –2x
4
5
6
(d) y = –3x – 1
x –3 0 3
y = –3x – 1 8 –1 –10
–2
–3 1
–1 2 3
0
1
7
8
2
3
4
5
6
y
–7
–6
–5
–10
–9
–8
–4
–3
–2
–1
x
y = –3x – 1
60

1
(e) y = –
1
4
x
x –3 0 3
y = –
1
4
x 3
4
0 –
3
4
y = –
1
4
x
–2
–3 1
–1 2 3
0
0.1
0.2
0.3
0.6
0.4
0.7
0.5
0.8
y
–0.5
–0.4
–0.3
–0.8
–0.2
–0.7
–0.1
–0.6
x
(f) y = 5 –
1
2
x
x –3 0 3
y = 5 –
1
2
x 6
1
2
5 3
1
2
y
–2
–3
–4 1
–1 4
2 3
x
0
1
2
3
5
4
6
7
8
y = 5 –
1
2
x
6. (a)
1 4
2 3
0
2
14
12
16
18
20
22
24
4
6
8
10
y
–6
–4
–2
x
y = 5x + 2
y = 5x – 2
y = 5x – 6
y = 5x
(b) They are parallel lines.
7. (a)
–2
–3
–4 1
–1 4
2 3
0
2
4
6
y
–8
–6
–4
–2
x
y = –
1
2
x + 3
y = –
1
2
x + 2
y = – 1
2
x – 2
y = – 1
2
x – 5
(b) They are parallel lines.
8. (a) (i) Amount of money left after 2 days
= 30 – 5 × 2 = $20
(ii) Amount of money left after 3 days
= 30 – 5 × 3 = $15
(iii)Amount of money left after 4 days
= 30 – 5 × 4 = $10
(iv) Amount of money left after 5 days
= 30 – 5 × 5 = $5
(b) x 0 1 2 3 4 5 6
y 30 25 20 15 10 5 0
61

1
(c)
1
0
10
20
25
15
5
30
2 4
3 5 6
y($)
x
(day)
y = –5x + 30
Intermediate
9. (a)
B(7, 1)
A(1, 1)
C(4, 8)
4 8
2 6 10
0
4
2
8
6
10
y
x
Isosceles triangle
(b)
F(7, 1)
D(1, 3)
E(3, 5)
4 8
2 6 10
0
4
2
6
x
y
Right-angled triangle
(c)
I(8, 2)
H(7, –1)
J(2, 5)
G(1, 1)
4 8
2 6 10
0
4
2
–2
–4
y
x
Quadrilateral
(d)
M(0, 3)
N(3, –1)
L(–2, 4)
K(–5, 3)
4 6
2
–2
–2
–4 0
4
2
6
y
x
Trapezium
(e)
O(1, 3)
L(–1, –2)
P(–1, 4)
Q(–3, 3)
4
2
–2
–2
–4 0
4
2
6
y
x
Kite
(f)
S(3, 0)
U(–2, –1)
R(0, 2)
T(1, –3)
4
2
–2
–2
–4
–4
0
4
2
y
x
Square
(g)
Y(2, 3)
V(–1, –3)
X(6, 2)
W(3, –4)
4
2
–2
–4
–2
–6
–4
0
4
2
y
x
Parallelogram
(h)
A(–4, –1)
D(–6, –4)
C(–4, –7)
B(–2, –4)
0
y
–4
–8
–10 –2
–2
–6
–4
–8
–6
x
Rhombus
62

1
10.
C(6, 2)
B(6, 5)
A(–2, 1)
4 8
2
–2 6
0
4
2
6
y
x
Area of triangle ABC =
1
2
× base × height
=
1
2
× 3 × 8
= 12 square units
11.
–2
–3 1
–1 2 3
0
1
2
3
4
5
y
–1
x
The points can be joined to form a straight line that
slopes upwards from left to right.
12. (i) When x = 5, y =
3
5
× 5 –
1
2
= 2
1
2
.
(ii) When x = 3
1
2
, y =
3
5
× 3
1
2
–
1
2
= 1
3
5
.
(iii)When x = –
2
3
, y =
3
5
× –
2
3





 –
1
2
= –
9
10
.
13. (i) When y = 12,
12 = 15 –
3
4
x
3
4
x = 15 – 12
3
4
x = 3
3x = 12
x = 4
(ii) When y = 21,
21 = 15 –
3
4
x
3
4
x = 15 – 21
3
4
x = –6
3x = –24
x = −8
(iii)When y = −60,
−60 = 15 –
3
4
x
3
4
x = 15 + 60
3
4
x = 75
3x = 300
x = 100
14. For the line y = 3x + 2,
(a) When x = 1, y = 5, then
LHS = 5
RHS = 3 × 1 + 2 = 5
Since LHS = RHS, then A(1, 5) lies on the line.
(b) When x = 3, y = 12, then
LHS = 12
RHS = 3 × 3 + 2 = 11
Since LHS ≠ RHS, then B(3, 12) does not lie
on the line.
(c) When x = 0, y = 2, then
LHS = 2
RHS = 3 × 0 + 2 = 2
Since LHS = RHS, then C(0, 2) lies on the line.
(d) When x = −2, y = 4, then
LHS = 4
RHS = 3 × (−2) + 2 = −4
Since LHS ≠ RHS, then D(−2, 4) does not lie
on the line.
(e) When x = –
1
3
, y = 1, then
LHS = 1
RHS = 3 × –
1
3





 + 2 = 1
Since LHS = RHS, then E –
1
3
, 1





 lies on
the line.
63

1
15. For the line y = –
1
2
x – 2,
(a) When x = 2, y = −1, then
LHS = −1
RHS = –
1
2
× 2 – 2 = –3
Since LHS ≠ RHS, then A(2, −1) does not lie
on the line.
(b) When x = −4, y = 0, then
LHS = 0
RHS = –
1
2
× (–4) – 2 = 0
Since LHS = RHS, then B(−4, 0) lies on the line.
(c) When x =
2
3
and y = –
7
3
, then
LHS = –
7
3
RHS = –
1
2
×
2
3
– 2 = –
7
3
Since LHS = RHS, then C
2
3
, –
7
3





 lies on
the line.
(d) When x = –
1
2
, y = –
7
4
, then
LHS = –
7
4
RHS = –
1
2
× –
1
2





 – 2 = –
7
4
Since LHS = RHS, then D –
1
2
, –
7
4





 lies
on the line.
(e) When x = 10, y = −3, then
LHS = −3
RHS = –
1
2
× 10 – 2 = –7
Since LHS ≠ RHS, then E(10, −3) does not lie
on the line.
16. (a) When x = −3,
y =
× +
3 (–3) 7
2
=
–2
2
= –1
When x = 0,
y =
× +
3 0 7
2
=
7
2
When x = 3,
y =
× +
3 3 7
2
=
16
2
= 8
x –3 0 3
y –1
7
2
8
–3 –2 –1
–1
1
2
3
4
5
6
7
8
0 1 2 3
y
x
y =
3x + 7
2
(b) When x = −3,
y =
× +
–2 (–3) 1
2
=
7
2
When x = 0,
y =
× +
–2 0 1
2
=
1
2
When x = 3,
y =
× +
–2 3 1
2
= −
5
2
x –3 0 3
y
7
2
1
2
–
5
2
–3 –2 –1
–1
–2
2
1
3
4
0 1 2 3
y
x
y =
–2x + 1
2
64

1
(c) When x = −3,
2y + 3 × (−3) = 4
2y – 9 = 4
2y = 4 + 9
2y = 13
y =
13
2
When x = 0,
2y + 3 × 0 = 4
2y = 4
y = 2
When x = 3,
2y + 3 × 3 = 4
2y + 9 = 4
2y = 4 – 9
2y = −5
y = −
5
2
x –3 0 3
y
13
2
2 –
5
2
–3 –2 –1
–1
–2
1
2
3
4
5
6
0 1 2 3
y
x
2y + 3x = 4
(d) When x = −3,
5y – 2 × (−3) = 8
5y + 6 = 8
5y = 8 – 6
5y = 2
y =
2
5
When x = 0,
5y – 2 × 0 = 8
5y = 8
y =
8
5
When x = 3,
5y – 2 × 3 = 8
5y – 6 = 8
5y = 8 + 6
5y = 14
y =
14
5
x –3 0 3
y
2
5
8
5
14
5
–3 –2 –1
2
1
3
0 1 2 3
y
x
5y + 2x = 8
17. (a)
x –3 0 1
y = 2x + 5 y = 2 × (–3) + 5
= –1
y = 2 × 0 + 5
= 5
y = 2 × 1 + 5
= 7
(b)
1
0
1
7
6
8
2
3
4
5
y
–2
–1
–1
–2
–3
x
y = 2x + 5
(c) From the graph, the value of x is about −5
3
4
.
(Note: It is necessary to extrapolate the graph so
that we can find the value of x when y is less than
−6.)
65

1
18. (a)
x –4 0 4
y = 3x – 2 y = 3 × (–4) – 2
= –14
y = 3 × 0 – 2
= –2
y = 3 × 4 – 2 = 10
y = 5x – 2 y = 5 × (–4) – 2
= –22
y = 5 × 0 – 2
= –2
y = 5 × 4 – 2 = 18
y = – 1
2
x – 2
y = – 1
2
× (–4) – 2
= 0
y = – 1
2
× 0 – 2
= –2
y = – 1
2
× 4 – 2
= –4
y = –2 y = –2 y = –2 y = –2
–2
–3
–4 1
–1 4
2 3
0
8
10
16
12
20
18
14
2
4
6
y
–10
–14
–12
–16
–18
–8
–6
–4
–2
x
y = –
1
2
x – 2
y = 5x – 2
y = 3x – 2
y = –2
(b) All the lines pass through the point (0, −2).
19. (a)
x –6 0 6
x = 1 N.A. N.A. N.A.
x + 5 = 0 N.A. N.A. N.A.
y = – 1
2
x – 2
y = – 1
2
× (–6) – 2
= 1
y = – 1
2
× 0 – 2
= –2
y = – 1
2
× 6 – 2
= –5
y = 3x – 7
y = 3 × (–6) – 7
= –25
y = 3 × 0 – 7
= –7
y = 3 × 6 – 7
= 11
–2
–3
–5
–6 –4 1
–1 4
2 3 6
5
0
8
10
12
2
4
6
y
–10
–18
–14
–22
–12
–20
–16
–24
–26
–8
–6
–4
–2
x
x = 1
x + 5 = 0
y = 3x – 7
y = –
1
2 x – 2
(b) The shape of the figure formed by the lines is a
trapezium.
(c) In order to find the area bounded by the lines, locate
the coordinates of the points of intersection of the
lines.
From the graph, the coordinates of points of
intersections of the lines are
(1, −2.5), (1, −4), (−5, 0.5) and (−5, −22)
Area bounded by the lines
=
1
2
× (1.5 + 22.5) × 6
= 72 square units
66

1
20. (a) (i) Amount of money the house owner has to pay
= 35 + 1 × 15 = $50
(ii) Amount of money the house owner has to pay
= 35 + 2 × 15 = $65
(iii)Amount of money the house owner has to pay
= 35 + 3 × 15 = $80
(iv) Amount of money the house owner has to pay
= 35 + 4 × 15 = $95
(b) x 0 1 2 3 4
y 35 50 65 80 95
(c)
1
0
10
50
90
30
70
20
60
100
40
80
2 4
3
y
x
y = 15x + 35
(d) (i) From the graph, the amount of money charged
if he spends 2.5 hours on the job
= $72.50
(ii) From the graph, the number of hours that the
electrician spends on the job
≈ 3.65 hours
21. (a) From the graph, the values of C can be obtained.
N 50 100 150 200
C 150 200 250 300
(b) (i) From the graph, the value of m is 100.
(ii) He has to pay $m for the operation cost of
printing the newsletters.
(c) From the graph, the amount of money Raj has to
pay is $175.
(d) From the graph, the maximum number of
newsletters he can print is 155.
67

1
Chapter 7 Number Patterns
Basic
1. (a) Rule: Add 5 to each term to get the next term. The
next two terms are 26 and 31.
(b) Rule: Subtract 3 from each term to get the next
term. The next two terms are 19 and 16.
(c) Rule: Multiply each term by 10 to get the next term.
The next two terms are 10 000 and 100 000.
(d) Rule: Multiply each term by 5 to get the next term.
The next two terms are 250 and 1250.
(e) Rule: Multiply the previous term by the term
number to get the next term. The next two terms
are 24 × 5 = 120 and 120 × 6 = 720.
(f) Rule: Take the cube of each term number to get the
next term. The next two terms are 53
= 125 and
63
= 216.
(g) Rule: Subtract 5 from each term to get the next
term. The next two terms are 32 and 27.
(h) Rule: Denote 64 = 82
as the first term. Subtract
1 from the base of each term and square it to get
the next term. The next two terms are 42
= 16 and
32
= 9.
(i) Rule: Add the previous term by its term number
to get the next term. The next two terms are
12 + 5 = 17 and 17 + 6 = 23.
(j) Rule: Add the square of the term number to each
term to get the next term. The next two terms are
34 + 52
= 59 and 59 + 62
= 95.
(k) Rule: Add the term number to the previous term
to get the next term. The next two terms are
30 + 5 = 35 and 35 + 6 = 41.
(l) Denote 7 as the zero term.
Rule: Add each term by 2 to the power of its term
number to get the next term. The next two terms
are 22 + 24
= 38 and 38 + 25
= 70.
(m) Denote 90 as the first term.
Rule 1: Subtract 10 from each odd term to get the
next odd term.
Rule 2: Add 10 to each even term to get the next
even term. The next two terms are 60 and 40.
(n) Rule: Denote 1024 = 210
as the first term. Subtract
1 from the power of each term to get the next term.
The next two terms are 25
= 32 and 24
= 16.
2. (a) Since the common difference is 5, Tn = 5n + ?.
The term before T1 is c = T0 = 12 – 5 = 7.
General term of the sequence, Tn = 5n + 7
(b) Since the common difference is –6, Tn = –6n + ?.
The term before T1 is c = T0 = 83 + 6 = 89.
General term of the sequence, Tn = –6n + 89.
(c) Since the common difference is 7, Tn = 7n + ?.
The term before T1 is c = T0 = 2 – 7 = –5.
General term of the sequence, Tn = 7n – 5.
(d) Since the common difference is 6, Tn = 6n + ?.
General term of the sequence, Tn = 6n + 1.
(e) Since the common difference is –4, Tn = –4n + ?.
General term of the sequence, Tn = –4n + 43.
(f) To find the formula, consider the following:
1, 2, 4, 8, 16, …
as 20
, 21
, 22
, 23
, 24
, …
General term of the sequence, Tn = 2n – 1
,
n = 1, 2, 3, …
(g) To find the formula, consider the following:
2, 6, 18, 54, 162, …
as 2 × 30
, 2 × 31
, 2 × 32
, 2 × 33
, 2 × 34
, …
General term of the sequence, Tn = 2 × 3n – 1
,
n = 1, 2, 3, …
(h) To find the formula, consider the following:
12, 36, 108, 324, 972, …
as 4 × 3, 4 × 32
, 4 × 33
, 4 × 34
, 4 × 35
, …
General term of the sequence, Tn = 4 × 3n
,
n = 1, 2, 3, …
(i) To find the formula, consider the following:
2000, 1000, 500, 250, 125, …
as
4000
2
,
4000
22 ,
4000
23 ,
4000
24 ,
4000
25 , …
General term of the sequence, Tn =
4000
2n ,
n = 1, 2, 3, …
3. (i) The next three terms of the sequence are 48, 96
and 192.
(ii) The next three terms of the sequence are 52, 100,
196.
Add 4 to the sequence in part (i).
4. (i) The next two terms of the sequence are 96 and 192.
(ii) To find the formula, consider the following:
3, 3 × 2, 6 × 2, 12 × 2, 24 × 2, …
3 × 20
, 3 × 2, 3 × 22
, 3 × 23
, 3 × 24
, …
General term of the sequence, Tn = 3 × 2n – 1
(iii)Let 3 × 2m – 1
= 1536
2m – 1
=
1536
3
= 512
By trial and error, 29
= 512
m – 1 = 9
m = 9 + 1 = 10
68

1
5. (i) The next two terms of the sequence are
1
54
=
1
625
and
1
65
=
1
7776
.
1
10
,
1
21
,
1
32
,
1
43
, …
General term of the sequence, Tn = −
n
1
n 1 ,
n = 1, 2, 3, …
(iii)When n = 10, T10 = −
1
1010 1 =
1
109
.
6. (a) When n = 1, 3(1) + 1 = 4
When n = 2, 3(2) + 1 = 7
When n = 3, 3(3) + 1 = 10
The first three terms are 4, 7 and 10.
(b) When n = 1, 2(1) – 7 = –5
When n = 2, 2(2) – 7 = −3
When n = 3, 2(3) – 7 = – 1
The first three terms are –5, −3 and −1.
(c) When n = 1, (1)2
– 1 = 0
When n = 2, (2)2
– 2 = 2
When n = 3, (3)2
– 3 = 6
(d) When n = 1, 2(1)2
– 3(1) + 5 = 4
When n = 2, 2(2)2
– 3(2) + 5 = 7
When n = 3, 2(3)2
– 3(3) + 5 = 14
(e) When n = 1,
−
(1)(1 1)
2
=
(1)(0)
2
= 0
When n = 2,
−
(2)(2 1)
2
=
(2)(1)
2
= 1
When n = 3,
−
(3)(3 1)
2
=
(3)(2)
2
= 3
(f) When n = 1,
+
2
1 1
= 1
When n = 2,
+
2
2 1
=
2
3
When n = 3,
+
2
3 1
=
2
4
=
1
2
The first three terms are 1,
2
3
and
1
2
.
7. (i) E
D E
D E
D E
D E
D E
D E
D E
E
(ii) Letter Number of Letters
A 2(1) – 1 = 1
B 2(2) – 1 = 3
C 2(3) – 1 = 5
D 2(4) – 1 = 7
E 2(5) – 1 = 9
 
nth
letter Tn
(iii)For the letter J, 2(10) – 1 = 19.
(iv) Since the common difference is 2, Tn = 2n + ?.
The term before T1 is c = T0 = 1 – 2 = – 1.
2n – 1 = 29
2n = 29 + 1
2n = 30
n = 15
When n = 15, it is the letter O.
Intermediate
8. (a) 18, 24
(b) 9, 16
(c) 250, 50
(d) 16, 23
(e) 3, 5
(f)
16
17
,
22
23
(g)
17
1
,
1
23
9. (a) The next three terms are 39, 51 and 65.
(b) The prime numbers are 11 and 29.
(c) For the two numbers to have HCF as 13, the two
numbers must have a common factor 13 and the
other factor less than 13. The other factor must be
different for the two numbers.
The possible numbers are 13, 26, 39, 52, 65, …
The two numbers whose HCF is 13 from this
sequence are 39 and 65.
(d) By prime factorisation, 195 = 3 × 5 × 13.
Thus the 3 numbers whose LCM is 195 may be
3 × 5, 3 × 13 and 5 × 13.
The three numbers whose LCM is 195 from this
sequence are 15, 39 and 65.
69

1
10. For the sequence 2, 5, 8, 11, … the next few terms are
14, 17, 20, 23, 26, 29, 32, 35, 38, …
For the sequence 3, 8, 13, 18, … the next few terms
are 23, 28, 33, 38, 43, …
By listing, the next two numbers which will occur in
both sequences are 23 and 38.
11. (i) The next two terms of the sequence are 642 and
621.
(ii) Since the common difference is –21,
Tn = –21n + ?.
General term of the sequence, Tn = 768 – 21n.
(iii) 768 – 21r = 390
21r = 768 – 390
= 378
r = 18
12. (i) a = 26 + 9 = 37, b = 37 + 13 = 50 and
c = 50 + 3 = 53
2, 5, 10, 17, 26, …
1+1, 4 + 1, 9 + 1, 16 + 1, 25 + 1, …
12
+ 1, 22
+ 1, 32
+ 1, 42
+ 1, 52
+ 1, …
General term of the sequence, Tn = n2
+ 1.
(iii)Add 3 to the odd number terms of sequence A to
get the corresponding odd number term in sequence
B.
Subtract 1 from the even number terms of sequence
A to obtain the corresponding even number terms
of sequence B.
13. (a) When n = 1, 2(1)2
– 3(1) + 5 = 4
When n = 2, 2(2)2
– 3(2) + 5 = 7
When n = 3, 2(3)2
– 3(3) + 5 = 14
When n = 4, 2(4)2
– 3(4) + 5 = 25
The first four terms of the sequence are 4, 7, 14
and 25.
(b) (i) Comparing the two sequences, the common
difference between two sequences is −3.
Since the formula for the sequence in part (a) is
2n2
– 3n + 5, then the formula for the sequence
is 2n2
– 3n + 5 – 3 = 2n2
– 3n + 2.
(ii) When n = 385,
2(385)2
– 3(385) + 2
= 295 297.
14. (i) 5th
line: n = 5, 6 × 5 – 10 = 20
(ii) Note that the product is the value of n and the value
of 1 more than n.
a = 29
The value of b is an even number and it is the
product of n and 2.
b = 28 × 2 = 56
The value of c is 29 × 28 – 56 = 756.
(iii)When n = 50,
51 × 50 – 50 × 2 = 2450
15. (i) 6th
line:
×
1
6 7
=
1
6
–
1
7
7th
line:
×
1
7 8
=
1
7
–
1
8
(ii) 272 = p × q
Notice that q is 1 more than p.
By trial and error, 16 × 17 = 272
p = 16 and q = 17
(iii)
1
100
–
1
101
=
×
1
100 101
=
1
10 100
16. (i) 1 + 2 + 3 + 4 + … + 99 + 100 + 99 + … + 3 + 2 + 1
= (100)2
= 10 000
(ii) 1 + 2 + 3 + … + (n – 1) + n + (n – 1) + … + 3 + 2
+ 1 = 7056
n2
= 7056
n = 84
17. (i) 7th
line: 73
– 7 = 336 = (7 – 1) × 7 × (7 + 1)
(ii) 1320 is divisible by 10. Thus the factors of 1320
are 10, 11 and 12.
1320 = (11 – 1) × 11 × (11 + 1)
n = 11
(iii)193
– 19 = (19 – 1) × 19 × (19 + 1) = 6840
18. (a) (i) The next four terms are 15 + 6 = 21,
21 + 7 = 28, 28 + 8 = 36 and 36 + 9 = 45.
(ii) The next four terms are 35 + 21 = 56,
56 + 28 = 84, 84 + 36 = 120, and
120 + 45 = 165.
(b) (i) 9th
line: 93
– 9 = 720 = 6 × 120
10th
line: 103
– 10 = 990 = 6 × 165
(ii) k = 6
p = 6 × 84 = 504
Notice that the number of terms follow the
number of terms for the sequence 0, 1, 4, 10,
…, 84. Since the 8th
term is 84, then
83
– 8 = 504 = 6 × 84.
m = 8
70

1
19. (a) 3rd
line: (1 + 2 + 3)2
= 36 = (1)3
+ (2)3
+ (3)3
4th
line: (1 + 2 + 3 + 4)2
= 100 = (1)3
+ (2)3
+ (3)3
+ (4)3
(b) (i) When l = 7, (1)3
+ (2)3
+ (3)3
+ (4)3
+ (5)3
+
(6)3
+ (7)3
= (1 + 2 + 3 + 4 + 5 + 6 + 7)2
= (28)2
= 784
(ii) When 1 = 19,
(1)3
+ (2)3
+ (3)3
+ (4)3
+ (5)3
+ (6)3
+ …
+ (19)3
= (1 + 2 + 3 + 4 + 5 + 6 + … + 19)2
= (190)2
= 36 100
(c) (1 + 2 + 3 + … + n)2
= 2025 = (45)2
We observe that
45 = 40 + 5 = 4 × 10 + 5
(1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5
= 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
n = 9
(d) (1)3
+ (2)3
+ (3)3
+ … + (m)3
= 782
(1 + 2 + 3 + 4 + 5 + … + m)3
= 782
1 + 2 + 3 + 4 + 5 + … + m = 78
Consider
78
= 6 × 13
= (1 + 12) + (2 + 11) + (3 + 10) + (4 + 9)
+ (5 + 8) + (6 + 7)
= 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
m = 12
20. 12
+ 12
+ 22
+ 32
= 3 × 5
12
+ 12
+ 22
+ 32
+ 52
= 5 × 8
Notice that numbers along this column follow the
sequence 1, 1, 2, 3, 5, 8, 13, …
(i) 7th
line: 12
+ 12
+ 22
+ 32
+ 52
+ 82
+ 132
= 13 × 21
(ii) 12
+ 12
+ 22
+ 32
+ 52
+ … + l2
+ m2
= 55 × n
Since the left-hand side follows the given sequence,
then l = 34 and m = 55.
The right-hand side of the equation follows the
given sequence too.
n = 89
21. (a)
Figure 4
(b)
Figure
Number (n)
1 2 3 4 5 ... n
Number of
Buttons
5 8 11 14 17 ... ...
(c) Since the common difference is 3, Tn = 3n + ?.
(d) (i) When n = 38, T38 = 3(38) + 2 = 116
The number of buttons needed to form
Figure 38 is 116.
(ii) 3n + 2 = 470
3n = 470 – 2
3n = 468
n = 156
Figure 156 is made up of 470 buttons.
(e) 3n + 2 = 594
3n = 594 – 2
3n = 592
n = 197
1
3
Since 197
1
3 is not a positive integer, it is not
possible for a figure in the sequence to be made up
of 594 buttons.
22. (a)
Figure
Number
Number of
Dots
Number of Small
Right-Angled Triangles
1 4 2
2 9 8
3 16 18
4 25 32
: : :
10 121 200
: : :
19 400 722
: : :
n x y
(b) (i) x = (n + 1)2
(ii) y = 2n2
71

1
23. (a)
110 209 308 407 506 605 704 803 902
121 220 319 418 517 616 715 814 913
132 231 330 429 528 627 726 825 924
143 242 341 440 539 638 737 836 935
154 253 352 451 550 649 748 847 946
165 264 363 462 561 660 759 858 957
176 275 374 473 572 671 770 869 968
187 286 385 484 583 682 781 880 979
198 297 396 495 594 693 792 891 990
(b) These are some of the possible patterns.
For each column, from top cell to bottom cell, add
11 to each term to get the next term.
For each row, from left cell to right cell, add 99 to
each term to get the next term.
For each diagonal, from left to right, add 10 to each
term to get the next term.
(c) From the table,
550 ÷ 11 = 50 = 52
+ 52
+ 02
803 ÷ 11 = 73 = 82
+ 02
+ 32
The two multiples are 550 and 803.
24. (a)
Figure 4
(b) Figure 1 8
Figure 2 14
Figure 3 20
Figure 4 26
Since the common difference is 6, Tn = 6n + ?.
General term of the sequence, Tn = 6n + 8,
n = 0, 1, 2, …
p = 8 + 6n, n = 0, 1, 2, ...
(c) When n = 45, T45 = 6(45) + 8 = 278
278 people can be seated when 45 tables are
placed together.
(d) 6n + 8 = 245
6n = 245 – 8 = 237
n = 39
1
2
Since n = 39
1
2 is not a positive integer, Kate is
not able to follow the arrangement in part (b) with
all the seats fully occupied.
25. (a)
Design 4
(b)
Figure
Number n
Number
of Circles
Number
of
Squares
Number
of Straight
Cines
1 1 4 4
2 2 6 7
3 3 8 10
4 4 10 13
5 5 12 16
: : : :
10 10 22 31
: : : :
23 23 48 70
(c) (i) n
(ii) Since the common difference is 2,
Tn = 2n + ?.
General term of the sequence of the number
of squares, Tn = 2n + 2 = 2(n + 1).
(iii) Since the common difference is 3,
Tn = 3n + ?.
General term of the sequence of the number
of straight lines, Tn = 3n + 1.
(d) (i) When n = 105, 3(105) + 1 = 316
316 straight lines are needed to form
Figure 105.
(ii) 2n + 2 = 30
2n = 30 – 2
2n = 28
n = 14
Figure 14 is formed using 30 squares.
(e) 2n + 2 = 50
2n = 50 – 2 = 48
Since 48 is divisible by 2, there is a figure formed
using 50 squares.
3n + 1 = 75
3n = 75 – 1 = 74
Since 74 is not divisible by 3, there is no figure
formed using 75 straight lines.
72

1
26. (i)
Figure 5
(ii) When n = 5,
Height of figure = 5
Number of squares = 5 + 4 + 3 + 2 + 1
=
+
5(5 1)
2
= 15
When n = 6,
Height of figure = 6
Number of squares = 6 + 5 + 4 + 3 + 2 + 1
=
+
6(6 1)
2
= 21
When n = n,
Number of squares = n + (n – 1) + (n – 2)
+ … + 3 + 2 + 1
=
+
n n
( 1)
2
Advanced
27. (a) Observe that the pattern is
676 = 26, 625 = 25, 576 = 24,
529 = 23, 484 = 22, 441 = 21
The missing terms are 529 and 484 .
(b) Observe that the pattern is
3375
3
= 15, …, 729
3
= 9, 343
3
= 7,
125
3
= 5
This follows that the cube root of a number gives
cold numbers.
The missing terms are 133
3
= 2197
3
and
113
3
= 1331
3
.
(c) Observe that the pattern is alternate cube and
square of the numbers in the sequence.
Add the term number to the previous term.
The missing terms are (23 + 7)3
= 303
and
(30 + 8)2
= 382
.
(d) Observe the pattern as taking the cube of prime
numbers.
, 73
and 173
.
(e) Observe that the pattern is taking the square of the
prime numbers.
, 172
and 112
.
28. (a) Since the common difference is 11, Tn = 11n + ?.
11n – 6 = 100
11n = 106
n ≈ 9.6
Thus the largest two-digit number occurs when n
= 9.
When n = 9, 11(9) – 6 = 93.
(b) To find the formula of the general term, consider
the following:
8 , 27, 64, 125, …
23
, 33
, 43
, 53
, …
General term of the sequence = n3
,
n = 2, 3, 4, 5, ….
n3
= 1000
n = 10
Thus the first four-digit number occurs when
n = 10. When n = 9, the largest three-digit number
occurs.
When n = 9, 93
= 729.
29. (a) To find the formula of the general term, consider
the following:
4, 9, 16, 25, …
22
, 32
, 42
, 52
, …
General term of the sequence = n2
,
n = 2, 3, 4, 5, ….
n2
= 100
n = 10
The smallest three-digit number is 100.
(b) To find the formula of the general term, consider
the following
2 = 2 × 30
6 = 2 × 31
18 = 2 × 32
54 = 2 × 33
General term = 2(3)n – 1
Let 2(3)n – 1
= 1000
3n – 1
= 500
By trial and error, 35
= 243 and 36
= 729 and so
n – 1 = 6
So the smallest four-digit number is
2(3)6
= 1458.
73

1
30. (a) 5 points
Set 5
(b) When n = 4,
number of lines formed = 6
When n = 5,
When n = 10,
When n = 23,
(c) (i) By observation, the number of points is the
same as the set number, n.
Thus the number of points needed to form Set
n = n.
(ii) To find the formula of the number of lines,
consider the following:
When n = 1, 0 =
−
1(1 1)
2
When n = 2, 1 =
−
2(2 1)
2
When n = 3, 3 =
−
3(3 1)
2
When n = 4, 6 =
−
4(4 1)
2
When n = 5, 10 =
−
5(5 1)
2
General term of the number of lines
=
−
n n
( 1)
2
(d) (i) When n = 16, number of lines =
−
16(16 1)
2
= 120
(ii) When n = 35, number of lines =
−
35(35 1)
2
= 595
(e) (i) Let
−
n n
( 1)
2
= 190
By trial and error, n must be 17, 18, 19, 20, …
n = 20
(ii) Let
−
n n
( 1)
2
= 1000
When n = 45, number of lines =
−
45(45 1)
2
= 990
When n = 46, number of lines =
−
46(46 1)
2
= 1035
There is no set of points having 1000 lines
formed.
31. (a) Term is obtained by adding the two terms
immediately above.
(b) (i) The next two rows are the 6th
and 7th
rows.
6th
row: 1 5 10 10 5 1
7th
row: 1 6 15 20 15 6 1
(ii) Sum of the terms in row 1 = 1
Sum of the terms in row 2 = 1 + 1 = 2
Sum of the terms in row 3 = 1 + 2 + 1 = 4
Sum of the terms in row 4 = 1 + 3 + 3 + 1 = 8
Sum of the terms in row 5
= 1 + 4 + 6 + 4 + 1 = 16
= 1 + 5 + 10 + 10 + 5 + 1 = 32
= 1 + 6 + 15 + 20 + 15 + 6 + 1
= 64
Yes, these sums form a pattern equal to 2n – 1
,
where n is the nth
row.
(c) (i) Sum of terms in 11th
row = 210
= 1024
(ii) Sum of terms in kth
row = 2k – 1
(d) (i) Number of terms in kth
row = k
(ii) The first two terms in the kth
row are 1 and
k – 1.
New Trend
32. (a) (i) To find the formula of the general term,
consider the following:
81 = 88 − 7(1)
74 = 88 − 7(2)
67 = 88 − 7(3)
60 = 88 − 7(4)
General term, Tn = 88 − 7n
(ii) T99 = 88 − 7(9)
= −605
(b) (i) n + 13
(ii) (n + 1)(n + 12) − n(n + 13)
= n2
+ 13n + 12 − n2
− 13n
= 12 (shown)
(iii)Sum of the numbers in the square
= n + (n + 1) + (n + 12) + (n + 13)
= 4n + 26
When 4n + 26 = 520
4n = 494
n = 123.5
Since 123.5 is not an integer, the sum of the
four numbers in the square cannot be 520.
74

1
33. (a) Common difference =
77 – 35
3
= 14
p = 35 − 14
= 21
q = 35 + 14
= 49
r = 77 −14
= 63
(b) Since the common difference is 14, Tn = 14n + ?.
The term before T1 is c = T0 = 21 − 14 = 7.
(c) When 14n + 7 = 170
14n = 163
n = 11
9
14
Since 11
9
14
is not an integer, 170 is not in the
sequence.
34. (a) (i) Next line is the 6th
line: 62
– 6 = 30.
(ii) 8th
line: 82
– 8 = 56
(iii)From the number pattern, we observe that
12
– 1 = 1(1 – 1)
22
– 2 = 2(2 – 1)
32
– 3 = 3(3 – 1)
42
– 4 = 4(4 – 1)
52
– 5 = 5(5 – 1)
:
nth
line: n2
– n = n(n – 1)
(b) 1392
– 139 = 139(139 – 1) = 19 182
35. (i) The next two terms of the sequence are 39 and 46.
(ii) Since the common difference is 7, Tn = 7n + ?.
(iii)When n = 101, T101 = 7(101) – 3 = 704.
(iv) 7n – 3 = 158.
7n = 158 + 3
= 161
n = 23
75

1
Revision Test B1
1. (a) 19 – 6x = 21 – 9x
–6x + 9x = 21 – 19
9x – 6x = 2
3x = 2
x =
2
3
(b) 2
1
3
x = 14
7
3
x = 14
3 ×
7
3
x = 3 × 14
7x = 42
x = 6
(c) 13 – 3x – 10 = 3x + 8 – 8x
3x – 3x = 8 – 5x
–3x + 5x = 8 – 3
2x = 5
x =
5
2
= 2
1
2
(d)
x – 3
4
–
x
2 – 5
2
=
1
4
x – 3
4
–
x
2(2 – 5)
4
=
1
4
x x
– 3 – 2(2 – 5)
4
=
1
4
x – 3 – 2(2x – 5) = 1
x – 3 – 4x + 10 = 1
7 – 3x = 1
6 = 3x
3x = 6
x = 2
2. (a) 3x < 20
x <
20
3
x < 6
2
3
5 6 7
6
2
3
(b) 2x ù −51
x ù −
51
2
x ù −25
1
2
–26 –25 –24
−25
1
2
3. (a) 3x < 43
x < 14
1
3
∴ The largest possible integer value of x is 14.
(b) 13x > 49
x > 3
10
13
∴ The smallest prime value of x is 5.
4. (a) (i) When a = 4,
4x = 15 – 3x
4x + 3x = 15
7x = 15
x =
15
7
= 2
1
7
(ii) When x = 4,
4a = 15 – 3(4)
4a = 15 – 12
4a = 3
a =
3
4
(b) 8p – 9q = 7q + 3p
8p – 3p = 7q + 9q
5p = 16q
p
q
=
16
5
2
5
×
p
q
=
2
5
×
16
5
p
q
2
5
= 1
7
25
5. (a) (2a + 15) ÷ 7 = 11
a
2 15
7
+
= 11
2a + 15 = 77
2a = 77 – 15
2a = 62
a = 31
(b) Let the first even number be n.
Then the second even number is (n + 2).
n + 2 + 4n = 72
5n = 72 – 2
5n = 70
n = 14
∴The two numbers are 14 and 14 + 2 = 16.
(c) Let Ethan’s age be x years.
76

1
Then Mr Lin’s age is (38 – x) years.
In three years’ time,
Ethan is (x + 3) years old.
Then Mr Lin is (38 – x + 3) = (41 – x) years old.
41 – x = 3(x + 3)
41 – x = 3x + 9
3x + x = 41 – 9
4x = 32
x= 8
∴Mr Lin is 30 years old and his son is
8 years old.
6.
–2
–3
–4 1
–1 4
2 3 6
5
0
1
7
8
9
10
11
12
13
14
2
3
4
5
6
y
–8
–7
–6
–5
–4
–3
–2
–1
x
y =
1
2 x – 2
y = –2x + 5
7. (i) The next two terms of the sequence are
8 + 8 = 16 and 16 + 16 = 32.
(ii) The next two terms of the sequence are
552 + 552 = 1104 and 1104 + 1104 = 2208.
77

1
Revision Test B2
1. (a)
1
4
(x – 2) – (x + 3) = 7(x – 1)
(x – 2) – 4(x + 3) = 28(x – 1)
x – 2 – 4x – 12 = 28x – 28
28x + 4x – x = – 2 – 12 + 28
31x = 14
x =
14
31
(b)
x
2( – 3)
3
–
x
5(3 – 1)
6
=
1
12
x
4( – 3)
6
–
x
5(3 – 1)
6
=
1
12
x x
4( – 3) – 5(3 – 1)
6
=
1
12
x x
4 – 12 – 15 5
6
+
=
1
12
12(–11x – 7) = 6
2(−11x – 7) = 1
−22x – 14 = 1
22x = –14 – 1
22x = −15
x = –
15
22
(c)
x
2
– 2
=
+
x
5
5
2(x + 5) = 5(x – 2)
2x + 10 = 5x – 10
5x – 2x = 10 + 10
3x = 20
x =
20
3
= 6
2
3
(d)
x
5
11 – 3
=
x
2
3 – 1
5(3x – 1) = 2(11 – 3x)
15x – 5 = 22 – 6x
15x + 6x = 22 + 5
21x = 27
x =
27
21
= 1
2
7
2. (a) 2x ø 18
x ø 9
(b)
x
2
5
ù –3
5 ×
x
2
5
ù –3 × 5
2x ù −15
x ù –
15
2
x ù –7
1
2
3.
f g
f g
3 –
2
+
=
4
5
5(3f – g) = 4( f + 2g)
15f – 5g = 4f + 8g
15f – 4f = 8g + 5g
11f = 13g
f
g
=
13
11
1
39
×
f
g
=
1
39
3
×
13
11
1
f
g
39
=
1
33
4. (i) V =
1
3
πr2
h
When r = 2, h = 5, π = 3.14,
V =
1
3
(3.14)(2)2
(5)
= 20.9 (to 3 s.f.)
(ii) V =
1
3
πr2
h
When V = 75, r = 3, π = 3.14,
75 =
1
3
(3.14)(3)2
h
471
50
h = 75
h = 75 × 50
471
= 7.96 (to 3 s.f.)
5. 6x ù 70
x ù
70
6
x ù 11
2
3
If x is a prime number, the smallest value of x is 13.
78

1
6. (a) Perimeter of square = 52 cm
4(2x + 5) = 52
8x + 20 = 52
8x = 52 – 20
8x = 32
x = 4
The length of the square is (2 × 4 + 5) = 13 cm.
Area of the square
= 132
= 169 cm2
(b) Let the number of type A eggs be x.
Then the number of type B eggs is 60 – x.
x(0.11) + (60 – x)(0.13) = 7
0.11x + 7.8 – 0.13x = 7
7.8 – 7 = 0.13x – 0.11x
0.02x = 0.8
x = 40
∴She bought 40 type A eggs and 20 type B eggs.
(c) Let the length of the field be x m.
Perimeter of fence = 2(x + 35) m
160 = 2(x + 35)
80 = x + 35
x = 80 – 35 = 45
∴The length of the field is 45 m.
(d) Let the number of paperback books be p.
Then the number of hardcover books is (50 – p).
4p + 1
1
2
(4)(50 – p) = 256
4p + 6(50 – p) = 256
4p + 300 – 6p = 256
6p – 4p = 300 – 256
2p = 44
p = 22
Number of hardcover books
= 50 – 22
= 28
7.
2
1
0
1
2
3
4
5
6
8
7
9
10
4 5 8
6
3 9
7
y
x
B(5, 10)
C(9, 6)
A(1, 2)
8. (a)
1
7
2
3
4
5
6
y
–7
–6
–5
–4
–3
–2
–1
–3
–4
–5 –2 3
1 2 5
4
0
x
y = 2x + 3
y = x – 1
y = 3 (4, 3)
(0, 3)
–1
(–4, –5)
(b) Area of triangle
=
1
2
× 4 × (3 + 5)
= 16 square units
9. 3, 7, 13, 21, …
3, 3 + 4, 3 + 4 + 6, 3 + 4 + 6 + 8, …
The next two terms of the sequence are
21 + 10 = 31 and 31 + 12 = 43
79

1
Mid-Year Examination Specimen Paper A
Part I
1. (a) False; 2.3 is a rational number.
(b) False; 189 ÷ 3 = 63 shows that 189 does not
satisfy the definition of prime numbers.
(c) True
2. (a) (i) 27.049 = 27 (to 2 s.f.)
(ii) 27.049 = 27.0 (to 1 d.p.)
(b) Using a calculator,
10
13
= 0.769 230 769
=
0.769 (to 3 d.p.)
3. (a) {[(–4) – (−2) × 7] + 9 × 5} ÷ 11
= {[(–4) – (−14)] + 9 × 5} ÷ 11
= {[(–4) + 14] + 9 × 5} ÷ 11
= {10 + 9 × 5} ÷ 11
= {10 + 45} ÷ 11
= 55 ÷ 11
= 5
(b)
3
1
2
– 2
1
3






2
3
1
2
+ 2
1
3
=
1
1
6






2
5
5
6
=
7
30
4. (a)
5
6
,
11
13
and
8
9
(b)
–2
–4
–4 –1.5 4.6
–1
–3 4
1 3
3
0
0
5
2
The ordering is 0, 3, –4, −1.5, 4.6 is –4, −1.5,
0, 3 and 4.6.
5. When a = 2, b = 0, c = 1 and d = −3,
(a)
+
abc bcd
acd abd
–
=
+
(2)(0)(1) – (0)(1)(–3)
(2)(1)(–3) (2)(0)(–3)
=
0
–6
= 0
(b)
ab
c
–
bc
d
+
ad
c
2
=
(2)(0)
(1)
–
(0)(1)
(–3)
+
(2)(–3)
2(1)
= −3
6. (a) (i) 3(x – 2y) – 7[x – 3(2y – 7x)]
= 3(x – 2y) – 7[x – 6y + 21x]
= 3(x – 2y) – 7[x + 21x – 6y]
= 3x – 6y – 7[22x – 6y]
= 3x – 6y – 154x + 42y
= 3x – 154x – 6y + 42y
= −151x + 36y
(ii)
2
3
(x – 4) –
2
5
(x + 3)
=
2
3
x –
8
3
–
2
5
x –
6
5
=
2
3
x –
2
5
x –
8
3
–
6
5
=
4
15
x – 3
13
15
(b) 3x + 18y + 27z
= 3(x + 6y + 9z)
7. (a) 2(x – 3) + 5(2x – 3) = 3
2x – 6 + 10x – 15 = 3
2x + 10x – 6 – 15 = 3
12x – 21 = 3
12x = 3 + 21
12x = 24
x = 2
(b)
3
4
(2x – 1) =
1
2
+
7
8
x
3
2
x –
3
4
=
1
2
+
7
8
x
3
2
x –
7
8
x =
1
2
+
3
4
5
8
x = 1
1
4
x = 1
1
4
÷
5
8
x = 2
8. (a) Cost price of T-shirts
= s × $p
= $ps
Selling price of T-shirts
= s × $q
= $qs
Profit
=
selling price – cost price
= $(qs – ps)
= $s(q – p)
80

1
(b) Let the present age of Kate’s brother be
x years old.
Then Kate is (x + 10) years old.
In 3 years’ time,
(x + 10) + 3 = 2(x + 3)
x + 13 = 2x + 6
2x – x = 13 – 6
x = 7
Kate is 17 years old and her brother is 7 years old.
9. (a) Let the first number be y.
Then the second number is (y + 8).
y + (y + 8) = 230
2y + 8 = 230
2y = 230 – 8
2y = 222
y = 111
The two numbers are 111 and 111 + 8 = 119.
(b) 12 = 22
× 3
28 = 22
× 7
112 = 24
× 7
HCF of 12, 28 and 112 = 22
= 4
LCM of 12, 28 and 112 = 24
× 3 × 7
= 336
Part II
Section A
1. (a) (i) 2
2
9
–
7
15
÷ 4
1
5
+
1
3
= 2
2
9
–
1
9
+
1
3
= 2
1
9
+
1
3
= 2
4
9
(ii) 4
1
2
+ 4
1
2
×
2
3
–
5
9
= 4
1
2
+ 3 –
5
9
= 7
1
2
–
5
9
= 6
17
18
(b) Express all the numbers in decimals.
3
7
= 0.428 571 428, 0.42, 0.428 282 828…,
0.424 242 42…, 0.428 428 428…
Arrange the numbers in ascending order.
0.42, 0.424 242 42…, 0.428 282 828…,
0.428 428 428…,
3
7
= 0.428 571 428
      
0.42, 0.428, 0.428 and
3
7
2. (a) (i) 12.57 + 3.893
= 62.409 288 58
= 62.4 (to 3 s.f.)
(ii) 15.762
–
1
0.026
× 76.8
= 15.762
– 2953.846 154
= −2705.468…
= −2710 (to 3 s.f.)
(b) Fraction remaining after the man saves part of
his salary
= 1 –
1
6
=
5
6
Fraction of the salary spent on rental
=
1
4
×
5
6
=
5
24
Fraction of salary spent on food and other
necessities
= 1 –
1
6
–
5
24
=
5
8
3. (a) 9a – {3a – 2[3a(2a + 1) – 2a(3a – 1)]}
= 9a – {3a – 2[6a2
+ 3a – 6a2
+ 2a]}
= 9a – {3a – 2[6a2
– 6a2
+ 3a + 2a]}
= 9a – {3a – 2[5a]}
= 9a – {3a – 10a}
= 9a – {–7a}
= 9a + 7a
= 16a
(b) 2ax – ay + 6ab – 3a
= a(2x – y + 6b – 3)
81

1
4. (a) (i) 2x + [7 – 3(x + 5)] = 4
2x + [7 – 3x – 15] = 4
2x + [7 – 15 – 3x] = 4
2x + [–8 – 3x] = 4
2x – 8 – 3x = 4
3x – 2x = – 8 – 4
x = −12
(ii)
x
3
7 – 5
=
x
5
3 – 2
3(3 – 2x) = 5(7 – 5x)
9 – 6x = 35 – 25x
25x – 6x = 35 – 9
19x = 26
x =
26
19
= 1
7
19
(b) (i) 6x ù 15
x ù 2
1
2
(ii) 11x < −65
x < −5
10
11
Section B
5. (i) 120k = 23
× 3 × 5 × k
For 120k to be a perfect cube,
then the smallest value of 120k
= (2 × 3 × 5)3
= (23
× 3 × 5) × 32
× 52
= 120 × 32
× 52
k = 32
× 52
= 225
(ii) 120 = 23
× 3 × 5
2800 = 24
× 52
× 7
HCF of 120 and 2800 = 23
× 5
= 40
LCM of 120 and 2800 = 24
× 3 × 52
× 7
= 8400
(iii) k
120
3
= ×
120 225
3
= 30
30 = 2 × 3 × 5
2800 = 24
× 52
× 7
HCF of 30 and 2800 = 2 × 5 = 10
LCM of 30 and 2800 = 24
× 3 × 52
× 7
= 8400
6. (a) –2 –
+
x
3
3
=
+
x
4
3
+ 5
+
x
4
3
+
+
x
3
3
= –2 – 5
+
x
7
3
= –7
–7(x + 3) = 7
x + 3 = –1
x = –4
(b) Let the price of the printer be $y.
y + 5
1
2
y = 2210
6
1
2
y = 2210
y = 340
The printer costs $340 and the desktop computer
costs 5
1
2
× 340 = $1870.
(c) Let the number of students who failed the test
be n.
Then the number of students who passed the test
will be 3n.
3n + n = 44
4n = 44
n = 11
The number of students who passed the test is
3 × 11 = 33.
(d) Let the first number be x.
Then the second number is x + 9.
x + (x + 9) = 63
2x + 9 = 63
2x = 63 – 9
2x = 54
x = 27
82

1
7. (a) When x = 2,
y = –5(2) – 2 = −12.
The value of p is −12.
(b)
–2
–3
–4 1
–1 2 3
0
8
10
14
12
16
18
20
2
4
6
y
–8
–6
–10
–14
–12
–16
–4
–2
x
y = –5x – 2
(c) (i) y ≈ –13.2
(d) (1, –10)
(c) (ii)
x ≈ –1.90
(c) (i) From the graph, y ≈ −13.2.
(ii) From the graph, x ≈ −1.90.
(d) The point (1, −10) is not a solution of the equation
y = –5x – 2 as the point does not lie on the line.
83

1
Mid-Year Examination Specimen Paper B
Part I
1.
–2
1 7
–1 4 6
1 3
6
0
2 5
4
3
5 7
2
2. 25x > 52
x > 2
2
25
Smallest prime value of x is 3.
3. (a) 5 × 12 – 25 ÷ 5 × 3
= 60 – 5 × 3
= 60 – 15
= 45
(b) (–3)2
× (–2)3
÷ (–3) × (–4)
= 9 × (–8) ÷ (–3) × (–4)
= –72 ÷ (–3) × (–4)
= 24 × (–4)
= –96
(c) 15 – 2{[12 – 7 × 10 ÷ 2 + 3]}
= 15 – 2{[12 – 70 ÷ 2 + 3]}
= 15 – 2{[12 – 35 + 3]}
=
15 – 2{−20}
= 15 + 40
= 55
4. (a) Express all the numbers in decimal.
13
11
= 1.181 818 182, 1.188 888, 1.818 181 81,
1.1832,
11
9
= 1.222 22…
Arrange the numbers in descending order.
11
9
= 1.222 22…, 1.818 181 81, 1.188 888, 1.1832,
13
11
= 1.181 818 182

11
9
,   
1.81, 1.18, 1.1832,
13
11
(b) (i) 24.056 = 24.06 (to 4 s.f.)
(ii) 0.000 142 254 = 0.000 142 25 (to 5 s.f.)
(iii) 150 000 (to 2 s.f.)
5. (a) (i)
2 2646
3 1323
3 441
3 147
7 49
7 7
1
2646 = 2 × 33
× 72
(ii) When k = 54,
2646
54
= 49 = 7
The largest prime number of
k
2646
is 7
when k = 54.
(b) 42 = 2 × 3 × 7
54 = 2 × 33
2646 = 2 × 33
× 72
So, the number that gives LCM 2646 must be divisible
by 72
= 49. Given that n > 54, n is either 2 × 72
or
3 × 72
. The next smallest number greater than 54 and
gives the LCM 2646 is 2 × 72
= 98.
6. When a = −1, b = 3, c = –4,
(a) (ab)2
– 4ca
= [(−1)(3)]2
– 4(–4)(−1)
=
(−3)2
– 16
= 9 – 16
= –7
(b)
a
b c
–
+
ab
ac
–
b
a b
–
=
(–1)
3 – (–4)
+
(–1)(3)
(–1)(–4)
–
3
(–1) – 3
=
–1
7
–
3
4
+
3
4
= –
1
7
7. (a) 5(2x – 3y) – 3[– 3(y – x) + 2y]
= 5(2x – 3y) – 3[– 3y + 3x + 2y]
= 5(2x – 3y) – 3[– 3y + 2y + 3x]
= 5(2x – 3y) – 3[– y + 3x]
= 10x – 15y + 3y – 9x
= 10x – 9x – 15y + 3y
= x – 12y
(b)
1
5
(–3x – 5) –
4
5
(–x – 3) + (x – 1)
= –
3
5
x – 1 +
4
5
x +
12
5
+ x – 1
= –
3
5
x +
4
5
x + x – 1 +
12
5
– 1
=
6
5
x +
2
5
8. (a) −mn – 5mnp + 3m
= m(−n – 5np + 3)
(b) 3ax – 2bx – 10cx + 5dx
= x(3a – 2b – 10c + 5d)
(c) 12pq – 2pr + 6pqr – 2p
= 2p(6q – r + 3qr – 1)
84

1
9. (a) 4(2x + 3) = 2(x – 3)
2(2x + 3) = x – 3
4x + 6 = x – 3
4x – x = – 3 – 6
3x = –9
x = −3
(b)
4
5
(–2x – 3) =
4
3
–
x
17
15
–
8
5
x –
12
5
=
4
3
–
17
15
x
17
15
x –
8
5
x =
4
3
+
12
5
−
7
15
x = 3
11
15
x = –8
(c)
x
5
2 –
=
+
x
4
3
5(x + 3) = 4(2 – x)
5x + 15 = 8 – 4x
5x + 4x = 8 – 15
9x = –7
x = –
7
9
10. (a) Let the first number be y.
Then the next consecutive number is (y + 1).
(y + 1) + 2y = 70
y + 2y + 1 = 70
3y = 70 – 1
3y = 69
y = 23
(b) Let Michael’s brother’s present age be n years.
Then Michael’s present age is 1
1
2
n years.
Six years ago,
Michael was 1
1
2
n – 6





 years old and his brother
was (n – 6) years old.
1
1
2
n – 6





= 2(n – 6)
1
1
2
n – 6





= 2n – 12
2n −1
1
2
n = 12 – 6
1
2
n = 6
n = 12
Michael is 1
1
2
(12) = 18 years old and his brother
is 12 years old.
11. (a) (i) Since the common difference is 3, Tn = 3n + ?.
The term before T1 is c = T0 = −2 – 3 = –5.
(ii) 3k – 5 = 304.
3k = 304 + 5
3k = 309
k = 103
The value of k is 103.
(b)
1
2
3
4
y
–7
–6
–5
–4
–3
–2
–1
–1
–2
–3 5
3 4 7
6
0
x
A (1, 3)
C (6, –1)
B (1, –5)
2
1
Length of base of ABC = length AB
= 3 – (–5) = 8
Perpendicular height from C = 6 – 1 = 5
Area of ABC
=
1
2
× base length × perpendicular height
=
1
2
× 8 × 5
= 20 square units
85

1
Part II
Section A
1. (a) 0.36 ÷ [0.36 – (2.16 ÷ 6 – 0.01 ÷ 0.25)] × 0.9
= 0.36 ÷ [0.36 – (0.36 – 0.04)] × 0.9
= 0.36 ÷ [0.36 – 0.32] × 0.9
= 0.36 ÷ 0.04 × 0.9
= 9 × 0.9
= 8.1
(b)
+
4.72 – 3.8 1.04
12.5 – 12.43
–
×
6.33 – 5.15 0.84
0.167
=
1.96
0.07
–
6.33 – 4.326
0.167
= 28 – 12
= 16
(c) 1.8 + 1
9
10
× (5.9 – 3.8)





 ÷ 1.41 – 1
2
5






= 1.8 + 1
9
10
× 2.1





 ÷ 1.41 – 1
2
5






= [1.8 + 3.99] ÷ 1.41 – 1
2
5






= 5.79 ÷ 0.01
= 579
2. (a) Height of water when it is at high tide = +2.8 m
Height of water when it is at low tide = −1.5 m
Difference between high tide and low tide
=
+2.8 – (−1.5)
= 4.3 m
(b)
1
2
m3
weighs
5
16
tonnes
1 m3
weighs
5
16
× 2 =
5
8
tonnes
3
1
5
m3
weighs
5
8
× 3
1
5
= 2 tonnes
(c) Length of ribbon being cut off = 3 × 2
5
12
= 7
1
4
m
Length of ribbon remaining
= 10
3
8
– 7
1
4
= 3
1
8
m
3. (a) 0.3(2x – 3) =
1
5
(0.7 + x) – 0.65x
0.3(2x – 3) = 0.2(0.7 + x) – 0.65x
0.6x – 0.9 = 0.14 + 0.2x – 0.65x
0.6x + 0.65x – 0.2x = 0.14 + 0.9
1.05x = 1.04
x = 0.990 (to 3 s.f.)
(b) –5 +
x
3
– 4
=
x
5
– 4
– 11
x
5
– 4
–
x
3
– 4
= –5 + 11
x
2
– 4
= 6
2 = 6(x – 4)
1 = 3(x – 4)
1 = 3x – 12
3x = 1 + 12
3x = 13
x =
13
3
= 4
1
3
4. (a) When p = −1, q = 2 and r = 8,
p
q
=
+
p q z
r
(3 – 2 5)
2
2
– q2
(–1)
2
=
+
z
(–1)(3(2) – 2 5)
2(8)
2
– (2)2
–
1
2
=
+
z
–(12 – 2 5)
16
– 4
–
1
2
+ 4 =
+
z
–(12 – 2 5)
16
3
1
2
=
+
z
–(12 – 2 5)
16
12
1
4
=
+ z
–12 2 – 5
16
16 × 12
1
4
= 16 ×
+ z
–12 2 – 5
16
196 = −12 – 5 + 2z
196 = −17 + 2z
2z = 196 + 17
2z = 213
z = 106
1
2
(b) 3xa – 3a – 3xb + 3b + 2ya – 2yb
= 3xa – 3xb – 3a + 3b + 2ya – 2yb
= 3x(a – b) – 3(a – b) + 2y(a – b)
= (a – b)(3x – 3 + 2y)
86

1
Section B
5. 40 = 23
× 5
98 = 2 × 72
500 = 22
× 53
(i) The greatest whole number that will divide 40, 98 and
500 exactly means the HCF of 40, 98 and 500.
HCF of 40, 98 and 500 = 2
(ii) The smallest whole that is divisible by 40, 98 and 500
means the LCM of 40, 98 and 500.
LCM of 40, 98 and 500 = 23
× 53
× 72
= 49 000
6. (a) Let the breadth of the rectangular field be x m.
Then the length of the field is 2x m.
Perimeter of field = 2(2x + x) m
360 = 2(2x + x)
360 = 2(3x)
6x = 360
x = 60
The breadth is 60 m and the length of the field is
60 × 2 = 120 m.
Area of field = 120 × 60
= 7200 m2
(b) (i) Perimeter of ABCD = Perimeter of PQR
2[(4x – 3) + (6x – 7)]
= 2x + (6x – 3) + (4x + 3)
2[4x – 3 + 6x – 7] = 2x + 6x + 4x – 3 + 3
2[4x + 6x – 3 – 7] = 12x
10x – 10 = 6x
10x – 6x = 10
4x = 10
x = 2.5
(ii) Length of rectangle ABCD = (6 × 2.5 – 7)
= 8 cm
Breadth of rectangle ABCD = (4 × 2.5 – 3)
= 7 cm
Area of rectangle = 8 × 7
= 56 cm2
Length of base of PQR = 2 × 2.5
= 5 cm
Perpendicular height of PQR = 6 × 2.5 – 3
= 12 cm
Area of PQR =
1
2
(12)(5)
= 30 cm2
(iii) No, even though the perimeter of the two figures
are the same.
7. (a)

1, 1, 2, 3, 5, 8, 13,...
+1 +1 +2 +3 +5 +8
The rule:
The next term can be obtained by adding the
previous two terms.
The next five terms are
13 + 8 = 21
21 + 13 = 34
34 + 21 = 55
55 + 34 = 89
89 + 55 = 144
(b) (i) 5th
line: 12
+ 12
+ 22
+ 32
+ 52
+ 82
= 8 × 13
(ii) 6th
line: 12
+ 12
+ 22
+ 32
+ 52
+ 82
+ 132
= 13 × 21
7th
line: 12
+ 12
+ 22
+ 32
+ 52
+ 82
+ 132
+ 212
= 21 × 34
8th
line: 12
+ 12
+ 22
+ 32
+ 52
+ 82
+ 132
+ 212
+
342
= 34 × 55
9th
line: 12
+ 12
+ 22
+ 32
+ 52
+ 82
+ 132
+ 212
+
342
+ 552
= 55 × 89
(iii) Adding the squares of the terms in the sequence
is the same as taking the product of the last term
in the sum, on the LHS, and the next term in the
sequence.
8. (a) When x = –4,
4p + 2(–4) = −1
4p – 8 = −1
4p = −1 + 8
4p = 7
p = 1.75
87

1
(b)
–2
–3
–4 1 4
–1 2 5
3 6
0
1
2
3
y
–3
–4
y = –3.5
–2
–1
x
x = 1.5
(c) (i)
(c) (ii)
6.5
(d) (1, –2)
4y + 2x = –1
(c) (i) From the graph, y ≈ −1.
(ii) Note: In this case, extrapolation is needed to
obtain the answer.
After extrapolating the graph, we find that x = 6.5.
(d) No. The point (1, −2) does not lie on the line with
equation 4y + 2x = −1.
88

1
Chapter 8 Percentage
Basic
1. (a) 18% =
18
100
=
9
50
(b) 85% =
85
100
=
17
20
(c) 125% =
125
100
= 1
1
4
(d) 210% =
210
100
= 2
1
10
(e) 0.25% =
0.25
100
=
×
×
0.25 100
100 100
=
25
10000
=
1
400
(f) 4.8% =
4.8
100
=
×
×
4.8 10
100 10
=
48
1000
=
6
125
(g) 1
1
3
% =
4
3
%
=
4
3
÷ 100
=
4
3
×
1
100
=
4
300
=
1
75
(h) 12
1
2
% =
25
2
%
=
25
2
÷ 100
=
25
2
×
1
100
=
25
200
=
1
8
2. (a) 9% =
9
100
= 0.09
(b) 99% =
99
100
= 0.99
(c) 156% =
156
100
= 1.56
(d) 0.05% =
0.05
100
= 0.0005
(e) 0.68% =
0.68
100
= 0.0068
(f) 1.002% =
1.002
100
= 0.010 02
(g) 2.4% =
2.4
100
= 0.024
(h) 14
2
5
% =
72
5
%
=
72
5
÷ 100
=
72
5
×
1
100
=
72
500
= 0.144
3. (a)
4
625
=
4
625
× 100%
= 0.64%
(b)
9
125
=
9
125
× 100%
= 7.2%
(c)
6
25
=
6
25
× 100%
= 24%
89

1
(d)
3
4
=
3
4
× 100%
= 75%
(e)
19
20
=
19
20
× 100%
= 95%
(f)
9
8
=
9
8
× 100%
= 112.5%
(g)
7
5
=
7
5
× 100%
= 140%
(h)
33
8
=
33
8
× 100%
= 412.5%
4. (a) 0.0034 = 0.0034 × 100%
= 0.34%
(b) 0.027 = 0.027 × 100%
= 2.7%
(c) 0.05 = 0.05 × 100%
= 5%
(d) 0.14 = 0.14 × 100%
= 14%
(e) 0.5218 = 0.5218 × 100%
= 52.18%
(f) 6.325 = 6.325 × 100%
= 632.5%
(g) 16.8 = 16.8 × 100%
= 1680%
(h) 332 = 332 × 100
= 33 200%
5. (a) Convert 1l to ml.
1l = 1000 ml
175
1000
× 100% = 17.5%
(b) Convert 1 day to hours.
1 day = 24 hours
6
24
× 100% = 25%
(c) Convert 1 hour to minutes.
1 hour = 60 minutes
20
60
× 100% = 33
1
3
%
(d) Convert $1.44 to cents.
$1.44 = 144 cents
80
144
× 100% = 55
5
9
%
(e) Convert 20 cm to mm.
20 cm = 20 × 10 = 200 mm
225
200
× 100% = 112.5%
(f) Convert 45 kg to g.
45 kg = 45 × 1000 = 45 000 g
45000
36000
× 100% = 125%
(g) Convert 2 years to months.
2 years = 2 × 12 = 24 months
24
18
× 100% = 133
1
3
%
(h) Convert $4.40 to cents.
$4.40 = 440 cents
440
99
× 100% = 444
4
9
%
6. Total amount of mixture = 8 + 42 = 50l
(i) Percentage of milk in the mixture
=
42
50
× 100%
= 84%
(ii) Percentage of water in the mixture
=
8
50
× 100%
= 16%
7. Percentage of latecomers in school A
=
25
1500
× 100%
= 1
2
3
% or 1.67% (to 3 s.f.)
Percentage of latecomers in school B
=
25
1800
× 100%
= 1
7
18
% or 1.39% (to 3 s.f.)
School A has 1.67% of students coming late whereas
school B has 1.39% of students coming late. Thus,
school B has a lower percentage of latecomers.
8. (a) 0.25% of 4000
=
0.25
100
× 4000
= 0.25 × 40
= 10
(b) 6% of 200 =
6
100
× 200
= 12
(c) 7.5% of $2500 =
7.5
100
× 2500
= 7.5 × 25
= $187.50
(d) 8% of 130 g =
8
100
× 130
= 10.4 g
90

1
(e) 20.6% of 15 000 people
=
20.6
100
× 15 000
= 20.6 × 150
= 3090 people
(f) 37
1
2
% of 56 cm
=
75
2
% of 56 cm
=
75
2
×
1
100
× 56
= 21 cm
(g) 45% of 4 kg
=
45
100
× 4
= 1.8 kg
(h) 66
2
3
% of 72 litres
=
200
3
% of 72 litres
=
200
3
×
1
100
× 72
= 48 litres
(i) 112
1
2
% of 200 m
=
225
2
% of 200 m
=
225
2
×
1
100
× 200
= 225 m
(j) 180% of 320
=
180
100
× 320
= 576
9. Method 1
Number of kilograms of zinc = 25% of 60
=
25
100
× 60
= 15
Number of kilograms of copper
= 60 – 15
= 45
The ingot of copper contains 45 kg of copper.
Method 2
Percentage of copper in ingot = 100% – 25% = 75%
Number of kilograms of copper in ingot = 75% of 60
=
75
100
× 60
= 45
The ingot of brass contains 45 kg of copper.
10. (a) Required value = 110% of $60
=
110
100
× 60
= $66
(b) Required value = 128% of 69 l
=
128
100
× 69
= 88.32 l
(c) Required value = 225% of 50 m
=
225
100
× 50
= 112.5 m
(d) Required value = 400% of 24 kg
=
400
100
× 24
= 96 kg
(e) Required value = 112
1
2
% of 32 g
=
225
2
% × 32
=
225
2
÷100





 × 32
=
225
2
×
1
100
× 32
= 36 g
(f) Required value = 100.03% of $400
=
100.03
100
× 400
= 100.03 × 4
= $400.12
(g) Required value = 100.5% of $4000
=
100.5
100
× 4000
= 100.5 × 40
= $4020
(h) Required value = 2600% of $1.50
=
2600
100
× 1.50
= $39
91

1
11. (a) Required value
= 99.4% of 1.25 km
=
99.4
100
× 1.25
= 1.2425 km
(b) Required value
= 95% of $88
=
95
100
× 88
= $83.60
(c) Required value
= 93% of $7500
=
93
100
× 7500
= $6975
(d) Required value
= 87
1
2
% of 64 g
=
175
2
% of 64 g
=
175
2
×
1
100
× 64
= 56 g
(e) Required value
= 86.5% of 78 kg
=
86.5
100
× 78
= 67.47 kg
(f) Required value
= 85% of 124 l
=
85
100
× 124
= 105.4 l
(g) Required value
= 58% of 350 m2
=
58
100
× 350
= 203 m2
(h) Required value
= 15% of 520
=
15
100
× 520
= 78
12. (a) Let the number be x.
12% of x = 48
12
100
× x = 48
x = 48 ÷
12
100
= 48 ×
100
12
x = 400
15
5
8
% of x = 555
125
8
% × x = 555
125
8
÷100





 × x = 555
125
8
×
1
100
× x = 555
5
32
× x = 555
x = 555 ÷
5
32
= 3552
(c) Let the number be x.
21% of x = 147
21
100
× x = 147
x = 147 ÷
21
100
= 147 ×
100
21
x = 700
(d) Let the number be x.
77.5% of x = 217
77.5
100
× x = 217
x = 217 ÷
77.5
100
= 217 ×
100
77.5
= 280
(e) Let the number be x.
124% of x = 155
124
100
× x = 155
x = 155 ÷
124
100
= 155 ×
100
124
= 125
92

1
13. (a) Let the number be x.
120% of x = 48
120
100
× x = 48
x = 48 ÷
120
100
= 48 ×
100
120
= 40
70% of x = 147
70
100
× x = 147
x = 147 ÷
70
100
=147 ×
100
70
= 210
(c) Let the number be x.
33
1
3
% of x = 432
100
3
% of x = 432
100
3
÷100





 × x = 432
100
3
×
1
100
× x = 432
1
3
x = 432
x = 432 ÷
1
3
= 1296
14. Increase in the number of buses operating
= 1420 – 1000
= 420
Percentage increase in the number of buses in operation
=
Increase
Original value
× 100%
=
420
1000
× 100%
= 42%
15. Decrease in the price of MP3 player
= $382 – $261.50
= $120.50
Percentage decrease in the price
=
Decrease
Original value
× 100%
=
120.5
382
× 100%
= 31.5% (to 3 s.f.)
16. 120% of Michael’s income = $120
120
100
× Michael’s income = $120
Michael’s income = 120 ÷
120
100
= 120 ×
100
120
= $100
17. Price of notebook in 2013 = 70% of $2000
=
70
100
× 2000
= $1400
Price of notebook in 2014 = 70% of $1400
=
70
100
× 1400
= $980
Intermediate
18. Let the total number of students taking Additional
Mathematics be x.
35% of x = 42
35
100
× x = 42
x = 42 ÷
35
100
= 42 ×
100
35
= 120
Number of students taking Additional Mathematics in
class C
= 120 – 42 – 40
= 38
19. Percentage of candidates who obtained grade C
= 100% – 18% – 38%
= 44%
Let the total number of candidates be x.
44% of x = 77
44
100
× x = 77
x = 77 ÷
44
100
= 77 ×
100
44
= 175
The total number of candidates is 175.
93

1
20. Amount of milk in the solution
= 30% of 125 l
=
30
100
× 125
= 37.5 l
Let the amount of water to be added be x l.
+ x
37.5
125
= 14%
+ x
37.5
125
=
7
50
875 + 7x = 1875
7x = 100
x = 142
6
7
Amount of water added = 142
6
7
l
21. 140% of price in first half of 2013 = $52 640
Price in first half of 2013 = 52 640 ÷
140
100
= 52 640 ×
100
140
= $37 600
98% of price in 2012 = $37 600
Price in 2012 = 37 600 ÷
98
100
= 37 600 ×
100
98
= $38 367.35
105% of original price of painting = $38 367.35
Original price of painting
= 38 367.35 ÷
105
100
= 38 367.35 ×
100
105
= $36 540.33 (to the nearest cent)
22. Number of girls in the club = 70% of 40
=
70
100
× 40
= 28
Number of boys in the club = 40 – 28
= 12
Let the number of new members who are girls be x and
the number of new members who are boys be y.
Then y – x = 6.
y = 6 + x
New percentage of girls in the club = 60%
+
+ +
x
x y
28
40
=
60
100
+
+ +
x
x y
28
40
=
3
5
5(28 + x) = 3(40 + x + y)
140 + 5x = 120 + 3x + 3y
140 – 120 + 5x – 3x = 3y
20 + 2x = 3y
Substitute y = 6 + x:
20 + 2x = 3(6 + x)
20 + 2x = 18 + 3x
20 – 18 = 3x – 2x
x = 2
y = 6 + 2
= 8
No. of members who are boys = 12 + 8
= 20
23. (i) 3 parts of the length AB = 3 cm
1 part of the length AB = 1 cm
7 parts, which is the length of AB = 7 cm
(ii) BC = 135% of AB
=
135
100
× 7
= 9.45 cm
(iii)AC = 85% of BC
=
85
100
× 9.45
= 8.0325 cm
24. (i) Selling price of the flat = 115% of $145 000
=
115
100
× 145 000
= $166 750
Amount gained by selling the flat
= 166 750 – 145 000
= $21 750
(ii) Selling price of the car = 88% of $50 000
=
88
100
× 50 000
= $44 000
Amount lost by selling his car
= 50 000 – 44 000
= $6000
(iii)Yes, he still gained an amount of
$21 750 – $6000 = $15 750
94

1
Advanced
25. (a) Zhi Xiang’s new monthly salary under scheme B
= 104.5% of $1500 + $50
=
104.5
100
× 1500 + 50
= 1567.5 + 50
= $1617.50
Zhi Xiang’s new salary as a percentage of his
present salary
=
1617.50
1500
× 100%
= 108% (to 3 s.f.)
(b) Tom’s new monthly salary under scheme A
= 106% of $1200
=
106
100
× 1200
= $1272
Tom’s new monthly salary under scheme B
= 104.5% of $1200 + $50
=
104.5
100
× 1200 + 50
= $1304
Since Tom’s salary will be higher under scheme
B, he should choose scheme B.
(c) Let Sharon’s current monthly wage be $x.
106% of $x = 104.5% of $x + $50
(106 – 104.5)% of x = 50
1.5% of x = 50
1.5
100
× x = 50
x = 50 ÷
1.5
100
= 50 ×
100
1.5
x = 3333.33 (to the nearest
cent)
Sharon’s salary is $3333.33.
New Trend
26. Let x be the total number of crayons.
Number of blue crayons =
4
9
x
Number of red crayons = 65% ×
5
9
x
=
65
100
×
5
9
x
=
13
36
x
Number of yellow crayons = x –
4
9
x –
13
36
x
=
7
36
x
7
36
x = 14
x = 72
There are 72 crayons altogether.
27. In 2013, the value of the bracelet
= 110% of $12 650
=
110
100
× 12 650
= $13 915
In 2014, the value of the bracelet
= 110% of $13 915
=
110
100
× 13 915
= $15 306.50
In 2015, the value of the bracelet
= 110% of $15 306.50
=
110
100
× 15 306.50
= $16 837.15
16 837.15 – 12 650
12 650
× 100% = 33.1%
The value of the bracelet in 2015 is $16 837.15 and the
overall percentage increase is 33.1%.
28. 103% of original bill = $82.70
103
100
× original bill = 82.70
Original bill = 82.70 ÷
103
100
= 82.70 ×
100
103
= $80.29
29. (a) Convert 3.96 m to cm.
3.96 m = (3.96 × 100) cm
= 396 cm
33
396
× 100% = 8
1
3
%
(b) 15 ÷ 0.3 = 50
50 glasses can be filled.
95

1
Chapter 9 Ratio, Rate and Speed
Basic
1. (a) 14 : 35
14 ÷ 7 : 35 ÷ 7
2 : 5
(b) 24 : 42
24 ÷ 6 : 42 ÷ 6
4 : 7
(c) 36 : 132
36 ÷ 6 : 132 ÷ 6
6 : 22
6 ÷ 2 : 22 ÷ 2
3 : 11
(d) 135 : 240
135 ÷ 5 : 240 ÷ 5
27 : 48
27 ÷ 3 : 48 ÷ 3
9 : 16
(e) 144 : 128
144 ÷ 16 : 128 ÷ 16
9 : 8
(f) 162 : 384
162 ÷ 6 : 384 ÷ 6
27 : 64
(g) 192 : 75
192 ÷ 3 : 75 ÷ 3
64 : 25
(h) 418 : 242
418 ÷ 2 : 242 ÷ 2
209 : 121
209 ÷ 11 : 121 ÷ 11
19 : 11
2. (a)
9
20
:
3
5
=
9
20
× 20 :
3
5
× 20
= 9 : 12
= 3 : 4
(b)
7
15
:
14
9
=
7
15
× 9 :
14
9
× 9
=
21
5
: 14
=
21
5
× 5 : 14 × 5
= 21 : 70
= 3 : 10
(c)
15
28
:
18
7
=
15
28
× 28 :
18
7
× 28
= 15 : 72
= 5 : 24
(d)
25
44
:
50
33
=
25
44
× 11 :
50
33
× 11
=
25
4
:
50
3
=
25
4
× 12 :
50
3
× 12
= 75 : 200
= 3 : 8
(e) 1
25
56
:
18
21
=
81
56
:
18
21
=
81
56
× 21 :
18
21
× 21
=
243
8
: 18
=
243
8
× 8 : 18 × 8
= 243 : 144
= 27 : 16
(f) 4
1
3
: 65 =
13
3
: 65
=
13
3
× 3 : 65 × 3
= 13 : 195
= 1 : 15
(g) 8
3
4
: 3
1
8
=
35
4
:
25
8
=
35
4
× 8:
25
8
× 8
= 70 : 25
= 14 : 5
(h) 2.4 : 1
1
5
= 2
4
10
: 1
1
5
=
12
5
× 5 :
6
5
× 5
= 12 : 6
= 2 : 1
3. (a) 0.09 : 0.21
0.09 × 100 : 0.21 × 100
9 : 21
3 : 7
(b) 0.192 : 0.064
0.192 × 1000: 0.064 × 1000
192 : 64
3 : 1
96

1
(c) 0.25 : 1.5
0.25 × 100 : 1.5 × 100
25 : 150
1 : 6
(d) 0.63 : 9.45
0.63 × 100 : 9.45 × 100
63 : 945
1 : 15
(e) 0.84 : 1.12
0.84 × 100 : 1.12 × 100
84 : 112
21 : 28
3 : 4
(f) 1.26 : 0.315
1.26 × 1000 : 0.315 × 1000
1260 : 315
4 : 1
(g) 1.44 : 0.48
1.44 × 100 : 0.48 × 100
144 : 48
3 : 1
(h) 1.8 : 0.4
1.8 × 10 : 0.4 × 10
18 : 4
9 : 2
4. (a) 6 parts = $336
1 part =
336
6
= $56
5 parts = 56 × 5 = $280
$56 : $280
(b) 14 parts = $336
1 part =
336
14
= $24
3 parts = 24 × 3 = $72
11 parts = 24 × 11 = $264
$72 : $264
(c) 16 parts = $336
1 part =
336
16
= $21
3 parts = 21 × 3 = $63
13 parts = 21 × 13 = $273
$63 : $273
(d) 8 parts = $336
1 part =
336
8
= $42
5 parts = 42 × 5 = $210
3 parts = 42 × 3 = $126
$210 : $126
(e) 12 parts = $336
1 part =
336
12
= $28
5 parts = 28 × 5 = $140
7 parts = 28 × 7 = $196
$140 : $196
(f) 14 parts = $336
1 part =
336
14
= $24
5 parts = 24 × 5 = $120
9 parts = 24 × 9 = $216
$120 : $216
(g) 24 parts = $336
1 part =
336
24
= $14
7 parts = 14 × 7 = $98
17 parts = 14 × 17 = $238
$98 : $238
(h) 21 parts = $336
1 part =
336
21
= $16
8 parts = 16 × 8 = $128
13 parts = 16 × 13 = $208
$128 : $208
(i) 21 parts = $336
1 part =
336
21
= $16
10 parts = 16 × 10 = $160
11 parts = 16 × 11 = $176
$160 : $176
(j) 24 parts = $336
1 part =
336
24
= $14
11 parts = 14 × 11 = $154
13 parts = 14 × 13 = $182
$154 : $182
5. (a) Convert $1 to cents.
$1 = 100 cents
45 cents : 100 cents
=
45
100
=
9
20
45 cents : $1 = 9 : 20
97

1
(b) Convert 1.25 m to cm.
1.25 m = 1.25 × 100 = 125 cm
25 cm : 125 cm
=
25
125
=
1
5
25 cm : 1.25 m = 1 : 5
(c) Convert 0.25 km to m.
0.25 km = 0.25 × 1000 = 250 m
250 m : 75 m
=
250
75
=
10
3
0.25 km : 75 m = 10 : 3
(d) Convert 0.2 kg to g.
0.2 kg = 0.2 × 1000 = 200 g
200 g : 40 g
=
200
40
=
5
1
0.2 kg : 40 g = 5 : 1
(e) Convert 1 hour to minutes.
1 hour = 60 minutes
35 min : 60 min
=
35
60
=
7
12
35 minutes : 1 hour = 7 : 12
(f) Convert 2 cm to mm.
2 cm = 2 × 10 = 20 mm
15 mm : 20 mm
=
15
20
=
3
4
15 mm : 2 cm = 3 : 4
(g) Convert 3.2 hours to minutes.
3.2 hours = 3.2 × 60 = 192 minutes
192 min : 72 min
=
192
72
=
8
3
3.2 hours : 72 minutes = 8 : 3
(h) Convert
7
200
l to cm3
.
7
200
l =
7
200
× 1000 = 35 cm3
35 cm3
: 105 cm3
=
35
105
=
1
3

7
200
l : 105 cm3
= 1 : 3
6. (a) 57 : 19 : 133
57 ÷ 19 : 19 ÷ 19 : 133 ÷ 19
3 : 1 : 7
(b) 64 : 96 : 224
64 ÷ 32 : 96 ÷ 32 : 224 ÷ 32
2 : 3 : 7
(c) 108 : 36 : 60
108 ÷ 6 : 36 ÷ 6 : 60 ÷ 6
18 : 6 : 10
18 ÷ 2 : 6 ÷ 2 : 10 ÷ 2
9 : 3 : 5
(d) 644 : 476 : 140
644 ÷ 28 : 476 ÷ 28 : 140 ÷ 28
23 : 17 : 5
(e) 665 : 1995 : 1330
665 ÷ 35 : 1995 ÷ 35 : 1330 ÷ 35
19 : 57 : 38
19 ÷ 19 : 57 ÷ 19 : 38 ÷ 19
1 : 3 : 2
(f) 1015 : 350 : 455
1015 ÷ 35 : 350 ÷ 35 : 455 ÷ 35
29 : 10 : 13
7. (a) 3 : 9 = 4 : a
3
9
=
a
4
(express ratios as fractions)
3a = 36
a = 12
(b) 4 : 3 = a : 6
4
3
=
a
6
(express ratios as fractions)
3a = 24
a = 8
(c) 5 : 11 = 10 : a
5
11
=
a
10
5a = 110
a = 22
98

1
(d) 12 : 25 = a : 5
12
25
=
a
5
25a = 60
a =
60
25
= 2
2
5
(e) 14 : 9 = 7 : a
14
9
=
a
7
14a = 63
a = 4.5 or 4
1
2
(f) a : 5.7 = 8 : 12
a
5.7
=
8
12
12a = 45.6
a = 3.8 or 3
4
5
8. (i) Convert 1.68 cm to cm.
1.68 m = 1.68 × 100 = 168 cm
168 cm : 105 cm
=
168
105
=
8
5
The ratio of Rui Feng’s height to his brother’s
height is 8 : 5.
(ii) Total height of the boys (in cm)
= 168 + 105 = 273 cm
1.68 m : 273 cm
168 cm : 273 cm
=
168
273
=
8
13
The ratio of Rui Feng’s height to the total height
of both boys is 8 : 13.
9. Total number of parts = 126 + 42 = 168 parts
(i) Total number of parts : Number of parts of
pure gold
168 : 126
168 ÷ 42 : 126 ÷ 42
4 : 3
(ii) Total number of parts : Number of parts of
alloy B
168 : 42
168 ÷ 42 : 42 ÷ 42
4 : 1
Alloy B : Pure Gold
1 : 3
10. (a) For the ratio 1 : 2 : 6,
9 parts = $180
1 part =
180
9
= $20
6 parts = 20 × 6 = $120
The smallest share is $20 and the largest
share is $120.
(b) For the ratio 1 : 4 : 7,
12 parts = $180
1 part =
180
12
= $15
7 parts = 15 × 7 = $105
share is $105.
(c) For the ratio 2 : 3 : 5,
10 parts = $180
1 part =
180
10
= $18
2 parts = 18 × 2 = $36
5 parts = 18 × 5 = $90
share is $90.
(d) For the ratio 2 : 13 : 5,
20 parts = $180
1 part =
180
20
= $9
2 parts = 9 × 2 = $18
13 parts = 9 × 13 = $117
The smallest share is $18 and the largest share
is $117.
(e) For the ratio 3 : 1 : 11,
15 parts = $180
1 part =
180
15
= $12
11 parts = 12 × 11 = $132
is $132.
(f) For the ratio 4 : 11 : 3,
18 parts = $180
1 part =
180
18
= $10
3 parts = 10 × 3 = $30
11 parts = 10 × 11 = $110
is $110.
99

1
11. (a) 7 parts = $84
1 part =
84
7
= $12
18 parts = 12 × 18 = $216
Largest part is $216.
Total sum = (15 + 18 + 7) × 12 = $480
(b) 7 parts = $133
1 part =
133
7
= $19
18 parts = 19 × 18 = $342
Total sum = (15 + 18 + 7) × 19 = $760
(c) 7 parts = $301
1 part =
301
7
= $43
18 parts = 43 × 18 = $774
Total sum = (15 + 18 + 7) × 43 = $1720
(d) 7 parts = $3990
1 part =
3990
7
= $570
18 parts = 570 × 18 = $10 260
Largest part is $10 260.
Total sum = (15 + 18 + 7) × 570 = $22 800
12. (a) 11 parts = 187°
1 part =
187
11
= 17°
7 parts = 17 × 7 = 119°
Angle D = 360 – 187 = 173°
Ratio of angle C to angle D = 119 : 173
(b) 11 parts = 242°
1 part =
242
11
= 22°
7 parts = 22 × 7 = 154°
Angle D = 360 – 242 = 118°
(c) 11 parts = 275°
1 part =
275
11
= 25°
7 parts = 25 × 7 = 175°
Angle D = 360 – 275 = 85°
= 35 : 17
13. (a) Rate =
350
40
=
35
4
= 8.75 km/l
(b) Rate =
120
8
= $15/hour
(c) Rate =
×
82 100
300
=
82
3
= 27
1
3
cents/unit
(d) Rate =
320
8
= 40 words/min
(e) Rate =
60
12
= $5/tile
(f) Rate =
1760
15
= 117
1
3
cents/min
14. (i) Cost of 1 m2
of flooring =
$36
20
= $1.80
(ii) Cost of 55 m2
of flooring = $1.80 × 55 = $99
(iii)Area of flooring for a cost of $1 =
20
36
=
5
9
m2
Area of flooring for the cost of $63
=
5
9
× $63 = 35 m2
15. Amount required to travel a distance of 50 km
= $1.35 × 50 = $67.50
Amount that each child will have to pay =
$67.50
54
= $1.25
16. Convert 75 cm to m.
75 cm = 75 ÷ 100 = 0.75 m
Area of rectangular brass sheet = 1.5 × 0.75
= 1.125 m2
Area of 1 kg of brass sheet =
1.125
7.2
= 0.156 25 m2
Area of 12.8 kg of brass sheet = 0.156 25 × 12.8
= 2 m2
17. Time required for one man to finish the project
= 45 × 8
= 360 hours
Time required for (45 – 5) = 40 men to finish the
project =
360
40
= 9 hours
18. 5.55 p.m. + 40 min = 6.35 p.m.
= 18 35
19.
22 17
43 min 8 h 17 min
23 00 07 00 07 17
Total time taken = 43 min + 8 h + 17 min
= 9 h
100

1
20. (a) 84 km/h =
84 km
1 h
=
84 000 m
3600 s
= 23
1
3
m/s
(b) 15 m/s =
15 m
1 s
=
(15 ÷ 1000) km
(1 ÷ 3600) s
= 54 km/h
(c)
2
3
km/min =
2
3
km
1 min
=
2
3
km
(1 ÷ 60) h
= 40 km/h
(d) 120 cm/s =
120 ÷ 100
1 s
= 1.2 m/s
21. Convert 44 minutes to hours.
44 min =
44
60
=
11
15
h
Time taken to travel a distance of 1 km =
11
15
÷ 11
=
1
15
h
(a) (i) Time taken to travel a distance of 45 km
=
1
15
× 45
= 3 h
(ii) Time taken to travel a distance of 36 km
=
1
15
× 36
= 2
2
5
h or 2 h 24 min
(iii)Time taken to travel a distance of 20 km
=
1
15
× 20
= 1
1
3
h or 1 h 20 min
(b) Speed of the cyclist
=
×
×
11 1000
44 60
= 4
1
6
m/s
22. (i) Time taken for the journey
= 50 min + 3 h 24 min + 2 h 6 min
+ 1 h 30 min
=
5
6
h + 3
2
5
h + 2
1
10
h + 1
1
2
h
= 7
5
6
h or 7 h 50 min
(ii) Average speed =
Total distance travelled
Total time taken
=
687
7
5
6
= 87.7 km/h (to 3 s.f.)
23. (i) Convert 36 minutes to hours.
36 min =
36
60
=
3
5
h
Average speed =
Total time taken
=
27
3
5
= 45 km/h
(ii) Time at which Michael reaches the station
= 08 37 + 36 min
= 09 13
Time at which the train arrives at the station
= 09 42 + 11 min
= 09 53
Waiting time = 09 53 – 09 13 = 40 min
24. (i) Time taken by the car for the whole journey
= 15 10 – 08 45
= 6 h 25 min
(ii) Distance = speed × time
= 84 × 6
5
12
= 539 km
25. (i) Convert 54 minutes to hours.
54 min =
54
60
=
9
10
h
Distance travelled for the first part of the journey
= 70 ×
9
10
= 63 km
Distance travelled for the return journey
= 63 km
Time taken for the return journey
=
63
45
= 1
2
5
h or 1 h 24 min
101

1
(ii) Time at which Khairul starts to return to the original
point
= 0955 + 54 min + 40 min
= 1129
Time when Khairul arrives at the starting point
= 1129 + 1 h 24 min
= 1253
Intermediate
26. (a) 16 parts = $160
1 part =
160
16
= $10
9 parts = $10 × 9 = $90
Difference between the largest share and the
smallest share
= $90 – $10
= $80
(b) 20 parts = $160
1 part =
160
20
= $8
2 parts = $8 × 2 = $16
13 parts = $8 × 13 = $104
smallest share
= $104 – $16
= $88
(c) 40 parts = $160
1 part =
160
40
= $4
5 parts = $4 × 5 = $20
22 parts = $4 × 22 = $88
smallest share
= $88 – $20
= $68
(d) 80 parts = $160
1 part =
160
80
= $2
11 parts = $2 × 11 = $22
37 parts = $2 × 37 = $74
smallest share
= $74 – $22
= $52
27. (a) X : Y = 2 : 3 Y : Z = 5 : 4
= 10 : 15 = 15 : 12
X : Z = 10 : 12 = 5 : 6
(b) X : Y = 5 : 7 Y : Z = 13 : 10
= 65 : 91 = 91 : 70
X : Z = 65 : 70 = 13 : 14
(c) X : Y = 7 : 3 Y : Z = 11 : 21
= 77 : 33 = 33 : 63
X : Z = 77 : 63 = 11 : 9
(d) X : Y = 8 : 15 Y : Z = 21 : 32
= 56 : 105 = 105 : 160
X : Z = 56 : 160 = 7 : 20
28. Rice B is sold at $6.90 for 5 kg. Thus it is sold at $13.80
for 10 kg.
Ratio of prices of rice A and B
= $9.20 : $13.80
= 920 : 1380
= 2 : 3
29. A : B = 8 : 3 A : C = 5 : 12
= 40 : 15 = 40 : 96
The ratio of salaries A, B and C
= 40 : 15 : 96
30. Height of the hall =
28
7
× 6 = 24 m
Ratio of its breadth to its height
= 21 : 24
= 7 : 8
31. (i) Dimensions of second rectangle
= 32 ×
5
4
cm by 24 ×
5
4
cm
= 40 cm by 30 cm
Ratio of perimeters of original rectangle and
second rectangle
= 2(32 + 24) : 2(40 + 30)
= 112 : 140
= 4 : 5
(ii) Ratio of areas of original rectangle and
second rectangle
= 32 × 24 : 40 × 30
= 768 : 1200
= 16 : 25
32. (i) Time for which the car is parked
= 16 30 – 07 45
= 8 h 45 min or 8
3
4
h
(ii) Parking fee
= $2.50 + 14 × $0.80 + $0.80 + $0.80
= $15.30
102

1
33. (i) Amount each tourist spends for 4 days
=
$3600
9
= $400
Cost of staying in the hotel for one day
=
$400
4
= $100
Cost of staying in the hotel for 6 days
= $100 × 6 = $600
Amount 15 tourists spend for staying in the hotel
for 6 days
= $600 × 15
= $9000
(ii) Amount each tourist spends =
$3000
10
= $300
Number of days each tourist can stay in the hotel
=
$300
$100
= 3
34. (i) Charges due to the number of calls
= 493 × $0.1605
= $79.1265
Total charges for the month
= $82.93 + $79.1265
= $162.06 (to the nearest cent)
(ii) Charges due to calls = $93.523 – $82.93
= $10.593
Number of calls made =
$10.593
$0.1605
= 66
She made 66 calls.
35. No. of hours 1 man will take to complete 1200 m
= 8 × 20 × 50
= 8000 h
No. of hours 1 man will take to complete 1800 m
=
1800
1200
× 8000
= 12 000 h
No. of men needed to complete the work on time
=
×
12 000
10 10
= 120
Additional number of men to be employed
= 120 – 60
= 70
36. (i) Amount of time to work on the project per day
= 8.5 × 4
= 34 h
Time required to finish the work
=
272
34
= 8 days
It will take 8 days for 4 men to finish the work.
(ii) Amount to be paid to the men per day
= $8.50 × 8.5 × 4
= $289
Total amount to be paid for the whole project
= 8 × $289
= $2312
(iii)Let the number of overtime hours needed to
complete the project in 4 days by each worker be
x.
5[4(8.5 + x)] = 272
5(34 + 4x) = 272
170 + 20x = 272
20x = 272 – 170 = 102
x = 5.1
The number of overtime hours is 5.1 h.
(iv) Overtime hourly rate
= 1.5 × $8.50
= $12.75
Total amount to be paid to the 4 men if the
project is to be completed in 5 days
= 5{4[(8.5 × $8.50) + (5.1 × $12.75)]}
= $2745.50
37. Distance travelled by the wheel = 765 × 2.8
= 2142 m
Number of revolutions made by the wheel to travel a
distance of 2142 m
=
2142
1.7
= 1260 times
103

1
38. Convert 46 minutes to hours.
46 min =
46
60
=
23
30
h
Let the time taken to travel from Town Y to Z be
T hours.
Average speed =
Total time taken
80 =
80 + 48
T +
23
30
80 T +
23
30





 = 128
80T + 61
1
3
= 128
80T = 128 – 61
1
3
= 66
2
3
T =
5
6
h
Speed of the driver when he is driving from
Town Y to Z
=
80
5
6
= 96 km/h
39. (i) Time arrived at B = 1035 + 0019
= 1054
Time arrived at C = 1150 + (0019 – 0011)
= 1158
(ii) Time to travel from Town C to D
= 1320 – 1158
= 1 h 22 min
Average speed =
Total time taken
=
123
1
22
60
= 90 km/h
Advanced
40.
a b
– 2
10
=
b
6
6(a – 2b) = 10b
6a – 12b = 10b
6a = 10b + 12b
6a = 22b
a
b
6
= 22
a
b
=
22
6
=
11
3
The ratio of a : b = 11 : 3.
41. Let the distance travelled by the motorist be y km.
y = x × 2
1
2
= 2
1
2
x — (1)
y = (x + 4) × 2
1
2
–
15
60






= 2
1
4
(x + 4) — (2)
Substitute (1) into (2):
2
1
2
x = 2
1
4
(x + 4)
2
1
2
x = 2
1
4
x + 9
2
1
2
x – 2
1
4
x = 9
1
4
x = 9
x = 36
The value of x is 36.
42. Time taken for the van to travel a distance of
130 km
=
130
65
= 2 h
Time taken for the car to travel a distance of
130 km
= 2 –
35
60
= 1
5
12
h
Average speed =
Total time taken
=
130
1
5
12
= 91
13
17
km/h
104

1
New Trend
43. (a) (i)
7
3
=
Number of sedans
180
Number of sedans =
7
3
× 180 = 420
(ii) Number of vehicles altogether
=
180
3
× (7 + 3 + 2)
= 720
(b) Blue : Black : White
3 : 5
6 : 7
= 21 : 30 : 35
×5
×7
×5
Blue sedan : black sedan : white sedan
= 21 : 30 : 35
44. (a) 280 km/h =
280 km
1h
=
280 000m
3600s
= 77
7
9
m/s
(b) Time take for bullet train to pass through tunnel
completely
=
(20 500 + 250)m
77
7
9
m/s
= 266
11
14
s
= 4 min 27 s (to the nearest second)
45. Lixin gets 13 – 7 = 6 parts more than Nora.
(a) 6 parts = $78
1 part =
78
6
= $13
12 parts = $13 × 12 = $156
(b) 6 parts = $126
1 part =
126
6
= $21
12 parts = $21 × 12 = $252
(c) 6 parts = $360
1 part =
360
6
= $60
12 parts = $60 × 12 = $720
(d) 6 parts = $540
1 part =
540
6
= $90
12 parts = $90 × 12 = $1080
46. Time taken to fly from Singapore to Helsinki
=
9257
752
= 12.3125 h
= 12 h 0.1325 × 60 min
= 12 h 19 min (to the nearest minute)
47. (i) Distance travelled on 1 litre of petrol
=
128
12
= 10
2
3
km
Distance travelled on 30 litres of petrol
= 10
2
3
× 30
= 320 km
(ii) Amount of petrol required to travel a distance of
1 km
=
12
128
litres
Amount of petrol required to travel a distance
of 15 000 km
=
12
128
× 15 000
= 1406.25 litres
Amount the car owner has to pay
= 1406.25 × $2.03
48. (a) 180 km →50.4 litres
100 km →
50.4
180
× 100
= 28 litres
The fuel consumption of the bus is 28 l/100 km.
(b) (i) 7.6 litres →100 km
50 litres →
100
7.6
× 50
= 658 km (to 3 s.f.)
(ii) 100 km →7.6 litres
330 km →
7.6
100
× 330
= 25.08 litres
1 litre →$2.07
25.08 litres →$2.07 × 25.08
The petrol will cost Fred $51.92 for a journey
of 330 km.
105

1
Chapter 10 Basic Geometry
Basic
1. (a) x° + 90° + 38° = 180° (adj. /s on a str. line)
x° = 180° – 90° – 38°
= 52°
x = 52
(b) 2x° + 80° = 180° (adj. /s on a str line)
2x° = 180° – 80°
= 100°
x° = 50°
x = 50
(c) 2x° + (5x – 9)° = 180° (adj. /s on a str. line)
7x° – 9° = 180°
7x° = 189°
x° = 27°
x = 27
(d) (5x – 23)° + (7x – 13)° = 180° (adj. /s on a
5x° + 7x° – 23° – 13° = 180° str. line)
12x° – 36° = 180°
12x° = 180° + 36°
= 216°
x° = 18°
x = 18
(e) 2x° + 90° + 3x° = 180° (adj. /s on a str. line)
2x° + 3x° = 180° – 90°
5x° = 90°
x° = 18°
x = 18
(f) 3x° + 4x° + 2x° = 180° (adj. /s on a str. line)
9x° = 180°
x° = 20°
x = 20
2. (a) 4x° + 3x° + 2x° = 180° (vert. opp. /s;
9x° = 180° adj. /s on a str. line)
x° = 20°
x = 20
(b) 3x° + 49° + 62° = 180° (adj. /s on a str. line)
3x° = 180° – 49° – 62°
3x° = 69°
x° = 23°
3x° + z° = 180° (adj. /s on a str. line)
3(23°) + z° = 180°
69° + z° = 180°
z° = 180° – 69°
= 111°
y° + z° = 180° (adj. /s on a str. line)
y° + 111° = 180°
y° = 180° – 111°
= 69°
x = 23, y = 69 and z = 111
3. (a) (3x + 34)° = (5x – 14)° (alt. /s, AB // CD)
5x° – 3x° = 34° + 14°
2x° = 48°
x° = 24°
x = 24
(b) (7x – 12)° + (4x – 17)° = 180° (int. /s,
7x° + 4x° – 12° – 17° = 180° AB // CD)
11x° – 29° = 180°
11x° = 180° + 29°
11x° = 209°
x° = 19°
x = 19
(c) 4x° + 5x° = 180° (alt. /s, adj. /s on a str. line)
9x° = 180°
x° = 20°
x = 20
(d) (5x – 14)° + (3x – 10)° = 180° (alt. /s, adj. /s
5x° + 3x° – 14° – 10°= 180° on a str. line)
8x° – 24° = 180°
8x° = 180° + 24°
= 204°
x° = 25.5°
x = 25.5
(e) (5x – 15)° + (75 – x)° = 180° (vert. opp. /s,
5x° – x° – 15° + 75° = 180° int. /s,
4x° + 60° = 180° AB // CD)
4x° = 180° – 60°
= 120°
x° = 30°
x = 30
(f) (3x + 40)° = (5x – 20)° (corr. /s, AB // CD)
5x° – 3x° = 40° + 20°
2x° = 60°
x° = 30
(5x – 20)° = 2y° (vert. opp. /s)
5 × 30° – 20° = 2y°
2y° = 130°
y° = 65°
x = 30 and y = 65
106

1
Intermediate
4. (a) 3x° + (7x – 21)° + (4x – 9)° = 180° (adj. /s on
3x° + 7x° + 4x° – 21° – 9° = 180° a str. line)
14x° – 30° = 180°
14x° = 180° + 30°
= 210°
x° = 15°
x = 15
(b)
1
3
x + 8






°
+
3
4
x – 18






°
+
1
2
x°
= 180° (adj. /s on a str. line)
1
3
x° +
3
4
x° +
1
2
x° + 8° – 18° = 180°
1
7
12
x° = 180° + 10°
= 190°
x° = 120°
x = 120
(c) 1.8x° + (2x + 12)° + x° = 180° (adj. /s on
a str. line)
1.8x° + 2x° + x° = 180° – 12°
4.8x° = 168°
x° = 35°
x = 35
(d) (0.5x + 14)° + (x + 15)° + (0.2x + 15)°
= 180° (adj. /s on a str. line)
0.5x° + x° + 0.2x° + 14° + 15° + 15° = 180°
1.7x° + 44° = 180°
1.7x° = 136°
x° = 80°
x = 80
5. (a) 3x° + (7x – 20)° = 180° (adj. /s on a str. line)
3x° + 7x° = 180° + 20°
10x° = 200°
x° = 20°
3x° + y° = 180° (adj. /s on a str. line)
3(20°) + y° = 180°
60° + y° = 180°
y° = 180° – 60° = 120°
x = 20 and y = 120
(b) (4x – 5)° + (8x – 41)° + 3x° + (3x + 10)°
= 360° (/s at a point)
4x° + 8x° + 3x° + 3x° – 5° – 41° + 10° = 360°
18x° – 36° = 360°
18x° = 360° + 36°
= 396°
x° = 22°
x = 22
(c) y° + 70° = 180° (adj. /s on a str. line)
y° = 180° – 70°
= 110°
28° + (3x – 5)° + 70° = 180° (adj. /s on a str. line)
3x° + 28° – 5° + 70° = 180°
3x° + 93° = 180°
3x° = 180° – 93°
= 87°
x° = 29°
x = 29 and y = 110
6. (a) Draw a line PQ through E that is parallel to AB and
CD.
A
C
P
44°
y°
z°
x°
83°
B
D
Q
E
z° = 44° (alt. /s, PQ // CD)
y° = 83° – 44°
= 39°
x° = y° = 39° (alt. /s, PQ // AB)
x = 39
(b) Draw a line PQ through E that is parallel to AB and
CD.
A
C
P
41°
y°
z°
x°
56°
B
D
Q
E
41° + z° = 180° (int. /s, PQ // CD)
z° = 180° – 41°
= 139°
56° + y° = 180° (int. /s, PQ // AB)
y° = 180° – 56°
= 124°
x° = y° + z°
= 124° + 139°
= 263°
x = 263
107

1
(c) Draw a line PQ through E that is parallel to AB and
CD.
A
C
P
145°
y°
z°
x°
123°
B
D
Q
E
123° + y° = 180° (int. /s, PQ // AB)
y° = 180° – 123°
= 57°
145° + z° = 180° (int. /s, PQ // CD)
z° = 180° – 145°
= 35°
x° = y° + z°
= 57° + 35°
= 92°
x = 92
(d) Draw a line PQ through E that is parallel to AB and
CD.
A
C
P
130°
y°
z°
x°
85°
B
D
Q
E
130° + z° = 180° (int. /s, PQ // CD)
z° = 180° – 130°
= 50°
z° + y° = 85°
y° = 85° – z°
= 85° – 50°
= 35°
y° + x° = 180° (int. /s, PQ // AB)
x° = 180° – y°
= 180° – 35°
= 145°
x = 145
(e)
A
(4x + 89)°
(7x + 14)°
B
E
G
C D
F
/CGE = (4x + 89)° (vert. opp. /s)
(4x + 89)° + (7x + 14)° = 180° (int. /s,
AB // CD)
4x° + 7x° + 89° + 14° = 180°
11x° + 103° = 180°
11x° = 180° – 103°
= 77°
x° = 7°
x = 7
(f) Draw a line PQ through E that is parallel to AB and
CD.
A
(2x + 10)°
(3x – 14)°
C
P
y°
z°
266°
B
Q
D
E
/AEC = 360° – 266° = 94° (/s at a point)
y° + (2x + 10)° = 180°
y° = 180° – (2x + 10)°
= 180° – 2x° – 10°
= 170° – 2x°
z° + (3x – 14)° = 180°
z° = 180° – (3x – 14)°
= 180° – 3x° + 14°
= 194° – 3x°
y° + z° = 94°
170° – 2x° + 194° – 3x° = 94°
2x° + 3x° = 170° + 194° – 94°
5x° = 270°
x° = 54°
x = 54
108

1
(g) Draw a line PQ through E that is parallel to AB and
CD.
A
(3x + 7)°
(7x – 30)°
C
P
y°
z°
131°
B
Q
D
E
z° + 131° = 180° (int. /s, PQ // CD)
z° = 180° – 131°
= 49°
y° + z° + (7x – 30)° = 360° (/s at a point)
y° + 49° + (7x – 30)° = 360°
y° = 360° – 49° – (7x – 30)°
= 360° – 49° – 7x° + 30°
= 341° – 7x°
(3x + 7)° + y° = 180° (int. /s, PQ // AB)
(3x + 7)° + 341° – 7x° = 180°
3x° + 7° + 341° – 7x° = 180°
4x° = 168°
x° = 42°
x = 42
(h) Draw a line PQ through E, and a line SR through
F, that is parallel to AB and CD.
C
Q
y°
w°
118°
87°
34°
E
F
A
P
B
D
R
S x°
118° + w° = 180° (int. /s, PQ // AB)
w° = 180° – 118°
= 62°
w° + y° = 87°
y° = 87° – w°
= 87° – 62°
= 25°
/RFE + y° = 180° (int. /s, SR // PQ)
/RFE = 180° – y°
= 180° – 25° = 155°
/CFR + 34° = 180° (int. /s, SR // CD)
/CFR = 180° – 34°
= 146°
x° = 155° + 146° = 301°
x = 301
7. (a)
A
J H
C
E F
D
z° 58°
316°
y°
x°
w°
B
G
w° + 316° = 360° (/s at a point)
w° = 360° – 316° = 44°
w° = x° (alt. /s, EG // HF)
x° = 44°
Extend the line EG to meet the line CD at J.
z° = 58° (corr. /s, JG // HF)
y° = 58° (alt. /s, AB // CD)
x = 44 and y = 58
(b) Extend the line AB to meet the line EC at F.
273°
54°
x° z° y°
w°
A
B
E
C
D
F
w° + 273° = 360° (/s at a point)
w° = 360° – 273° = 87°
y°= w° = 87° (corr. /s, AB // CD)
z° + y°= 180° (adj. /s on a str. line)
z° = 180° – y°
= 180° – 87°
= 93°
x° = 54° + z° (ext. / of BEF)
= 54° + 93°
= 147°
x = 147
109

1
(c) Draw a line PQ through E that is parallel to
AB and CD.
A
Q
P
C
E
D
z°
56°
158°
y°
x°
w°
B
w° + 158° = 360° (/s at a point)
w° = 360° – 158°
= 202°
y°= 56° (alt. /s, PQ // CD)
z° + y° = w° = 202°
z° = 202° – y°
= 202° – 56°
= 146°
x° + z° = 180° (int. /s, PQ // AB)
x° = 180° – z°
= 180° – 146°
= 34°
x = 34
(d) Draw a line PQ through E that is parallel to AB and
CD.
A
C
E
F
D
z°
139°
67°
y°
w°
x°
B
Q
P
w° + 139° = 180° (int. /s, PQ // AB)
w° = 180° – 139°
= 41°
w° + y° = 67°
y° = 67° – w°
= 67° – 41°
= 26°
z° = y° = 26° (alt. /s, PQ // CD)
x° + z° = 180° (adj. /s on a str. line)
x° = 180° – z°
= 180° – 26°
= 154°
x = 154
(e) Draw a line PQ through E that is parallel to AB and
CD.
A
C
E
P Q
D
F
95°
37°
2x°
y°
w°
z°
B
z° = 37° (alt. /s, PQ // CD)
y° + z° = 95°
y° = 95° – z°
= 95° – 37°
= 58°
w° + y° = 180° (int. /s, PQ // AB)
w° = 180° – y°
= 180° – 58°
= 122°
2x° + w° = 360° (/s at a point)
2x° = 360° – w°
= 360° – 122°
= 238°
x° = 119°
x = 119
(f) Draw a line PQ through E that is parallel to AB and
CD.
A
B
P
C
E
Q
D
226°
34° y°
z°
w°
(x + 15)°
w° + 226° = 360° (/s at a point)
w° = 360° – 226°
= 134°
110

1
y° + w° = 180° (int. /s, PQ // DC)
y° = 180° – w°
= 180° – 134°
= 46°
z° = 34° (alt. /s, PQ // BA)
(x + 15)° = y° + z°
(x + 15)° = 46° + 34° = 80°
x° = 80° – 15°
= 65°
x = 65
(g) Draw a line PQ through E that is parallel to AB and
CD.
D
C
A
B
Q
P
E
z°
x°
y°
w°
63°
194°
w° + 194° = 360° (/s at a point)
w° = 360° – 194°
= 166°
y° + w° = 180° (int. /s, PQ // BA)
y° = 180° – w°
= 180° – 166°
= 14°
z° + y° = 63°
z° = 63° – y°
= 63° – 14°
f = 49°
x° = z° = 49° (alt. /s, PQ // DC)
x = 49
(h) Draw a line PQ through E that is parallel to AB and
CD.
28°
249°
B
A C
P
Q
D
(2x + 13)°
w°
y°
z°
w° + 249° = 360° (/s at a point)
w° = 360° – 249°
= 111°
y° + w° = 180° (int. /s, PQ // CD)
y° = 180° – w°
= 180° – 111°
= 69°
z° = 28° (alt. /s, AB // PQ)
(2x + 13)° = y° + z°
(2x + 13)° = 69° + 28° = 97°
2x° = 97° – 13°
= 84°
x° = 42°
x = 42
8. (a)
z°
a°
w°
A C
F
E
D
B
58°
(y + 15)°
(2x + 12)°
w° + 58° = 180° (adj. /s on a str. line)
w°= 180° – 58°
= 122°
a°= w° = 122° (alt. /s, DF // AC)
(y + 15)° + a° = 360° (/s at a point)
y° + 15° + 122° = 360°
y° = 360° – 15° – 122°
= 223°
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1
z° + a° = 180° (int. /s, AB // CE)
z° = 180° – a°
= 180° – 122°
= 58°
(2x + 12)° + z° = 360° (/s at a point)
2x° + 12° + 58° = 360°
2x° = 360° – 12° – 58°
= 290°
x° = 145°
x = 145 and y = 223
(b) Draw a line PQ through C that is parallel to ED
and AB.
x° z°
w°
y°
114°
118°
P
F
E
D
C
Q
B
A
z° + 114° = 180° (int. /s, PQ // AB)
z° = 180° – 114°
= 66°
w° + z° = 118°
w° = 118° – 66°
= 52°
y° + w° = 180° (int. /s, PQ // ED)
y° + 52° = 180°
y° = 180° – 52°
= 128°
Draw another line SR through B that is parallel to
AF and CD.
x°
b° a°
y°
114°
118°
S
F
E
D
C
R
B
A
a° + 118° = 180° (int. /s, SR // DC)
a° = 180° – 118°
= 62°
b° + a° = 114°
b° = 114° – a°
= 114° – 62°
= 52°
x° + b° = 180° (int. /s, SR // FA)
x° = 180° – b°
= 180° – 52°
= 128°
x = y = 128
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1
Advanced
9. (i)
B
A
S
R
J
I
P
Q
K
G
F E
H
D
C
70°
25°
125°
80°
/ADE = 70° (alt. /s, AB // ED)
/FED + /ADE = 180° (int. /s, FE // AD)
/FED = 180° – /ADE
= 180° – 70°
= 110°
/CED + /FEB = /FED
/CED = /FED – /FEB
= 110° – 80°
= 30°
(ii) /EFG + /FED = 180° (int. /s, FG // ED)
/EFG = 180° – /FED
= 180° – 110°
= 70°
(iii)/HGF = /EFG (alt. /s, HG // FE)
= 70°
Draw a line PQ through H that is parallel to FG
and KJ.
/GHP = /HGF = 70° (alt. /s, PQ // FG)
/IHP = 125° – 70°
= 55°
Draw a line SR through I that is parallel to PQ
and KJ.
/HIR = /IHP = 55° (alt. /s, SR // PQ)
/JIR = /IJK = 25° (alt. /s, SR // KJ)
/HIJ = 55° + 25°
= 80°
Reflex /HIJ = 360° – 80°
= 280°
10. Draw a line PQ through F that is parallel to AB and
CD.
C
F
E
P
A
D
B
Q
z°
a°
x°
b°
y°
w°
a° = z° (alt. /s, PQ // CD)
b° + a° = y°
b° = y° – a°
= y° – z°
Draw a line SR through E that is parallel to AB,
PQ and CD.
C
F
E
P
S
A
D
B
Q
R
z°
a°
x°
b°
y°
w°
d°
c°
c° + b° = 180° (int. /s, SR // PQ)
c° = 180° – b°
= 180° – (y° – z°)
= 180° – y° + z°
d° + w° = 180° (int. /s, SR // AB)
d° = 180° – w°
x° = d° + c°
= 180° – w° + 180° – y° + z°
= 360° – w° – y° + z°
x = 360 – w – y + z
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New Trend
11. (i) WPX = 180° − 65° − (180° − 145°) (vert. opp.
/s, adj. /s on a str. line, / sum of )
= 80°
(ii) Reason 1
Converse of interior angles theorem
Since WYZ + YWX = 180°, then AB // CD
(converse of int. /s)
Reason 2
Converse of corresponding angles postulate
PWX = 180° – 145° (adj. /s on a str. line)
= 35°
Since PWX = WYZ, then AB // CD (converse of
corr. /s)
(iii)DZR = BXZ (corr. /s, AB // CD)
= 65°
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Chapter 11 Triangles, Quadrilaterals and
Polygons
Basic
1. (a) 2x° + 46° + 82° = 180° ( sum of )
2x° = 180° – 46° – 82°
= 52°
x° = 26°
x = 26
(b) x° + 58° + 58° = 180° ( sum of )
x° = 180° – 58° – 58°
= 64°
x = 64
(c) x° + x° + 70° = 180°
2x° = 180° – 70°
= 110°
x° = 55°
x = 55
(d) 3x° = 63°
x° = 21°
x = 21
(e) 3y° = 48° (base s of isos. )
y° = 16°
2x°+ 3y° + 48° = 180° ( sum of )
2x° = 180° – 3y°– 48°
= 180° – 3(16°) – 48°
= 180° – 48° – 48°
= 84°
x° = 42°
x = 42 and y = 16
2. (a) x° + 39° = 123° (ext.  of )
x° = 123° – 39°
= 84°
x = 84
(b) y° + 40° = 180° (adj. s on a str. line)
y° = 180° – 40°
y° = 140°
4x° + 3x° = y° = 140° (ext.  of )
7x° = 140°
x° = 20°
x = 20 and y = 140
(c) 26° + 26° = x° (ext.  of )
x° = 52°
x = 52
(d) CAD = 180° – 110° (adj. s on a str. line)
= 70°
CDA = CAD = 70° (base s of isos. ACD)
x° + 70° = 110° (ext.  of )
x° = 110°– 70°
= 40°
x = 40
(e) EAD = x° (vert. opp. s)
x° + 72° + 50° = 180° ( sum of )
x° = 180° – 72° – 50°
= 58°
x = 58
(f) DAC = 60° (s of equilateral ACD)
2y° + 2y° = 60° (base s of isos. ACB,
4y° = 60° ext.  of )
y° = 15°
y = 15
3. (a) CAB = 46° (alt. s, DE // AB)
x° + 46° = 91°(ext.  of ACB)
x° = 91°– 46°
= 45°
x = 45
(b) CBA = 3x° (alt. s, CD // AB)
3x° + 2x° + 55° = 180° ( sum of ACB)
5x° = 180° – 55°
= 125°
x° = 25°
x = 25
(c) BDA = x° (base s of isos. ABD)
x° + x° + x° + 63° = 180° ( sum of ADC)
3x° = 180° – 63° = 117°
x° = 39°
x = 39
(d) DBA = 58° (alt. s, DE // AB)
x° + 58° = 79° (ext.  of ACB)
x° = 79° – 58°
= 21°
y°+ 79° + 3x° = 180° ( sum of ACD)
3x° + y° = 180° – 79° = 101°
y° = 101° – 3x°
= 101° – 3(21°)
= 38°
x = 21 and y = 38
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4. (a) 2x° + 62° = 134° (ext.  of )
2x° = 134° – 62°
= 72°
x° = 36°
BCE = 2x° (alt. s, CE // AB)
y° + 134° + 2x° = 360° (s at a point)
y° = 360° – 134° – 2(36°)
= 360° – 134° – 72°
= 154°
x = 36 and y = 154
(b) ACD = 180° – 109° (int. s, ED // AF)
= 71°
x° + 24° = 71° (ext.  of ABC)
x° = 71° – 24°
= 47°
x = 47
(c) y° + 63° = 142° (ext.  of )
y° = 142° – 63°
= 79°
ADF + 63° = 180° (int. s, EF // AC)
ADF = 180° – 63°
= 117°
x° = ADF = 117° (vert. opp. s)
x = 117 and y = 79
(d) DEC = y° (alt. s, ED // AC)
4x° + y° = 180° (adj. s on a str. line)
y° = 180° – 4x°
ECD= 180° – y° – 36°
= 144° – y°
144° – y° + 2x° = 4x° (ext.  of DEC)
144° – (180° – 4x°) + 2x° = 4x°
2x° = 36°
x° = 18°
y° = 180° – 4(18°)
= 108°
x = 18 and y = 108
5. (i) BAC = 36° (base s of isos. ABC)
ACD= ABC + BAC (ext.  of ABC)
= 36°+ 36°
= 72°
ADC = ACD = 72° (base s of isos. ACD)
CAD = 180° – 72° – 72° ( sum of ACD)
= 36°
(ii) ADE = CAD + ACD (ext.  of ACD)
= 72° + 36°
= 108°
6. (a) x° + 29° = 90° (DAB is a right angle)
x° = 90° – 29°
= 61°
y° = BAC = 29° (alt. s, DC // AB)
x = 61 and y = 29
(b) x° =
180° – 118°
2
(base s of isos. )
= 31°
CBD + 31° = 90° (CBA is a right angle)
CBD = 90° – 31°
= 59°
y°+ 59° = 118° (ext.  of )
y° = 118° – 59°
= 59°
x = 31 and y = 59
(c) x° + x° = (3x – 18)° (ext.  of )
2x° = 3x° – 18°
x° = 18°
y° + x° + 90° – x° + x°= 180° ( sum of ABD)
y° = 180° – 90° – x°
= 180° – 90° – 18°
= 72°
x = 18 and y = 72
(d) 2x° + 2x° = 180° – (162 – 3x)° (ext.  of )
4x° = 180° – 162°+ 3x°
x° = 18°
DAC = 90° – 2(18°)
= 54°
y° + 54° = (162 – 3x)° (ext.  of )
y° = (162 – 3x)° – 54°
= (162 – 3(18))° – 54°
= 108° – 54°
= 54°
x = 18 and y = 54
7. (a) y° = 120° (opp. s of //gram)
x° + 24° + y° = 180° ( sum of ABC)
x° = 180° – 24° – y°
= 180° – 24° – 120°
= 36°
x = 36 and y = 120
(b) 7x° + 5x° = 180° (int. s, DC // AB)
12x° = 180°
x° = 15°
2y° = 5x° (opp. s of //gram)
2y° = 5(15°)
2y° = 75°
y° = 37.5°
x = 15 and y = 37.5
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(c) DAB = 180° – 68° (int. s, AD // BC)
= 112°
x° + 112° = 139° (ext.  of )
x° = 139° – 112°
= 27°
y° + x° = 68° (opp. s of //gram)
y° = 68° – x°
= 68° – 27°
= 41°
x = 27 and y = 41
8. (a) y° =
180° – 58°
2
(base s of isos. ABC)
= 61°
x° = 180° – 31° – 31° (base s of isos. ACD)
= 118°
x = 118 and y = 61
(b) x° + 33° + 56° = 180° ( sum of ABD)
x° = 180° – 33° – 56°
= 91°
y° = 33°
x = 91 and y = 33
(c) (x + 5)° + 90° + 28° = 180° ( sum of )
x° + 5° + 90° + 28° = 180°
x° = 180° – 5° – 90° – 28°
= 57°
(y – 6)° + 47° + 90° = 180°( sum of )
y° – 6° + 47° + 90° = 180°
y° = 180° + 6° – 47° – 90°
= 49°
x = 57 and y = 49
(d) (x – 5)° + 63° + 90° = 180° ( sum of )
x° – 5° + 63° + 90° = 180°
x° = 180° + 5° – 63° – 90°
= 32°
(2y – 3)° + 37° + 90° = 180°( sum of )
2y° – 3° + 37° + 90° = 180°
2y° = 180° + 3° – 37° – 90°
= 56°
y° = 28°
x = 32 and y = 28
9. (a) Sum of interior angles of a polygon with 7 sides
= (n – 2) × 180°
= (7 – 2) × 180°
= 900°
(b) Sum of interior angles of a polygon with 17 sides
= (n – 2) × 180°
= (17 – 2) × 180°
= 2700°
(c) Sum of interior angles of a polygon with 22 sides
= (n – 2) × 180°
= (22 – 2) × 180°
= 3600°
(d) Sum of interior angles of a polygon with 30 sides
= (n – 2) × 180°
= (30 – 2) × 180°
= 5040°
10. (a) Sum of interior angles of a polygon with 4 sides
= (n – 2) × 180°
= (4 – 2) × 180°
= 360°
a° + 125° + 65° + 92° = 360°
a° = 360° – 125° – 65° – 92°
= 78°
a = 78
(b) Sum of interior angles of a polygon with 4 sides
= (n – 2) × 180°
= (4 – 2) × 180°
= 360°
2b° + 105° + 75° + b° = 360°
2b° + b° = 360° – 105° – 75°
3b° = 180°
b° = 60°
b = 60
(c) Sum of interior angles of a polygon with 5 sides
= (n – 2) × 180°
= (5 – 2) × 180°
= 540°
c° + (2c – 15)° + 130° + 65° + 120° = 540°
c° + 2c° = 540° + 15° – 130° – 65° – 120°
3c° = 240°
c° = 80°
c = 80
11. Let the number of sides of the regular polygon be n.
Size of each interior angle =
(n – 2) × 180°
n
(a)
(n – 2) × 180°
n
= 108°
(n – 2) × 180 = 108n
180n – 108n = 2 × 180
72n = 360
n = 5
(b)
(n – 2) × 180°
n
= 156°
(n – 2) × 180 = 156n
180n – 156n = 2 × 180
24n = 360
n = 15
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Size of each exterior angle =
360°
n
.
(a)
360°
n
= 5°
5n = 360
n = 72
(b)
360°
n
= 6°
6n = 360
n = 60
(c)
360°
n
= 8°
8n = 360
n = 45
(d)
360°
n
= 18°
18n = 360
n = 20
13. (a) Size of each exterior angle =
360°
6
= 60°
(b) Size of each exterior angle =
360°
8
= 45°
(c) Size of each exterior angle =
360°
24
= 15°
(d) Size of each exterior angle =
360°
72
= 5°
14. (a) The sum of interior angles of the polygon is 1620°.
i.e. (n – 2) × 180° = 1620°
Number of sides of the polygon
=
1620
180
+ 2 = 11
(b) The sum of interior angles of the polygon is 3600°.
i.e. (n – 2) × 180° = 3600°
=
3600
180
+ 2 = 22
(c) The sum of interior angles of the polygon is 4500°.
i.e. (n – 2) × 180° = 4500°
=
4500
180
+ 2 = 27
(d) The sum of interior angles of the polygon is 7020°.
i.e. (n – 2) × 180° = 7020°
=
7020
180
+ 2 = 41
15. (i) The sum of exterior angles of a triangle = 360°.
(2x + 10)° + (3x – 5)° + (2x + 40)° = 360°
2x° + 3x° + 2x° = 360° – 10° + 5° – 40°
7x° = 315°
x° = 45°
x = 45
(ii) The largest exterior angle gives the smallest
interior angle.
The largest exterior angle
= (3x – 5)° or (2x + 40)°
= (3 × 45 – 5)° or (2 × 45 + 40)°
= 130°
The smallest interior angle = 180° – 130° = 50°
(iii) The smallest exterior angle gives the largest
interior angle.
The smallest exterior angle = (2x + 10)°
= (2 × 45 + 10)°
= 100°
The largest interior angle = 180° – 100° = 80°
16. (i) Sum of interior angles of a quadrilateral
= (n – 2) × 180°
= (4 – 2) × 180°
= 360°
(2x + 15)° + (2x – 5)° + (3x + 75)° + (3x – 25)°
= 360°
2x° + 2x° + 3x° + 3x° = 360° – 15° + 5° – 75°
+ 25°
10x° = 300°
x° = 30°
x = 30
(ii) Smallest interior angle
= (2x – 5)°
= (2 × 30 – 5)°
= 55°
(iii) Largest interior angle gives the smallest exterior
angle
Largest interior angle
= (3x + 75)°
= (3 × 30 + 75)°
= 165°
Smallest exterior angle = 180° – 165° = 15°
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17. (i) Sum of interior angles of a hexagon
= (n – 2) × 180°
= (6 – 2) × 180°
= 720°
(2x + 17)° + (3x – 25)° + (2x + 49)° + (x + 40)° +
(4x – 17)° + (3x – 4)° = 720°
2x° + 3x° + 2x° + x° + 4x° + 3x°
= 720° – 17° + 25° – 49° – 40° + 17° + 4°
15x° = 660°
x° = 44°
x = 44
(ii) Smallest interior angle of the hexagon
= (x + 40)°
= (44 + 40)°
= 84°
(iii) The largest interior angle gives the smallest
exterior angle.
The largest interior angle
= (4x – 17)°
= (4 × 44 – 17)°
= 159°
The smallest exterior angle = 180° – 159° = 21°
18. (i) The sum of exterior angles of a pentagon = 360°.
2x° + (2x + 5)° + (3x + 10)° + (3x – 15)° + (x + 30)°
= 360°
2x° + 2x° + 3x° + 3x° + x°
= 360° – 5° – 10° + 15° – 30°
11x° = 330°
x° = 30°
x = 30
(ii) The largest exterior angle gives the smallest
interior angle.
= (3x + 10)°
= (3 × 30 + 10)°
= 100°
The smallest interior angle
= 180° – 100° = 80°
(iii) The smallest exterior angle gives the largest
interior angle.
Smallest exterior angle
= 2x°
= 2(30°) = 60°
The largest interior angle = 180° – 60° = 120°
19. (i) The sum of interior angles of a quadrilateral = 360°
30 parts = 360°
1 part = 12°
9 parts = 12 × 9 = 108°
The largest interior angle = 108°.
(ii) 6 parts = 12° × 6 = 72°
The smallest interior angle = 72°.
= 180° – 72°
= 108°
Intermediate
20. (a) y° = 61° + 59° (ext.  of )
= 120°
GDE = 61° (vert. opp. s)
x°= 69° + 61°
= 130° (ext.  of )
x = 130 and y = 120
(b) x°= 110° + 40° (ext.  of )
= 150°
ADB + x° = 180° (adj. s on a str. line)
ADB = 180° – x°
= 180° – 150°
= 30°
y° = 30° + 90° (ext.  of )
= 120°
x = 150 and y = 120
21. (a) BEF = 180° – 84° = 96° (adj. s on a str. line)
Sum of angles in a quadrilateral is 360°.
x° + 92° + 118° + 96° = 360°
x° = 360° – 92° – 118° – 96°
= 54°
y° + x° + 92° = 180° ( sum of )
y° = 180° – x° – 92°
= 180° – 54° – 92°
= 34°
x = 54 and y = 34
(b) EBA = 53° (corr. s, CD // AB)
y° + 53° = 360° (s at a point)
y° = 360° – 53°
= 307°
FED = 53° (base s of isos. )
x°= 53° + 53° (ext.  of , corr. s)
= 106°
x = 106 and y = 307
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(c) x° + 25° + 121° = 180° (corr. s, adj.
s on a str. line)
x° = 180° – 25° – 121°
= 34°
y° + x° + 78° = 180°( sum of )
y° = 180° – x° – 78°
= 180° – 34° – 78°
= 68°
x = 34 and y = 68
(d) x° + 124° = 180° (corr. s, adj. s on a str. line)
x° = 180° – 124°
= 56°
ABD = 103° (corr. s)
y° + 103° = 180° (adj. s on a str. line)
y° = 180° – 103°
= 77°
z° + y° = 124° (ext.  of )
z° = 124° – y°
= 124° – 77°
= 47°
x = 56, y = 77 and z = 47
(e) Draw a line PQ through C that is parallel to AE
and BD.
132°
A
E
C D
B
Q
3x°
P
y° w°
2x°
w° = 2x° (opp. s of //gram)
y° = 132° – w°
= 132° – 2x°
3x° + y° = 180° (int. s, EA // QP)
3x° + 132 – 2x° = 180°
x° = 180° – 132°
= 48°
x = 48
(f) Draw a line PQ through E that is parallel to AB
and CD.
Q
P
A
B
C
E
x°
z°
y°
a°
w°
D
247°
245°
w°+ 247° = 360°(s at a point)
w° = 360° – 247°
= 113°
a° + w° = 180°(int. s, AB // PQ)
a° = 180° – w°
= 180° – 113°
= 67°
y° + 245° = 360° (s at a point)
y° = 360°– 245°
= 115°
z° + y° = 180° (int. s, CD // PQ)
z° = 180° – y°
= 180° – 115°
= 65°
x° = 180° + 67°+ 65°
= 312°
x = 312
22. Sum of angles in a triangle = 180°
(2x – 5)° + 3x –
1
2






°
+ 30 –
1
2
x






°
= 180°
2x° + 3x° –
1
2
x° = 180° + 5° +
1
2
°
– 30°
4.5x° = 155.5°
x° = 34
5
9
°
x = 34
5
9
23. (a) 64° + ADC = 180° (int. s, DC // AB)
ADC = 180° – 64°
= 116°
x° =
1
2
× 116°
= 58°
y° = x° (alt. s, DA // CB)
= 58°
x = y = 58
120

1
(b) 108° + BAD = 180° (int. s, BC // AD)
BAD = 180° – 108°
= 72°
x° =
1
2
× 72°
= 36°
y° = x° (alt. s, DC // AB)
= 36°
x = y = 36
(c) (y – 5)° = 36° (alt. s, DC // AB)
y° = 36° + 5°
y° = 41°
x° = (y – 5)°
= (41 – 5)°
x° = 36°
x = 36 and y = 41
(d) (3x – 30)° = (2x + 15)° (opp. s of //gram)
3x° – 2x° = 15° + 30°
x° = 45°
(3x – 30)° + BCD = 180° (int. s, BA // CD)
BCD = 180° – (3x – 30)°
= 180° – (3 × 45 – 30)
= 75°
y° =
1
2
× 75°
= 37.5°
z° = y° (alt. s, BA // CD)
= 37.5°
x = 45, y = 37.5 and z = 37.5
24.
110°
X Q
R
P
S
(i) PXS =
180° – 110°
2
= 35° (adj. s on a str. line)
PSX = 180° – 90° – 35° (sum of )
= 55°
(ii) XRS = QXR (alt. s PQ // SR)
= 35°
25. P
S
Q
R
108°
40°
(i) PSQ = 40°(alt. s, PS // QR)
(ii) PSR = 180° – 108° (int. s, PQ // SR)
= 72°
QSR = 72° – 40°
= 32°
26.
114°
A
C
D B
(i) ABD =
114°
2
= 57°
(ii) ACD = ACB
= 180° – 90° – 57°
(The diagonals of a rhombus bisect at 90°)
= 33°
27. Q
R
P
66°
S
(i) QRS = 66° (PS = QR)
(ii) SPQ = 180° – 66° (int. s, PQ // SR)
= 114°
PQS = PSQ
=
180° – 114°
2
(base s of isos. PQS)
= 33°
121

1
28.
66°
42°
B
C
A
D
(i) ACD =
180° – 66°
2
(base s of isos. ACD)
= 57°
(ii) ABC = 180° – 42° – 42° (base s of isos. ABC)
= 96°
29. (i) 115° + PQR = 180°(int. s, PQ // SR)
PQR = 180° – 115°
= 65°
UQV = 65° – 35°
= 30°
TVQ + 30° = 110° (ext.  of QUV)
TVQ = 110° – 30°
= 80°
(ii) SXY = 110° (corr. s, TV // WY)
110° + SXW = 180° (adj. s on a str. line)
SXW = 180° – 110°
= 70°
30. (i) 66° + ADC = 180° (int. s, AB // DC)
ADC = 180° – 66°
= 114°
CDQ + 145° + 114° = 360° (s at a point)
CDQ = 360° – 145° – 114°
= 101°
(ii) 114° + BCD = 180° (int. s, AD // BC)
BCD = 180° – 114°
= 66°
(iii) DCP = CDQ = 101° (alt. s, PC // DQ)
PCB = 101° – 66°
= 35°
31. (i) DAC = 33°
QBC = 66° (corr. s, AD // BC)
(ii) 66° + ADC = 180° (int. s, AB // DC)
ADC = 180° – 66°
= 114°
ADB = 114° ÷ 2 = 57°
DBC = ADB (alt. s, AD // BC)
= 57°
(iii) BCD = 66° (opp. s in a //gram)
72° + 66° + Reflex BCR = 360° (s at a point)
Reflex BCR = 360° – 72° – 66°
= 222°
32. (i) AEB= 180° – 53° – 90° ( sum of )
= 37°
PEQ = AEB (vert. opp. s)
= 37°
(ii) QED = 90° – 37° = 53°
EDR = 180° – 53° (int. s, QE // RD)
= 127°
(iii) EDC = 360° – 126° – 127° (s at a point)
= 107°
BCD + 107° = 180° (int. s, ED // AC)
BCD = 180° – 107°
= 73°
33. (i) ABQ = 45° – 21°
= 24°
BAQ =
180° – 24°
2
(base s of isos. ABQ)
= 78°
(ii) Since BQ = BA, BQ = BC,
ABC = 45° + 21°
= 66°
BCQ =
180° – 66°
2
(base s of isos. QBC)
= 57°
DCQ = 90° – 57°
= 33°
(iii) DPC = 180° – 45° – 33° ( sum of )
= 102°
QPB = 102° (vert. opp. s)
34. (i) QAD = 60° (s of equilateral )
BAD + 90° + 135° + 60° = 360° (s at a point)
BAD = 360° – 90° – 135° – 60°
= 75°
(ii) Sum of interior angles of a quadrilateral = 360°.
CDA + 106° + 100° + 75° = 360° (s at a point)
CDA = 360° – 106° – 100° – 75°
= 79°
(iii) PDQ + 60° + 79° = 180° (adj. s on a str. line)
PDQ = 180° – 60° – 79°
= 41°
122

1
35. (i) ABC = 180° – 68° – 68° (base s of isos.
= 44° ABC)
(ii) ACD = 68°
ADC = 180° – 68° – 68° ( sum of )
= 44°
60° + 90° + ADP = 180° ( sum of )
ADP = 180° – 90° – 60°
= 30°
PDQ + ADP = ADC
PDQ = ADC – ADP
= 44° – 30°
= 14°
(iii) DQR = PDQ + DPR
= 14° + 90°
= 104°
36. (i) TBD = 180° – 81° (int. s, BD // TE)
= 99°
DBC = 61° (alt. s, ED // BC)
ABT = 180° – 99° – 61° (adj. s on a str. line)
= 20°
(ii) TED = 180° – 61° (int. s, DB // ET)
= 119°
(iii) BCD = 180° – 61° – 61°
= 58°
37. (i) BAD = 180° – 88° (int. s, BC // AD)
= 92°
DAE = 180° – 92° (adj. s on a str. line)
= 88°
AED =
180° – 88°
2
(base s of isos. ADE)
= 46°
(ii) FAB = 180° – 162° (adj. s on a str. line)
= 18°
FAD = FAB + BAD
= 18° + 92°
= 110°
(iii) ADC = 88° (opp. s in a //gram)
Sum of angles in a quadrilateral = 360°
FCD + 48° + 110° + 88° = 360°
FCD = 360° – 48° – 110° – 88°
= 114°
BCF = FCD – BCD
= 114° – 92°
= 22°
38. Let the number of sides of the polygon be n.
Sum of interior angles = (n – 2) × 180°
Sum of exterior angles = 360°
(n – 2) × 180° = 2 × 360°
180n – 360 = 720
180n = 720 + 360
= 1080
n = 6
The number of sides of the polygon is 6.
(n – 2) × 180°
n
Size of each exterior angle =
360°
n
(n – 2) × 180°
n
= 35 ×
360°
n
(n – 2) × 180° = 35 × 360°
180n – 360 = 12 600
180n = 12 600 + 360
= 12 960
n = 72
40.
T
C
D
E
A B
Size of each exterior angle of the pentagon =
360°
5
= 72°
CBT = BCT = 72°
BTC = 180° – 72° – 72° ( sum of )
= 36°
41. (i) Size of each interior angle =
(12 – 2) × 180°
12
= 150°
ABC = 150°
(ii) BCA =
180° – 150°
2
= 15°
ACD + BCA = BCD = 150°
ACD = 150° – BCA
= 150° – 15°
= 135°
123

1
42. (a) Let the number of sides of the polygon be n.
Sum of interior angles = (n – 2) × 180°
(n – 2) × 180° = 124° + (n – 1) × 142°
180n – 360 = 124 + 142n – 142
180n – 142n = 124 – 142 + 360
38n = 342
n = 9
The number of sides of the polygon is 9.
(b) Size of each angle in a pentagon ABCDE
=
(5 – 2) × 180°
5
= 108°
Size of each angle in a hexagon CDZYXW
=
(6 – 2) × 180°
6
= 120°
(i) WCD = size of each angle in a hexagon
= 120°
(ii) BCD = size of each angle in a pentagon
= 108°
(iii) Since CB = CW, BCW is an isosceles triangle.
BCW = 360°– 108° – 120° (s at a point)
= 132°
CBW =
180° – 132°
2
(base s of isos. BCW)
= 24°
43. (i) CBA = 180° – 18° (adj. s on a str. line)
= 162°
(ii) Size of each interior angle =
(n – 2) × 180°
n
162 =
(n – 2) × 180°
n
162n = (n – 2) × 180°
162n = 180n – 360
180n – 162n = 360
18n = 360
n = 20
The value of n is 20.
(iii) BCY =
180° – 162°
2
= 9°
CBY = (360° – 162° – 162°) ÷ 2
= 18°
BYC = 180° – 18° – 9° ( sum of )
= 153°
44. (i) Size of each interior angle =
(n – 2) × 180°
n
.
174° =
(n – 2) × 180°
n
174n = (n – 2) × 180
174n = 180n – 360
180n – 174n = 360
6n = 360
n = 60
The value of n is 60.
(ii) PBC =
(5 – 2) × 180°
5
(base s of isos. PBQ)
= 108°
ABP = 360° – 108° – 174° (s at a point)
= 78°
(iii) PBQ =
180° – 108°
2
= 36°
QBC = 108° – 36°
= 72°
BQC = 180° – 72° – 72° (base s of isos. BQC)
= 36°
(iv) DCR = 360° – 108° – 174° (s at a point)
= 78°
CDR =
180° – 78°
2
(base s of isos. CDR)
= 51°
Advanced
45. A
C
B
D
e°
h°
i°
c°
b°
a°
d°
f°
x°
g°
F
F
E
(i) For BCD,
a° = x° (base s of isos. BCD)
b° = 180° – x° – x° ( sum of BCD)
= 180° – 2x°
For ADB,
c° = 180° – b° (adj. s on a str. line)
= 180° – (180° – 2x°)
= 180° – 180° + 2x°
= 2x°
e° = c° (base s of isos. ADB)
d° = 180° – c° – e° ( sum of ABD)
= 180° – 2x° – 2x°
= 180° – 4x°
124

1
For ADE,
f° = 180° – x°– (180° – 4x°) (adj. s on a str. line)
= 180° – x° – 180° + 4x°
= 3x°
g° = 180° – 3x° – 3x° ( sum of ADE)
= 180° – 6x°
For AEF,
h° = 180° – 2x° – (180° – 6x°) (adj. s on a str. line)
= 180° – 2x° – 180° + 6x°
= 4x°
i° = h° = 4x° (base s of isos. AEF)
i° + x° + 90° = 180° ( sum of CEF)
4x° + x° + 90° = 180°
5x° = 180°– 90° = 90°
x° = 18°
x = 18
(ii) Let n be the number of isosceles triangles that can be
formed.
From (i),
(n + 1)x° + 90° = 180°
When x° = 5°,
(n + 1)5° + 90° = 180°
5(n + 1) = 90
n + 1 = 18
n = 17
There are 17 isosceles triangles that can be formed
when x = 5.
46. (i) Let polygon A have a sides and polygon B have b
sides.
360
a
+
360
b
= 80
360(a + b) = 80ab
9(a + b) = 2ab
9a + 9b = 2ab
9a = 2 − 9b
b =
9a
2a – 9
A possible solution is polygon A has 5 sides and
polygon B has 45 sides.
(ii) Sum of exterior angles of any polygon = 360°.
When the exterior angle of their shared side decreases,
the corresponding exterior angle of each polygon
decreases.
Number of sides =
360°
size of each exterior angle
Number of sides increases as size of each interior
angle in both polygons decreases.
New Trend
47. Let the first angle be x°.
x + (x − 10) + 4(x − 10) + x +
120
100
x





 = 360
x + x − 10 + 4x − 40 + 2.2x = 360
8.2x = 410
x = 50
The angles of the quadrilateral are 50°, 40°, 160° and
110°.
48. Sum of interior angles of a polygon with 6 sides
= (n – 2) × 180°
= (6 – 2) × 180°
= 720°
d° + 125° + d° + 3d° + 70° + 110° = 720°
d° + d° + 3d° = 720° – 125° – 70° – 110°
5d° = 415°
d° = 83°
d = 83
49. Size of each interior angle of the decagon
=
(10 – 2) × 180°
10
= 144°
Size of each interior angle of the hexagon
=
(6 – 2) × 180°
6
= 120°
x° = 360° − 144° − 120° (s at a point)
= 96°
x = 96
50. x° + 63° = 180° (int. s, AD // BC)
x° = 180° – 63°
= 117°
y° = 61° (alt. s, DC // AB)
AD = BC = 7.5 cm
z = 7.5
x = 117, y = 61 and z = 7.5
51. (a) Size of each interior angle of a regular 24-sided
polygon =
(24 – 2) × 180°
24
= 165°
(b) Let the number of sides of the polygon be n.
Sum of interior angles = (n − 2) × 180°
(n − 2) × 180° = 172° + 2(158°) + (n − 3)p°
488 + (n − 3)p = 180n − 360
(n − 3)p = 180n − 848
p =
180n – 848
n – 3
125

1
52. (a) Size of each interior angle =
(n – 2) × 180°
n
150° =
(n – 2) × 180°
n
150n = (n − 2) × 180
150n = 180n − 360
180n − 150n = 360
30n = 360
n = 12
(b) Size of each exterior angle =
360°
n
=
360°
9
= 40°
(n – 2) × 180°
n
(n – 2) × 180°
n
= 165.6°
(n – 2) × 180 = 165.6n
180n – 165.6n = 2 × 180
14.4n = 360
n = 25
126

1
Chapter 12 Geometrical Constructions
Basic
1.
A B
8.4 cm
2.
Q
6 cm
P
127

1
3. R
P
Q
88°
4.
P Q
R
8 cm
76° 58°
(i) PRQ = 46°
(ii) PR = 9.4 cm
(iii) QR = 10.8 cm
128

1
5.
A B
X
C
8 cm
46° 64°
Length of AX = 5.2 cm
6.
A B
C
8.3 cm
9.2 cm 7.9 cm
ABC = 69°
129

1
7.
P Q
R
11.5 cm
10.8 cm
9.5 cm
QPR = 50°
PQR = 61°
8.
P Q
R
8.5 cm
4.6 cm
54°
Length of QR = 6.9 cm.
130

1
9.
X Y
Z
6.6 cm
98°
9.2 cm
YXZ = 37°
10. A
B C
5 cm
7.2 cm 7.2 cm
ABC = 70°
131

1
11.
Q R
P
9.6 cm 9.6 cm
12 cm
PQR = 51°
12.
X Y
Z
6.8 cm
6.8 cm
54°
Length of YZ = 8.0 cm
132

1
13.
A B
C
7 cm
7 cm
60°
60° 60°
7 cm
14.
L
N
M
6.9 cm
74° 49°
Length of LN = 6.2 cm
133

1
15.
P
Q
S
R
4.5 cm
5.6 cm
72°
Length of diagonal PR = 8.2 cm
Length of diagonal QS = 6 cm
16.
A
D
B
C
10.3 cm
6.3 cm
105°
Length of diagonal AC = 13.4 cm
Length of diagonal BD = 10.6 cm
134

1
17.
A
D
B
C
6.4 cm
7.6 cm
115°
Length of BD = 11.8 cm
BDA = 29°
18.
P
S
Q
R
50 mm
68 mm
Length of diagonal PR = 84 mm
Length of diagonal QS = 84 mm
135

1
19.
A
D
B
C
9 cm
7.8 cm
Length of BD = 11.9 cm
ABD = 41°
20.
P
S
Q
R
10.4 cm
32°
Length of PS = 6.5 cm
Length of PR = 12.3 cm
136

1
21.
W
X
Z
Y
7.4 cm
7.4 cm
Length of diagonal WY = 10.5 cm
22.
A
D
C
B 6.5 cm
56°
Length of diagonal BD = 11.5 cm
Length of diagonal AC = 6.1 cm
137

1
23.
Q
R S
P
8.4 cm
75°
Length of diagonal QS = 13.3 cm
24.
H
I
J
K 7.2 cm
110°
Length of diagonal HJ = 11.8 cm
Length of diagonal KI = 8.4 cm
138

1
25.
A
D
C
B
6.4 cm
116°
11.6 cm
Length of AB = 3.8 cm
Length of AD = 10.1 cm
26.
3.6 cm
124°
3.6 cm
6.9 cm
P
S
Q
R
139

1
27.
6 cm
44°
6 cm
9 cm
4.5 cm
A B
C
D
ADC = 82°
28.
Q
68°
9.8 cm
7.2 cm
R
P
S
140

1
29.
D
85°
75°
10.8 cm
7.4 cm
G F
E
Length of GE = 12.5 cm
Length of GF = 8.2 cm
30.
W 11.9 cm
8.2 cm
13.9 cm
Z
Y
X
Length of WZ = 8.1 cm
XWZ = 63°
141

1
Intermediate
31.

6 cm
7 cm
6.5 cm
A
C
X
B
(ii)
(i) ABC = 58°
(ii) Length of BX = 5.6 cm
32.
13 cm
5 cm
(ii)
(ii)
12 cm
D E
F
G
(i) ∠DEF = 67°
(ii) Length of GF = 6.5 cm
142

1
33.
I M J
K
(ii)
64° 55°
8 cm
(i) Length of IK = 7.5 cm
Length of JK = 8.1 cm
(ii) Length of IM = 0.9 cm
34.
8.5 cm
9.2 cm (ii)
75°
D E
G
F
(i) The angle that is facing the longest side is DEF.
The size of DEF = 75°.
(ii) Length of DG = 8.2 cm
143

1
35.
8.5 cm
9.2 cm
(iv)
(iii)
(ii)
75°
D E
X
K
F
(i) Length of EF = 6.3 cm
(ii) Length of DX = 6.2 cm
(iv) Length of DK = 5.4 cm
36.
9.4 cm
(ii)
(iii)
5.2 cm
3.8 cm
80°
A
K
H
B
C
D
(i) Length of AD = 7 cm
BAD = 47°
(ii) Length of HB = 7 cm
(iii) Length of KB = 8.4 cm
(iv) Length of HK = 4.5 cm
144

1
37.
6.8 cm
4 cm
5 cm
(iii)
(iii)
110°
(ii) arc of circle of radius 3.5 cm
S
K
P Q
H
R
(i) Length of QR = 4.8 cm
PQR = 104°
(iii) Length of RH = 4.9 cm
Length of HK = 5.8 cm
38.
A
R P
(ii)
(i)
(iii)
(iv)
Q
C
D
B
68°
10 cm
7.2 cm
(i) Length of diagonal AC = 14.3 cm
(ii) Length of PC = 3.0 cm
(iii) Length of DR = 0.3 cm
(iv) Perpendicular height of Q to the base of the parallelogram
= 5.0 cm
145

1
39. D
X
C
B A
10 cm
(iii)
105° 85°
8.4 cm
10.8 cm
(i) Length of CD = 11.5 cm
(ii) ADC = 82°
(iii) Length of CX = 7.4 cm
146

1
40.
8.5 cm
9.8 cm
42°
110°
6.5 cm
P
S
Q
K
R
H
(ii)
(i) Length of RS = 8.3 cm
(ii) Length of QK = 4.2 cm
Length of HK = 6 cm
(iii) Ratio of QK: KH = 4.2 : 6
= 7 : 10
147

1
41.
7.6 cm
(ii)
(ii)
5.3 cm
110°
82°
105°
A
D
K
B
C
(i) Length of CD = 11.6 cm
Length of BC = 7.6 cm
(ii) Length of AK = 8.7 cm
Length of BK = 6.8 cm
42.
6.2 cm
8.2 cm
38°
7.2 cm
5 cm
(ii)
(i)
(iv)
(iii)
P
S
X
Y
R
Q

(i) Length of SQ = 8.2 cm
PRS = 60°
(ii) Length of PX = 5.3 cm
(iii) Length of RY = 3.7 cm
(iv) Length of XY = 2.3 cm
148

1
43.
6.2 cm
5.7 cm
4.8 cm
(i)
(ii)
(iv)
(iii)
(ii)
85°
112°
P
S
K
R
M
U H
Q
(i) Length of QR = 7.2 cm
Length of QS = 8.1 cm
(ii) Length of SH = 3.6 cm
Length of KQ = 6.6 cm
(iii) Length of RM = 8.4 cm
(iv) Length of UM = 3.7 cm
149

1
Advanced
44.
14 cm
13 cm 12 cm
(ii)
(i)
(i)
(ii)
(ii)
(iii)
A B
X
C
(ii) The length of AX, of BX and of CX = 7.6 cm
150

1
45.
A B
K
C
13.2 cm
14.2 cm
(iii) (b)
(iii) (a)
(iv)
(ii)
(ii)
56°
(i) Length of AC = 12.9 cm
(iii) (a) Shortest distance of K from AB = 4.0 cm
(b) Shortest distance of K from BC = 4.0 cm
151

1 152
New Trend
46.
(iv)
(v)
(iii)
(ii)
X
P
13.4 cm
12.4 cm
6.4 cm
A B
C
(i) The angle that is facing the longest side is ABC.
ABC = 84°
(iv) The point X is equidistant from the points B and C, and equidistant from the lines AB and BC.
(v) Point P is on the perpendicular bisector to the right of the angle bisector, closer to BC than BA.
152

1
Revision Test C1
1. Mr Lee’s salary in February without allowance
=
108
100
× 1800
= $1944
His salary in February with allowance
= 1944 + 20
= $1964
Increase in salary = $1964 – $1800 = $164
Percentage increase
=
164
1800
× 100%
= 9
1
9
%
2. (a) Total parts required to manufacture an article
= 9 + 5 + 3
= 17
Cost of labour
=
9
17
× 918
= $486
(b) (i) Time at which the train arrives at Station B
= 1056 + 1 hour 32 minutes
= 1156 + 32 minutes
= 0000 + 28 minutes
= 0028
(ii) Speed of train
=
Distance
Time taken
=
161
1
32
60
= 105 km/h
3. (a) (i) Time taken to plant a row of lettuce
= 5 × 7
= 35 man-days
Time taken to plant a row a cabbage
= 2 × 4
= 8 man-days
Total time taken to complete the job
= 35 + 8
= 43 man-days
(ii) Total cost to complete the job
= 7 × $40 + 4 × $50
= $280 + $200
= $480
(b) Cost of planting 3 rows of lettuce
= 3 × $40
= $120
Cost of planting the cabbage
= $610 – $120
= $490
Maximum number of rows of cabbage that can
be planted
=
490
50
= 9.8
≈ 9 rows
4. B
A
C
F
D
E
28°
88°
x°
DCF = 28° (alt. s, FC // DE)
BCF = 88° – 28° = 60°
x° = 180° – 60° (int. s, AB // FC)
x° = 120°
∴ x = 120
5. 112° + (27° + x°) = 180° (int. s, CD // AB)
x° = 180° – 112° – 27°
= 41°
49° + y° + x° = 180° ( sum of )
y° = 180° – 49° – 41°
= 90°
∴ x = 41 and y = 90
6. (a) (i) TQR = 180° – 120° (adj. s on a str. line)
= 60°
3x° + 60° = 5x° (ext.  of )
(ii) 60 + 3x = 5x
5x – 3x = 60
2x = 60
x = 30
(b) (i) Sum of angles of a pentagon
= (5 – 2) × 180°
= 540°
3x + 4x + 5x + (3x – 20) + (5x – 50) = 540
3x + 4x + 5x + 3x + 5x – 20 – 50 = 540
20x – 70 = 540
20x = 610
x = 30.5
153

1
(ii) Largest interior angle
= (5x)°
= (5 × 30.5)°
= 152.5°
(iii)Smallest exterior angle
= 180° – 152.5°
= 27.5°
7. (i) and (ii)
C
B
A
X 5 cm
30°
10 cm
(ii) From the diagram,
the length of CX = 5.0 cm.
154

1
Revision Test C2
1. Length of RQ = 10 – 6
= 4 cm
Length of RQ after decrease
= 91% × 4
=
91
100
× 4
= 3.64 cm
Length of PR = 10 – 3.64 = 6.36 cm
Increase in the length of PR
= 6.36 – 6
= 0.36 cm
Percentage increase in the length of PR
=
0.36
6
× 100%
= 6%
2. (i) Extra distance travelled
= 560 – 280
= 280 km
Extra charge
= 280 × 0.25
= $70
Total hire charges
= ($75 × 3) + $25 + $70
= $320
(ii) Charges based on the total distance travelled
= $415 – ($75 × 4) – $25
= $90
Extra distance travelled
=
90
0.25
= 360 km
= 280 + 360
= 640 km
(iii)Amount that is chargeable
= (320 – 280) × 0.25
= $10
Hire amount
= $185 – $25 – $10
= $150
Number of days that he hired the car
=
150
75
= 2 days
3. (a) Hourly rate of the tutor
=
124
2
1
2
= $49.60
Amount charged for a lesson that lasts 3
3
4
hours
= 49.6 × 3
3
4
= $186
(b) 6 printers can print 200 copies in 1
1
2
hours.
1 printer can print 200 copies in 1
1
2
× 6 = 9 hours.
8 printers can print 200 copies in 9 ÷ 8 = 1
1
8
hours.
∴ 8 printers can print 800 copies in
1
1
8
× 4 = 4
1
2
hours.
4. DFE = 360° – 308° (s at a point)
= 52°
x° = 52° (corr. s, CD // EF)
(76 + x)° + y° = 180° (int. s, AB // CD)
y° = 180° – 76° – 52°
= 52°
∴ x = 52 and y = 52
5. P
R
A B
Q
116°
a°
b°
2x°
130°
S
T
Draw a line AB through T that is parallel to PQ
and RS.
116° + b° = 180° (int. s, PQ // AB)
b° = 180° – 116°
= 64°
a° + 64° = 130° (vert. opp. s)
a° = 130° – 64°
= 66°
2x° + 66° = 180° (int. s, AB // RS)
2x° = 180° – 66°
= 114°
x° = 57°
∴ x = 57
155

1
6. (i) If AB // CD, then
DFG = 180° – 125° (adj. s on a str. line)
= 55°
FGB = EFG (corr. s, AB // CD)
= 125°
GJK = 180° – 65° (adj. s, n a str. line)v
= 115°
FKJ = KJB (alt. s, AB // CD)
= 65°
DFG = FGB = 55° + 125°
= 180°
GJK = FKJ = 115° + 65°
= 180°
By the converse of interior angle theorem, AB is parallel to CD.
(ii) x° + KJB = 180° (int. s, Ab // Cd)
x° = 180° – 65°
= 115°
y° = FGB (vert. opp. s)
= 125°
x° + y° = 115° + 125°
= 240°
x + y = 240
7. 88° + 99° + [(n – 2) × 163°] = (n – 2) × 180°
187° + 163n° – 326° = 180n° – 360°
187° – 326° + 360° = 180n° – 163n°
17n° = 221°
n° = 13°
∴ n = 13
156

1
8. (i) and (ii)
D
C
X
A B
9 cm
6 cm
(ii) Length of the perpendicular line from D to AB (DX) = 5 cm
157

1
Chapter 13 Perimeter and Area of Plane Figures
Basic
1. (a) 7.3 cm2
= 7.3 × 10 × 10
= 730 mm2
(b) 4.65 m2
= 4.65 × 10 000
= 46 500 cm2
(c) 3650 mm2
= 3650 ÷ 100
= 36.5 cm2
(d) 200 000 cm2
= 200 000 ÷ 10 000
= 20 m2
(e) 50 000 mm2
= 50 000 ÷ 100 ÷ 10 000
= 0.05 m2
2. (a) Breadth of rectangle =
48
8
= 6 cm
Perimeter of rectangle = 2(6 + 8)
= 28 cm
(b) Breadth of rectangle =
0.9
1.2
= 0.75 m
Perimeter of rectangle = 2(0.75 + 1.2)
= 3.9 m
(c) Length of rectangle =
1.76
0.8
= 2.2 cm
Perimeter of rectangle = 2(0.8 + 2.2)
= 6 cm
3. Perimeter of square = 4 × length of square
48 = 4 × length of square
Length of square =
48
4
= 12 cm
Area of square = 12 × 12
= 144 cm2
4. Circumference of a circle = 2pr
Area of a circle = pr2
Diameter Radius Circumference Area
(a) 2 × 10
= 20 cm
10 cm 2 × 3.142 × 10
= 62.8 cm
(to 3 s.f.)
3.142 × 102
= 314 cm2
(to 3 s.f.)
(b) 2 × 0.7495
= 0.150 m
(to 3 s.f.)
0.471 ÷ (2 × 3.142)
= 0.07495
= 0.0750 m (to 3 s.f.)
0.471 m 3.142 × 0.074952
= 0.0177 m2
(to 3 s.f.)
(c) 1.2 m 1.2 ÷ 2
= 0.6 m
2 × 3.142 × 0.6
= 3.77 m (to 3 s.f.)
3.142 × 0.62
= 1.13 m2
(to 3 s.f.)
(d) 3.999 × 2
= 8.00 cm
(to 3 s.f.)
50.24 ÷ 3.142
= 3.999 cm
= 4.00 cm (to 3 s.f.)
2 × 3.142 × 3.999
= 25.1 cm
(to 3 s.f.)
50.24 cm2
(e) 2 × 11.996
= 24.0 cm
(to 3 s.f.)
452.16 ÷ 3.142
= 11.996 cm
= 12.0 cm (to 3 s.f.)
2 × 3.142 × 11.996
= 75.4 cm
(to 3 s.f.)
452.16 cm2
(f) 2 × 14
= 28 cm
14 cm 2 × 3.142 × 14
= 88.0 cm
3.142 × 142
= 616 cm2
(to 3 s.f.)
(g) 2 × 4.2
= 8.4 cm
4.2 cm 2 × 3.142 × 4.2
= 26.4 cm
(to 3 s.f.)
3.142 × 4.22
= 55.4 cm2
(to 3 s.f.)
(h) 2 × 19.987
= 40.0 m
(to 3 s.f.)
125.6 ÷ (2 × 3.142)
= 19.987 m
= 20.0 m (to 3 s.f.)
125.6 m 3.142 × 19.9872
= 1260 m2
(to 3 s.f.)
(i) 84 mm 84 ÷ 2
= 42 mm
2 × 3.142 × 42
= 264 mm
(to 3 s.f.)
3.142 × 422
= 5540 mm2
(to 3 s.f.)
(j) 2 × 21.0057
= 42.0 cm
(to 3 s.f.)
132 ÷ (2 × 3.142)
= 21.0057 cm
= 21.0 cm (to 3 s.f.)
132 cm 3.142 × 21.00572
= 1390 cm2
(to 3 s.f.)
(k) 2 × 12.4920
= 25.0 cm
(to 3 s.f.)
78.5 ÷ (2 × 3.142)
= 12.4920 cm
= 12.5 cm (to 3 s.f.)
78.5 cm 3.142 × 12.49202
= 490 cm2
(to 3 s.f.)
(l) 56 cm 56 ÷ 2
= 28 cm
2 × 3.142 × 28
= 176 cm
(to 3 s.f.)
3.142 × 282
= 2460 cm2
(to 3 s.f.)
(m) 2 × 38.9752
= 78.0 mm
(to 3 s.f.)
244.92 ÷ (2 × 3.142)
= 38.9752 mm
= 39.0 mm (to 3 s.f.)
244.92 mm 3.142 × 38.97522
= 4770 mm2
(to 3 s.f.)
(n) 60 cm 60 ÷ 2
= 30 cm
2 × 3.142 × 30
= 189 cm (to 3 s.f.)
3.142 × 302
= 2830 cm2
(to 3 s.f.)
(o) 2 × 4.9984
= 10.0 cm
(to 3 s.f.)
78.5 ÷ 3.142
= 4.9984 cm
= 5.00 cm
2 × 3.142 × 4.9984
= 31.4 cm
(to 3 s.f.)
78.5 cm2
5. (a) (i) Perimeter of figure
= 2 + 3 + 1 + 2 + 1 + 1
= 10 cm
(ii) Area of figure
= (2 × 1) + (2 × 1)
= 2 + 2
= 4 cm2
(b) (i) Perimeter of figure
= 3 + 9 + 3 + 3 + 3 + 3 + 3 + 3
= 30 cm
158

1
(ii) Area of figure
= (9 × 3) + (3 × 3)
= 27 + 9
= 36 cm2
(c) (i) Perimeter of figure
= 12 + 6 + 6 + 6 + 12 + 6 + 6 + 6
= 60 cm
(ii) Area of figure
= 2(12 × 6)
= 2(72)
= 144 cm2
(d) (i) Perimeter of figure
= 14 + 7 + 7 + 7 + 14 + 7 + 7 + 7
= 70 cm
(ii) Area of figure
= 2(14 × 7)
= 2(98)
= 196 cm2
6. (a) (i) Perimeter of figure
=
1
2
× 2 × 3.142 ×
49
2











 + 49
= 76.979 + 49
= 125.979
= 126 cm (to 3 s.f.)
(ii) Area of figure
=
1
2
× 3.142 ×
49
2






2
= 942.992 75
= 943 cm2
(to 3 s.f.)
(b) (i) Perimeter of figure
=
1
2
× 2 × 3.142 ×
21
2











 + 20 + 21 + 20
= 32.991 + 61
= 93.991
= 94.0 cm (to 3 s.f.)
(ii) Area of figure
= (20 × 21) +
1
2
× 3.142 ×
21
2






2








= 420 + 173.202 75
= 593.202 75
= 593 cm2
(to 3 s.f.)
(c) (i) Perimeter of figure
=
1
2
× 2 × 3.142 ×
21
2












+
1
2
× 2 × 3.142 ×
14
2











 + 21 + 14
= 32.991 + 21.994 + 21 + 14
= 89.985
= 90.0 cm (to 3 s.f.)
(ii) Area of figure
= (14 × 21) +
1
2
× 3.142 ×
21
2






2








+
1
2
× 3.142 ×
14
2






2








= 294 + 173.202 75 + 76.979
= 544.181 75
= 544 cm2
(to 3 s.f.)
(d) (i) Perimeter of figure
= 2 × 3.142 ×
28
2











 + 16 + 16
= 87.976 + 16 + 16
= 119.976
= 120 cm (to 3 s.f.)
(ii) Area of figure
= 28 × 16
= 448 cm2
(Note: The semicircle removed from the
rectangle can be replaced by the semicircle
that is placed beside the rectangle. Therefore,
the area of the figure is that of a rectangle of
28 cm by 16 cm.)
7. Length of the pool with the walkway
= 20 + 1.5 + 1.5
= 23 m 20 m
17 m
1.5 m
Breadth of the pool with the walkway
= 17 + 1.5 + 1.5
= 20 m
Area of pool with walkway
= 23 × 20
= 460 m2
Area of the swimming pool
= 20 × 17
= 340 m2
Area of walkway = 460 – 340
= 120 m2
159

1
8. (i) Perimeter of the shaded region
= 40 + 40 + 2 × 3.142 ×
28
2






= 80 + 87.976
= 167.976
= 168 cm (to 3 s.f.)
(ii) Area of the shaded region
= (40 × 28) – 3.142 ×
28
2






2








= 1120 – 615.832
= 504.168
= 504 cm2
(to 3 s.f.)
9. (i) Perimeter of quadrant
=
1
4
× 2 × 3.142 × 10





 + 10 + 10
= 15.71 + 20
= 35.71
= 35.7 cm (to 3 s.f.)
(ii) Area of quadrant
=
1
4
× 3.142 × 102
= 78.55
= 78.6 cm2
(to 3 s.f.)
10.
Base Height Area
(a) 10 cm 12 cm 10 × 12 = 120 cm2
(b) 100 ÷ 5
= 20 m
5 m 100 m2
(c)
5.2 mm
50.96 ÷ 5.2
= 9.8 mm
50.96 mm2
11.
Parallel
side 1
Parallel
side 2
Height Area
(a)
5 cm 11 cm 4 cm
1
2
(5 +11) × 4
= 32 cm2
(b)
6 m 14 m
65 ÷
1
2
(6 + 14)






= 6.5 m
65 m2
(c)
2 mm
(34.65 ÷ 8.25)
× 2 – 2
= 6.4 mm
8.25 mm 34.65 mm2
12. (a) The figure shown is a trapezium.
Area of the trapezium
=
1
2
(11 + 13) × 9
= 108 cm2
(b) The figure shown is a parallelogram.
Area of parallelogram
= 16 × 9
= 144 cm2
(c) If we rearrange the figure, it turns out to be a
parallelogram
Area of the figure
= 18 ×
1
2
× 16






= 144 cm2
(d) The figure is a rhombus and it is a special case
of parallelogram.
Area of rhombus
= 32 ×
1
2
× 18






= 288 cm2
(e) The figure is a trapezium.
Area of trapezium
=
1
2
(8.3 + 11.7) × 7.2
= 72 cm2
(f) The figure is a trapezium and a rectangle.
Area of figure
=
1
2
(9 + 26) × (32 − 10)





 + (26 × 10)
= 385 + 260
= 645 cm2
(g) The figure is made up of two trapeziums.
Area of figure
=
1
2
(9 + 23) × 10





 +
1
2
(9 + 17) × 7






= 160 + 91
= 251 cm2
13. (a) Area of the figure
=
1
2
× 11 × 14
= 77 cm2
Area of figure =
1
2
× k × 16
77 =
1
2
× k × 16
77 = 8k
k =
77
8
= 9
5
8
160

1
(b) Area of parallelogram = 16 × x
144 = 16x
x = 9
(c) Area of ABCD =
1
2
(18 + 24) × h
273 =
1
2
(18 + 24) × h
273 = 21h
h = 13
(d) Area of ABCD =
1
2
(32 + y) × 24
912 =
1
2
(32 + y) × 24
38 =
1
2
(32 + y)
76 = 32 + y
y = 76 – 32
= 44
(e) Area of trapezium =
1
2
(27 + 37) × x
480 =
1
2
(27 + 37) × x
960 = 64x
x = 15
14. Let the perpendicular height be h cm.
Area of parallelogram = (4 + 3) × h
35 = 7h
h = 5
Area of nPQT =
1
2
× 4 × 5
= 10 cm2
15. (i) Area of parallelogram ABCD
= 28 × 22
= 616 cm2
(ii) Area of parallelogram ABCD = 18 × AB
616 = (18 × AB) cm2
AB = 34
2
9
cm
Perimeter of parallelogram
= 2 22 + 34
2
9






= 112
4
9
cm
16. Let the length of the other parallel side be y cm.
Area of trapezium =
1
2
(6 + y) × 5
45 =
1
2
(6 + y) × 5
90 = 5(6 + y)
18 = 6 + y
y = 18 – 6
= 12
The length of the other parallel side is 12 cm.
17. (a) Area of shaded region
=
1
2
× 4.6 × 8





 +
1
2
× 6.5 × 8






= 18.4 + 26
= 44.4 cm2
(b) Area of circle with radius 10 cm
= 3.142 × 102
= 314.2 cm2
Area of circle with radius 6 cm
= 3.142 × 62
= 113.112 cm2
Area of shaded region
= 314.2 – 113.112
= 201.088
= 201 cm2
(to 3 s.f.)
(c) Area of circle of radius 10 cm
= 3.142 × 102
= 314.2 cm2
Area of square
= 14.14 × 14.14
= 199.9396 cm2
= 314.2 – 199.9396
= 114.2604
= 114 cm2
(to 3 s.f.)
(d) Area of circle with diameter 32 cm
= 3.142 ×
32
2






2
= 804.352 cm2
Area of circle with diameter 20 cm
= 3.142 ×
20
2






2
= 314.2 cm2
= 804.352 – 314.2
= 490.152
= 490 cm2
(to 3 s.f.)
161

1
(e) Area of square = 16 × 16 = 256 cm2
Area of circle of diameter 16 cm
= 3.142 ×
16
2






2
= 201.088 cm2
= 256 – 201.088
= 54.912
= 54.9 cm2
(to 3 s.f.)
(f) Area of shaded region
= (3.5 × 4.6) + [(3.8 + 3.5 + 3.7) × 3.4]
= 16.1 + 11 × 3.4
= 16.1 + 37.4
= 53.5 cm2
(g) Area of rectangle
= 13 × 11
= 143 cm2
Area of triangle with perpendicular height of
5 cm
=
1
2
× 11 × 5
= 27.5 cm2
Area of triangle with perpendicular height of
4 cm
=
1
2
× 11 × 4
= 22 cm2
= 143 – 27.5 – 22
= 93.5 cm2
(h) Area of circle of radius 80 mm (8 cm)
= 3.142 × 82
= 201.088 cm2
(Note: The centre of the rectangle is not at the
centre of the circle.)
Area of rectangle
= 5.6 × 8.5
= 47.6 cm2
= 201.088 – 47.6
= 153.488
= 153 cm2
(to 3 s.f.)
Intermediate
18. (a) Let the length of the square be n cm.
n2
= 900
Thus n = 900 = 30 cm
Perimeter of square = 4 × 30
= 120 cm
(b) Let the length of the square be x cm.
12.8 = 4x
x = 3.2
Area of the square = (3.2)2
= 10.24 cm2
19. (a) (i) Let the breadth of the rectangle be y cm.
2[y + (y + 8)] = 80
y + y + 8 = 40
2y = 40 – 8
2y = 32
y = 16
The length of the rectangle is (16 + 8)
= 24 cm.
(ii) Area of the rectangle
= 16 × 24
= 384 cm2
(b) Let the length of the rectangle be x m.
0.464 × x = 11.6
x = 25 m
= 2(25 + 0.464)
= 50.928 m
(c) Let the breadth of the rectangle be y cm.
Then the length of the rectangle is (3y) cm.
Perimeter of rectangle = 2(3y + y) cm
1960 = 2(3y + y)
980 = 4y
y = 245
The breadth is 245 cm and the length is 735 cm.
Area of the rectangle
= 735 × 245
= 180 075 cm2
= 180 075 ÷ 10 000
= 18.0075 m2
162

1
(d) Let the breadth of the rectangle be x cm.
Then the length of the rectangle is 2x cm.
Circumference of the wire
= 2 × 3.142 ×
35
2
= 3.142 × 35
= 109.97 cm
Circumference of the wire is the perimeter of the
rectangle.
109.97 = 2(x + 2x)
54.985 = x + 2x
3x = 54.985
x = 18.328 33 (to 5 d.p.)
Area of the rectangle
= 18.328 33 × 2(18.328 33)
= 672 cm2
(to 3 s.f.)
20. (a) Area of nACD =
1
2
× DC × AB
8.4 =
1
2
× 4 × AB
2AB = 8.4
AB = 4.2 cm
(b) Area of nABC
=
1
2
× BC × AB
=
1
2
× 6 × 4.2
= 12.6 cm2
21. (a) The height from X to the length PQ =
10
2
= 5 cm
Area of nPQX =
1
2
× 16 × 5
= 40 cm2
(b) Area of nPQR =
1
2
× (16 × 10)
= 80 cm2
Area of nQRX = 80 – 40
= 40 cm2
22. (a) Perimeter of quadrant = r + r + arc length PQ
50 = r + r + arc length PQ
Arc length PQ = 50 – r – r
= (50 – 2r) cm
1
4
× 2 ×
22
7
× r = 50 – 2r
11
7
× r = 50 – 2r
11
7
× r + 2r = 50
3
4
7
r = 50
r = 50 ÷ 3
4
7
= 14 cm
Area of quadrant =
1
4
×
22
7
× 142
= 154 cm2
(b) Circumference of wheel
= 2 × 3.142 ×
25
2






= 78.55 cm
Number of complete revolutions
=
200
78.55 ÷ 100
≈ 254.61
= 254 revolutions (to 3 s.f.)
Note: The answer cannot be 255 as the wheel has
made 254 revolutions but has not yet completed
the 255th
revolution.
(c) Distance moved by the tip of the hand for
26 minutes
=
26
60
× 2 × 3.142 × 8
= 21.8 cm (to 1 d.p.)
(d) Distance travelled in 5 minutes
= 90 ×
5
60
= 7.5 km
Circumference of car wheel
= 2 × 3.142 × 0.000 35
= 0.002 199 4 km
Number of revolutions made
= 7.5 ÷ 0.002 199 4
= 3410 (to 3 s.f.)
(e) Distance covered when the athlete runs round the
track once
=
4
8
= 0.5 km
= 500 m
Let the radius of the track be r m.
Circumference of the track = 2 × 3.142 × r
500 = 2 × 3.142 × r
r =
×
500
2 3.142
= 79.57 m (to 2 d.p.)
163

1
23. (a) Area of the rhombus
= length of the diagonal
× perpendicular height to the diagonal
90 = 18 × perpendicular height to the diagonal
Perpendicular height to the diagonal =
90
18
= 5 cm
Length of the other diagonal = 5 × 2
= 10 cm
(b) Perpendicular height of the rhombus to the
diagonal
=
24
2
= 12 cm
Area of rhombus
= 28 × 12
= 336 cm2
(c) (i) Area of trapezium
=
1
2
(sum of its parallel sides) × 12
210 =
1
2
(sum of its parallel sides) × 12
Sum of its parallel sides = 210 × 2 ÷ 12
= 35 cm
(ii) Let the length of the shorter side be n cm.
35 = 2
1
2
n + n
3
1
2
n = 35
n = 35 ÷ 3
1
2
= 10
Length of the longer side = 2
1
2
× 10
= 25 cm
24. (a) Length of arc PR
=
1
4
× 2 × 3.142 × 5
= 7.855 cm
Perimeter of the shaded region
= 5.66 + 7.855 + 3 + (4 + 5) + 4
= 29.515
= 29.5 cm (to 3 s.f.)
(b) Area of rectangle
= 9 × 8
= 72 cm2
Area of nAPQ
=
1
2
× 4 × 4
= 8 cm2
Area of quadrant BPR
=
1
4
× 3.142 × 52
= 19.6375 cm2
= 72 – 8 – 19.6375
= 44.3625
= 44.4 cm2
(to 3 s.f.)
25. Area of rectangle ABCD
= 60 × 28
= 1680 cm2
Area of semicircle BXC
=
1
2
× 3.142 ×
28
2






2
= 307.916 cm2
Area of nADX
=
1
2
× 28 × (60 – 14)
= 644 cm2
Area of the shaded region
= 1680 – 307.916 – 644
= 728.084
= 728 cm2
(to 3 s.f.)
26. (a) Circumference of the pond
= 2 × 3.142 × 3.2
= 20.1088 m
Circumference of the pond with concrete path
= 2 × 3.142 × (3.2 + 1.4)
= 2 × 3.142 × 4.6
= 28.9064 m
Perimeter of the shaded region
= 28.9064 + 20.1088
= 49.0152
= 49.0 m (to 3 s.f.)
164

1
(b) Area of the pond
= 3.142 × 3.22
= 32.174 08 m2
Area of the pond with concrete path
= 3.142 × 4.62
= 66.484 72 m2
Area of the shaded region
= 66.484 72 – 32.174 08
= 34.310 64
= 34.3 m2
(to 3 s.f.)
The area of the concrete path is 34.3 m2
.
27. (a) Area of shaded region A
= area of circle with radius 5 cm
= 3.142 × 52
= 78.55
= 78.6 cm2
(to 3 s.f.)
(b) Area of circle with radius 10 cm
= 3.142 × 102
= 314.2 cm2
Area of circle with radius 8 cm
= 3.142 × 82
= 201.088 cm2
Area of shaded region B
= 314.2 – 201.088
= 113.112
= 113 cm2
(to 3 s.f.)
28. (a) 1.25 m = 1.25 × 100
= 125 cm
The largest possible radius is 125.4 cm or
1.254 m.
(b) Smallest possible radius = 124.5 cm = 1.245 m
Smallest possible area
= 3.142 × (1.245)2
= 4.870 178 55
= 4.870 m2
(to 4 s.f.)
29. Area of quadrant
=
1
4
× 3.142 × 212
= 346.4055 cm2
Area of nOCA
=
1
2
× 13 × 21
= 136
1
2
cm2
= 346.4055 – 136
1
2
= 209.9055
= 210 cm2
(to 3 s.f.)
30. (a) Area of quadrant
=
1
4
× 3.142 × 402
= 1256.8 cm2
Area of triangle
=
1
2
× 40 × 40
= 800 cm2
= 1256.8 – 800
= 456.8
= 457 cm2
(to 3 s.f.)
(b) Area of square
= 24 × 24
= 576 cm2
Area of a circle with radius 12 cm
= 3.142 × 122
= 452.448 cm2
= 576 – 452.448
= 123.552
= 124 cm2
(to 3 s.f.)
(c) Area of semicircle with radius 5 cm
=
1
2
× 3.142 × 52
= 39.275 cm2
Area of triangle
=
1
2
× 6 × 8
= 24 cm2
= 39.275 – 24
= 15.275
= 15.3 cm2
(to 3 s.f.)
(d) Area of trapezium
=
1
2
(23 + 33) × 19
= 532 cm2
Area of triangle
=
1
2
× 33 × 19
= 313.5 cm2
= 532 – 313.5
= 218.5 cm2
165

1
(e) Area of semicircle with diameter 5 cm
=
1
2
× 3.142 ×
5
2






2
= 9.818 75 cm2
Area of semicircle with diameter 2 cm
=
1
2
× 3.142 ×
2
2






2
= 1.571 cm2
Area of semicircle with diameter 3 cm
=
1
2
× 3.142 ×
3
2






2
= 3.534 75 cm2
= 9.818 75 – 1.571 + 3.534 75
= 11.7825
= 11.8 cm2
(to 3 s.f.)
(f) Area of trapezium
=
1
2
× (19 + 29) × 21
= 504 cm2
= 3.142 ×
21
2






2
= 346.4055 cm2
= 504 – 346.4055
= 157.5945
= 158 cm2
(to 3 s.f.)
(g) Total shaded area
= area of semicircle with radius 12 cm
+ area of rectangle 23 cm by 12 cm
+ area of rectangle 17 cm by 12 cm
+ area of triangle
=
1
2
× 3.142 × 122





 + (12 × 23) + (17 × 12)
+
1
2
× 6 × 4






= 226.224 + 276 + 204 + 12
= 718.224
= 718 cm2
(to 3 s.f.)
(h) Place the quadrants and fill the gap. The figure is
then changed into a rectangle with dimensions 20
cm by 18 cm.
5 cm 5 cm
13 cm
20 cm
= area of rectangle with dimension 20 cm by
18 cm
= 20 × 18
= 360 cm2
(i)
A
B
14 cm
Area of region A
=
1
4
× 3.142 × 72





 –
1
2
× 7 × 7






= 38.4895 – 24.5
= 13.9895 cm2
Area of region B = area of region A
= 13.9895 cm2
= 2 × 13.9895
= 27.979
= 28.0 cm2
(to 3 s.f.)
31. (a) Since ABCD is a square, then
3x = 22
x = 7
1
3
(b) Area of shaded region
= area of square ABCD – area of PQRC
403 = (22 × 22) – y2
y2
= (22 × 22) – 403
= 484 – 403
= 81
y = 9
166

1
32. (a) (i) Perimeter of rectangle
= 2[(3x + 4) + (4x – 13)]
94 = 2[(3x + 4) + (4x – 13)]
94 = 2[3x + 4x + 4 – 13]
94 = 2[7x – 9]
94 = 14x – 18
14x = 94 + 18
14x = 112
x = 8
(ii) Length of rectangle = 3 × 8 + 4
= 28 cm
Breadth of rectangle = 4 × 8 – 13
= 19 cm
= 532 cm2
(b) Area of trapezium
=
1
2
× [(x + 5) + (3x + 1)] × 6
= 3[(x + 5) + (3x + 1)]
66 = 3[(x + 5) + (3x + 1)]
66 = 3[x + 3x + 5 + 1]
66 = 3[4x + 6]
66 = 12x + 18
12x = 66 – 18
12x = 48
x = 4
33. (a) Perimeter of semicircle
=
1
2
× 2 × 3.142 ×
2x
2





 + 2x
= 3.142x + 2x
= 5.142x cm
= 2[(x + 11) + (x – 3)]
= 2(x + x + 11 – 3)
= 2(2x + 8) cm
5.142x = 2(2x + 8)
5.142x = 4x + 16
5.142x – 4x = 16
1.142x = 16
x = 14.0105
= 14.0 (to 3 s.f.)
(b) Area of semicircle
=
1
2
× 3.142 ×
2 × 14.01
2






2
= 308.356 037 1 cm2
Length of rectangle = 14.01 + 11
= 25.01 cm
Breadth of rectangle = 14.01 – 3
= 11.01 cm
Area of rectangle = 25.01 × 11.01
= 275.3601 cm2
Difference in area = 308.356 037 1 – 275.3601
= 32.9959
= 33.0 cm2
(to 3 s.f.)
34. (a) Number of slabs needed along its length
=
×
25 100
25
= 100
(b) Number of slabs needed along its row
=
×
12 100
25
= 48
(c) Area of rectangular courtyard
= (25 × 100) × (12 × 100)
= 3 000 000 cm2
Area of each slab
= 25 × 25
= 625 cm2
Number of slabs needed to pave the whole
courtyard
=
3 000 000
625
= 4800
(d) Total cost of paving the courtyard
= $0.74 × 4800
= $3552
35. (a) Let the radius of the semicircle be r cm.
Area of semicircle
=
1
2
× 3.142 × r2
= 1.571r2
cm2
Area of triangle AFE
=
1
2
× 2r × r
= r2
cm2
Area of shaded region = 1.571r2
– r2
73 = 1.571r2
– r2
0.571r2
= 73
r2
= 127.845 884 4
r = 11.3 (to 3 s.f.)
Length of AE = 2 × 11.306 895
= 22.6138
= 22.6 cm (to 3 s.f.)
(b) Area of trapezium ABDE
=
1
2
× (48 + 22.6138) × 20
= 706.138
= 706 cm2
(to 3 s.f.)
167

1
36. (a) Let the length AB be h cm.
Area of quadrilateral ABCD
= 8 × h
= 8h cm2
Area of quadrilateral EFGH
= 10 × h
= 10h cm2
Area of nIJK
=
1
2
× 14 × h
= 7h cm2
Ratio of area of ABCD to area of EFGH to area of
nIJK
= 8h : 10h : 7h
= 8 : 10 : 7
(b) Area of nIJK = 56
1
2
× 14 × h = 56
7h = 56
h = 8
The quadrilateral LMNO is a trapezium.
Area of quadrilateral LMNO
=
1
2
× (3 + 17) × 8
= 80 cm2
Advanced
37. (a) Perimeter of triangle ABC
= 2x + (x + 5) + (4x – 2)
= 2x + x + 4x + 5 – 2
= (7x + 3) cm
Perimeter of rectangle PQRS
= 2[(7x – 10) + (2x + 1)]
= 2(7x + 2x – 10 + 1)
= 2(9x – 9) cm
= 18(x – 1) cm
The equation is 1
1
2
(7x + 3) = 18(x – 1).
(b) 1
1
2
(7x + 3) = 18(x – 1)
3(7x + 3) = 36(x – 1)
21x + 9 = 36x – 36
36x – 21x = 9 + 36
15x = 45
x = 3
Perimeter of triangle ABC = 7 × 3 + 3
= 24 cm
Area of triangle ABC
=
1
2
× (2 × 3) × (3 + 5)
=
1
2
× 6 × 8
= 24 cm2
(c) Area of rectangle PQRS
= (2 × 3 + 1) × (7 × 3 – 10)
= 7 × 11
= 77 cm2
Difference between the area of triangle ABC and
the area of rectangle PQRS
= 77 – 24
= 53 cm2
38. Let the radius of each circle be r cm.
Area of each circle = pr2
36p = pr2
r2
= 36
r = 36 = 6
Length of CD = 6 + 6
= 12 cm
39. AM : MJ = 1 : 1
AM =
1
2
× 10
= 5 cm
Area of AMIB
=
1
2
× (5 + 10) × 10
= 75 cm2
LG = 2 cm (based on L divides DG in the
ratio of 1 : 2)
Area of BIGL
=
1
2
× (2 + 10) × 17
= 102 cm2
Area of nLGF
=
1
2
× 3 × 2
= 3 cm2
Total area of the shaded region
= 75 + 102 + 3
= 180 cm2
168

1
New Trend
40. (a) A = π
5r +5kr
2






2
= π
5r 1+ k
( )
2








2
=
25
4
πr2
(1 + k)2
(b) When k = 3,
Area of large circle =
25
4
πr2
(4)2
= 100πr2
cm2
=
100 πr2
2
+
1
2
π
5kr
2






2
−
1
2
π
5r
2






2
= 50πr2
+
1
2
π
225r2
4





 −
1
2
π
25r2
4






= 50πr2
+
225πr2
8
−
25πr2
8
= 75πr2
cm2
Area of unshaded region
=
100 πr2
2
+
1
2
π
5r
2






2
−
1
2
π
5kr
2






2
= 50πr2
+
25πr2
8
−
225πr2
8
= 50πr2
− 25πr2
= 25πr2
cm2
Difference in area = 75πr2
− 25πr2
= 50πr2
cm2
169

1
Chapter 14 Volume and Surface Area of
Prisms and Cylinders
Basic
1. (a) 6.2 m3
= 6.2 × 100 × 100 × 100
= 6 200 000 cm3
(b) 2.9 m3
= 2.9 × 100 × 100 × 100
= 2 900 000 cm3
(c) 35 000 cm3
= 35 000 ÷ 100 ÷ 100 ÷ 100
= 0.035 m3
(d) 75 cm3
= 75 ÷ 100 ÷ 100 ÷ 100
= 0.000 075 m3
(e) 97.8 l = 97.8 × 1000
= 97 800 cm3
(f) 1 cm3
= 1 ml
0.07 cm3
= 0.07 ml
2. (a) (i) Volume of cube = 53
= 125 cm3
(ii) Total surface area
= 6l2
= 6 × 52
= 150 cm2
(b) (i) Volume of cube = 2.43
= 13.824 cm3
= 6 × 2.42
= 34.56 cm2
(c) (i) Volume of rectangular cuboid
= 30 × 25 × 12
= 9000 cm3
(ii) Total surface area of cuboid
= 2[(30 × 25) + (30 × 12) + (25 × 12)]
= 2[750 + 360 + 300]
= 2820 cm2
(d) (i) Volume of rectangular cuboid
= 1.2 × 0.8 × 0.45
= 0.432 m3
(ii) Total surface area of cuboid
= 2[(1.2 × 0.8) + (1.2 × 0.45) + (0.8 × 0.45)]
= 2[0.96 + 0.54 + 0.36]
= 3.72 m2
3. (a) The base is a triangle with height 12 cm and base
length 16 cm.
Base area = area of triangle
=
1
2
× 12 × 16
= 96 cm2
Volume of prism = base area × height
= 96 × 14
= 1344 cm3
Total surface area of prism
= 96 + 96 + (16 × 14) + (12 × 14) + (14 × 20)
= 864 cm2
(b) The shape of the base is a cross.
Base area
= (14 × 14) – 4(5 × 5)
= 96 cm2
= 96 × 3
= 288 cm3
Total surface area of solid
= (2 × 96) + 8(5 × 3) + 4(3 × 4)
= 192 + 120 + 48
= 360 cm2
(c) The base is a triangle with height 8 cm and base
length 6 cm.
Base area =
1
2
× 8 × 6
= 24 cm2
= 24 × 14
= 336 cm3
Total surface area of the solid
= 24 + 24 + (10 × 14) + (14 × 8) + (6 × 14)
= 48 + 140 + 112 + 84
= 384 cm2
(d) The base is a U-shape.
Base area = (21 × 15) – (9 × 7)
= 315 – 63
= 252 cm2
= 252 × 10
= 2520 cm3
Total surface area of the solid
= (252 × 2) + 2(6 × 10) + 2(7 × 10) + (9 × 10)
+ (21 × 10) + 2(15 × 10)
= 504 + 120 + 140 + 90 + 210 + 300
= 1364 cm2
170

1
4. Volume of a closed cylinder = pr2
h
Total surface area of closed cylinder = 2pr2
+ 2prh
Diameter Radius Height Volume Total Surface Area
(a) 24 × 2
= 48 cm
24 cm 21 cm 3.142 × (24)2
× 21
= 38 000 cm3
(to 3 s.f.)
(2 × 3.142 × (24)2
)
+ (2 × 3.142 × 24 × 21)
= 3619.584 + 3167.136
= 6790 cm2
(to 3 s.f.)
(b) 1.45 × 2
= 2.9 cm
1.45 cm 1.4 cm 3.142 × (1.45)2
× 1.4
= 9.25 cm3
(to 3 s.f.)
(2 × 3.142 × (1.45)2
)
+ (2 × 3.142 × 1.45 × 1.4)
= 13.212 11 + 12.756 52
= 26.0 cm2
(to 3 s.f.)
(c) 28 × 2
= 56 cm
0.28 m
= 28 cm
45 cm 3.142 × (28)2
× 45
= 111 000 cm3
(to 3 s.f.)
(2 × 3.142 × (28)2
)
+ (2 × 3.142 × 28 × 45)
= 4926.656 + 7917.84
= 12 800 cm2
(to 3 s.f.)
(d) 18.2 × 2
= 36.4 cm
182 mm
= 18.2 cm
7.5 cm 3.142 × (18.2)2
× 7.5
= 7810 cm3
(to 3 s.f.)
(2 × 3.142 × (18.2)2
)
+ (2 × 3.142 × 18.2 × 7.5)
= 2081.512 16 + 857.766
= 2940 cm2
(to 3 s.f.)
(e) 4.998 × 2
= 10.0 cm
(to 3 s.f.)
(2826) ÷ (3.142 × 36)
= 4.998
= 5.00 cm (to 3 s.f.)
36 cm 2826 cm3
(2 × 3.142 × (4.998)2
)
+ (2 × 3.142 × 4.998 × 36)
= 156.97 + 1130.67
= 1290 cm2
(to 3 s.f.)
(f) 1.118 × 2
= 2.236 cm
(30.615) ÷ (3.142 × 7.8)
= 1.118 cm
= 1.12 cm (to 3 s.f.)
7.8 cm 30.615 cm3
(2 × 3.142 × (1.118)2
)
+ (2 × 3.142 × 1.118 × 7.8)
= 7.854 52 + 54.799
= 62.7 cm2
(to 3 s.f.)
(g) 19.994 × 2
= 40.0 cm
(to 3 s.f.)
(8164) ÷ (3.142 × 6.5)
= 19.994 cm
= 20.0 cm (to 3 s.f.)
65 mm
= 6.5 cm
8164 cm3
(2 × 3.142 × (19.994)2
)
+ (2 × 3.142 × 19.994 × 6.5)
= 2512.092 + 816.6749
= 3330 cm2
(to 3 s.f.)
(h) 5.6 × 2
= 11.2 cm
5.6 cm 532 ÷ (3.142 × 5.62
)
= 5.3992 cm
= 5.40 cm (to 3 s.f.)
532 cm3
(2 × 3.142 × (5.6)2
)
+ (2 × 3.142 × 5.6 × 5.3992)
= 197.066 24 + 190.00
= 387 cm2
(to 3 s.f.)
(i) 2.65 × 2
= 5.3 cm
2.65 cm 20.74 ÷ (3.142 × 2.652
)
= 0.940 cm (to 3 s.f.)
20.74 cm3
(2 × 3.142 × (2.65)2
)
+ (2 × 3.142 × 2.65 × 0.940)
= 44.129 39 + 15.6534
= 59.8 cm2
(to 3 s.f.)
(j) 15 × 2
= 30 cm
15 cm 5400 ÷ (3.142 × 152
)
= 7.6384 cm
= 7.64 cm (to 3 s.f.)
0.0054 m3
(2 × 3.142 × (15)2
)
+ (2 × 3.142 × 15 × 7.6384)
= 1413.9 + 719.996
= 2130 cm2
(to 3 s.f.)
5. (a) Let the height of the room be h m.
Volume of room = (12 × 9 × h) m3
540 = 12 × 9 × h
h = 5
∴The height of the room is 5 m.
171

1
(b) Let the length of the box be n cm.
60 = n × 4 × 2
∴ n = 7.5
The length of the box is 7.5 cm.
6. (a) Number of cubes that can be obtained along the
length
= 20 ÷ 4
= 5
Number of cubes that can be obtained along the
breadth
= 16 ÷ 4
= 4
height
= 8 ÷ 4
= 2
Therefore, the number of cubes that can be obtained
= 5 × 4 × 2
= 40
(b) Number of cubes that can be obtained along the
length
= 80 ÷ 4
= 20
breadth
= 25 ÷ 4
≈ 6
height
= 35 ÷ 4
≈ 8
= 20 × 6 × 8
= 960
(c) Number of cubes that can be obtained along the
length
= 120 ÷ 4
= 30
breadth
= 85 ÷ 4
≈ 21
height
= 50 ÷ 4
≈ 12
= 30 × 21 × 12
= 7560
7. Number of cubes that can be cut along the length
= 420 ÷ 20
= 21
Number of cubes that can be cut along the breadth
= 140 ÷ 20
= 7
Number of cubes that can be cut along the height
= 120 ÷ 20
= 6
Therefore, the number of cubes that can be cut
= 21 × 7 × 6
= 882
(Note: For questions 6 and 7, understand the difference
between “cut” and “melt” and “recast”.)
8. Total volume of water
= 37 + 20
= 57 m3
Let the depth of water in the trough be h m.
Volume of water = 8 × 3 × h
= 24 h m3
57 = 24h
∴ h = 2.375
The depth of the water, after 20 m3
of water is added,
is 2.375 m.
9. (i) Volume of air = volume of cuboid
= 12 × 7 × 3
= 252 m3
(ii) Number of students allowed staying in the
dormitory
= 252 ÷ 14
= 18
10. (i) Volume of hall
= volume of cuboid
+ volume of half-cylindrical ceiling
= (30 × 80 × 10) +
1
2
× 3.142 ×
30
2






2
× 80








= 24 000 + 28 278
= 52 278
= 52 300 cm3
(to 3 s.f.)
(ii) Total surface area of hall
= [2(30 × 10) + 2(80 × 10) + (80 × 30)]
+
1
2
[(2 × 3.142 × 152
) + (2 × 3.142 × 15 × 80)]
= [600 + 1600 + 2400] +
1
2
[1413.9 + 7540.8]
= 4600 + 4477.35
= 9077.35
= 9080 m2
(to 3 s.f.)
172

1
Intermediate
11. (a) Density of solid
=
mass
volume
=
45
8
= 5.625 g/cm3
(b) Density of solid
=
mass
volume
=
×
1.35 1000
250
= 5.4 g/cm3
(c) Density of solid
=
mass
volume
=
0.46 × 1000
78000 ÷ 10 ÷ 10 ÷ 10
= 5.90 g/cm3
(to 3 s.f.)
(d) Density of solid
=
mass
volume
=
×
0.325 1000
85
= 3.82 g/cm3
(to 3 s.f.)
12. Volume of block = volume of cube
= (28)3
= 21 952 cm3
Let one unit of the length of the block be y cm.
Then (5y) × (4y) × (3y) = 21 952
60y3
= 21 952
y3
= 365
13
15
y = 7.152
Longest side of the cuboid
= 5 × 7.152
= 35.761
= 35.8 cm (to 1 d.p.)
13. (a) Length of the square base
= 1225
= 35 cm
Length of the square base
= diameter of the cylindrical pillar
Base area of cylinder with diameter 35 cm
= 3.142 ×
35
2






2
= 962.2375 cm2
Volume of the pillar
= base area × height
= 962.2375 × (3.5 × 100)
= 336 783.125
= 337 000 cm3
(to 3 s.f.)
(b) Volume of block of wood
= base area × length
= 1225 × (3.5 × 100)
= 428 750 cm3
Volume of block left after making the pillar
= 428 750 – 336 783.125
= 91 966.875
= 92 000 cm3
(to 3 s.f.)
14. (a) (i) Convert 12 litres to cm3
.
12 l = 12 × 1000 = 12 000 cm3
Height of water
= volume of water ÷ base area of tank
= 12 000 ÷ (40 × 28)
= 10.714
= 10.7 cm (to 3 s.f.)
(ii) Surface area in contact with the water
= (40 × 28) + 2[(40 × 10.714)
+ (28 × 10.714)]
= 1120 + 2[428.56 + 299.992]
= 2577.104
= 2580 cm2
(to 3 s.f.)
(b) (i) Volume of tank
= 65 × 42 × 38
= 103 740 cm3
Volume of each cylindrical cup
= 3.142 × (3.5)2
× 12
= 461.874 cm3
Number of cups that can fill the tank
=
103740
461.874
≈ 224.61
= 224 complete cups
(Note: The answer is not 225 as the question
requires the number of complete cups.)
173

1
(ii) Volume of cup = 224 × 461.874
= 103 460 cm3
Volume of sugarcane left in the tank
= 103 740 – 103 460
= 280 cm3
15. (i) Let the length of the cube be l cm.
Total surface area of cube = 6l2
294 = 6l2
6l2
= 294
l = 7
Volume of cube = 73
= 343 cm3
(ii) Convert 343 cm3
to m3
.
343 cm3
= 343 ÷ 100 ÷ 100 ÷ 100
= 3.43 × 10– 4
m3
Density of solid cube
=
mass
volume
=
×
1.47
3.43 10– 4
= 4285.714
= 4290 kg/m3
(to 3 s.f.)
16. (a) (i) Base area = (8 × 3) +
1
2
(3 + 6) × 4
= 24 + 18
= 42 cm2
Volume of prism
= 42 × 6
= 252 cm3
= area of all the surfaces
= 42 + 42 + 2(6 × 3) + (5 × 6) + (6 × 2)
+ (6 × 7) + (8 × 6)
= 84 + 36 + 30 + 12 + 42 + 48
= 252 cm2
(iii)Mass of solid = density × volume
= 2.8 × 252
= 705.6 g
(b) (i) Base area =
1
2
(9 + 6) × 4
= 30 cm2
Volume of prism = 30 × 8
= 240 cm3
= 30 + 30 + (8 × 9) + (6 × 8) + (8 × 5)
+ (4 × 8)
= 60 + 72 + 48 + 40 + 32
= 252 cm2
(iii)Mass of solid = density × volume
= 2.8 × 240
= 672 g
17. (i) Area of face ABQP
=
1
2
(7 + 13) × 8
= 80 cm2
(ii) Base area = area of face ABQP = 80 cm2
Volume of solid
= 80 × 40
= 3200 cm3
(iii)Total surface area
= area of all the faces
= 2(80) + (13 × 40) + (7 × 40) + (8 × 40)
+ (10 × 40)
= 160 + 520 + 280 + 320 + 400
= 1680 cm2
18. A drawing of the cross-section of the swimming pool
is helpful in solving the problem.
40 m
10 m
3 m
1.2 m
Area of cross-section
=
1
2
× (1.2 + 3) × 40 + 10 × 1.2
= 96 m2
Volume of water in the pool when it is full
= area of cross-section × width
= 96 × 32
= 3072 cm3
19. (i) Let the radius of the base of the cylinder be r cm.
Circumference of base of cylinder = 2pr
88 = 2 × 3.142 × r
r = 14.004
Total surface area
= 2pr2
+ (circumference × height)
= (2 × 3.142 × 14.0042
) + (88 × 10)
= 1232.367 909 + 880
= 2112.367 909
= 2110 cm2
(to 3 s.f.)
(ii) Volume of cylinder
= pr2
h
= 3.142 × 14.0042
× 10
= 6161.840
= 6160 cm3
(to 3 s.f.)
174

1
20. Volume of water in container P
= 3.142 ×
3
2






2
× 24
= 169.668 cm3
Let the height of water in container Q be h cm.
Volume of water in container Q = 169.668 cm3
Base area of container Q
= 3.142 ×
8
2






2
= 50.272 cm2
50.272 × h = 169.668
h = 3
3
8
The height of water in container Q is 3
3
8
cm.
21. Volume of water in the cylinder when it is filled to the
brim
= 3.142 ×
10
2






2
× 30
= 2356.5 cm3
Volume of water in the tank before the ball bearings
are added
=
3
8
× 2356.5
= 883.6875 cm3
Volume of water and ball bearings
=
1
2
× 2356.5
= 1178.25 cm3
Volume of 8 ball bearings
= 1178.25 – 883.6875
= 294.5625 cm3
Volume of each ball bearing
= 294.5625 ÷ 8
= 36.820
= 36.8 cm3
(to 3 s.f.)
22. (i) Total surface area of an open cylinder
= pr2
+ 2prh
= (3.142 × 142
) + (2 × 3.142 × 14 × 30)
= 615.832 + 2639.28
= 3255.112
= 3260 cm2
(to 3 s.f.)
(ii) 1 m2
= 1 × 100 × 100 = 10 000 cm2
10 000 cm2
costs 750 cents
3255.112 cm2
costs 244.1334 cents
= 244 cents (to the nearest cent)
23. (a) Volume of metal cube = (46)3
= 97 336 cm3
Volume of each cylindrical rod
= 3.142 × 22
× 3.2
= 40.2176 cm3
Maximum number of rods that can be obtained
=
97 336
40.2176
≈ 2420
(b) Volume of the metal disc
= 3.142 × 82
× 3
= 603.264 cm3
Volume of each bar
= 3.142 × 12
× 4.2
= 13.1964 cm3
Maximum number of bars that can be obtained
=
603.264
13.1964
= 45.7143
≈ 45
(c) Volume of butter
= 3.142 × 32
× 10
= 282.78 cm3
Volume of each circular disc
= 3.142 × (1.5)2
× 0.8
= 5.6556 cm3
Maximum number of discs formed
=
282.78
5.6556
= 50
24. Volume of the metal
= 12 × 18 × 10
= 2160 cm3
Volume of each cylindrical plate
= 2160 ÷ 45
= 48 cm3
Let the thickness of each plate be t cm.
48 = 3.142 × 1.22
× t
t = 10.61 cm (to 2 d.p.)
∴The thickness of each plate is 10.61 cm.
25. (i) Internal curved surface area
= 2prh
= 2 × 3.142 × 9 × (12 × 100)
= 67 867.2 cm2
= 6.79 m2
(to 3 s.f.)
(ii) External radius of the pipe = 9 + 0.5
= 9.5 cm
Volume of metal
= [3.142 × (9.5)2
× 1200] – (3.142 × 92
× 1200)
= [3.142 × 1200](9.52
– 92
)
= 34 876.2
= 34 900 cm3
(to 3 s.f.)
175

1
26. (i) Convert 385 litres to cm3
.
385 litres = 385 × 1000 = 385 000 cm3
(ii) Base area = 3.142 × (70)2
= 15 395.8 cm2
Volume of water in tank = base area × height h
385 000 = 15 395.8 × h
h = 25.007
= 25.0 cm (to 3 s.f.)
(iii)Total surface area of the liquid in contact with the
cylindrical tank
= (3.142 × 702
) + (2 × 3.142 × 70 × 25.007)
= 15 395.8 + 11 000.08
= 26 395.88
= 26 400 cm2
(to 3 s.f.)
27. (a) Since water is discharged through the pipe at a rate
of 28 m/min, the volume of water discharged in 1
minute is the volume of water that fills the pipe to
a length of 28 m.
In 1 minute, volume of water discharged
= volume of pipe of length 28 m
= pr2
h
= 3.142 × (4.2 ÷ 100)2
× 28
= 0.155 189 664 cm3
Volume of water in rectangular tank
= 4 × 2.5 × 2.4
= 24 m3
Amount of time needed to fill the tank completely
=
24
0.155 189 664
= 154.6495 minutes
= 2 hours and 35 minutes (to the nearest minute)
(b) Since water is discharged through the pipe at a
rate of 3.4 m/s, the volume of water discharged in
1 second is the volume of water that fills the pipe
to a length of 3.4 m.
In 1 second, volume of water discharged
= volume of pipe of length 3.4 m
= pr2
h
= 3.142 × [(5.2 ÷ 2) ÷ 100]2
× 3.4
= 0.007 221 572 8 m3
Volume of cylindrical tank
= 3.142 × (2.3)2
× 1.6
= 26.593 888 8 m3
Amount of time needed to fill the tank
= 26.593 888 8 ÷ 0.007 221 572 8
= 3682.561 893 seconds
= 61 minutes (to the nearest minute)
(c) Base area of trapezium
=
1
2
× (7 + 5) × 2.5
= 15 m2
In 1 hour, volume of water discharged
= 15 × (12 × 1000)
= 180 000 m3
In 1 second,
volume of water discharged
= (180 000 ÷ 3600)
= 50 m3
In 5 seconds, the volume of water discharged
= 5 × 50 = 250 m3
(d) Since water is discharged through the pipe at a rate of
18 km/h, the volume of water discharged in 1 hour is
the volume of water that fills the pipe to a length of
18 km = 18 000 m.
= volume of pipe of length 18 km
= 3.142 × (4 ÷ 100)2
× 18 000
= 90.4896 m3
In 1
2
3
hours, the volume of water discharged
= 1
2
3
× 90.4896
= 150.816 m3
Volume of swimming pool = 50 × 25 × height h
150.816 = 1250 × h
∴ h = 0.120 652 8 m
= 12.1 cm (to 3 s.f.)
28. (a) Volume of rectangular cuboid
= 0.40 × 0.25 × 0.08
= 0.008 m3
Density of solid
=
mass
volume
=
33.6
0.008
= 4200 kg/m3
(b) Convert 10.5 kg to g.
10.5 kg = 10.5 × 1000 = 10 500 g
Volume of metal
=
10500
3.5
= 3000 cm3
3000 = 4 × 3 × x
x = 250
176

1
(c) Convert 22.44 kg to g.
22.44 kg = 22.44 × 1000 = 22 440 g
Volume of metal
=
22440
13.6
= 1650 cm3
Let the radius of the glass cylinder be x cm.
1650 = 3.142 × x2
× 21
x2
= 25.0068
x = 5.000 68
Diameter of the glass cylinder
= 2 × 5.000 68
= 10.0 cm (to 3 s.f.)
(d) Convert 14.5 kg to g.
14.5 kg = 14.5 × 1000 = 14 500 g
Volume of metal
=
14 500
3.8
= 3815.789 474 cm3
(to 6 d.p.)
Let the length of the rod be l cm.
3815.789 474 = 3.142 × 62
× l
∴ l = 33.7346
= 34 cm (to the nearest cm)
29. (i) Base area =
1
2
× (12 + 8) × 7
= 70 cm2
Volume of block = base area × length
= 70 × 28
= 1960 cm3
(ii) Total surface area of block
= 70 + 70 + (12 × 28) + (7 × 28) + (28 × 8.06)
+ (8 × 28)
= 70 + 70 + 336 + 196 + 225.68 + 224
= 1121.68 cm2
(iii)Mass of the block = density × volume
= 1.12 × 1960
= 2195.2 g
30. (i) Total surface area of the solid block
= (2 × 3.142 × 142
)
+ (2 × 3.142 × 14 × [1.2 × 100])
= 1231.664 + 10 557.12
= 11 788.784
= 11 800 cm2
(to 3 s.f.)
(ii) Volume of block
= 3.142 × (14)2
× (1.2 × 100)
= 73 899.84
= 73 900 cm3
(to 3 s.f.)
(iii)Convert 92.4 kg to g.
92.4 kg = 92.4 × 1000 = 92 400 g
Density of block
=
92 400
79899.84
= 1.250 34…
= 1.25 g/cm3
(to 3 s.f.)
31. (i) Volume of box
= 48 × 36 × 15
= 25 920 cm3
Number of items in box
= (48 ÷ 7) × (36 ÷ 7) × (15 ÷ 7)
≈ 6 × 5 × 2
= 60
Total volume of items = 60 × 73
= 20 580 cm3
Volume of sawdust
= 25 920 – 20 580
= 5340 cm3
(ii) Mass of sawdust = density × volume
= 0.75 × 5340
= 4005 g
32. (i) Total surface area of the cuboid
= 2[(30 × 25) + (30 × 15) + (25 × 15)]
= 2[750 + 450 + 375]
= 3150 cm2
(ii) Volume of cuboid
= 30 × 25 × 15
= 11 250 cm3
Volume of each coin
= 3.142 × 1.52
× (2.4 ÷ 10)
= 1.696 68 cm3
Number of coins that can be made
= 11 250 ÷ 1.696 68
≈ 6630
(Note: The answer is not 6631 as the number of
coins is 6630.6, which is less than 6631.)
(iii)Total volume of coins
= 6630 × 1.696 68
= 11 248.9884 cm3
Volume of molten metal left behind
= 11 250 – 11 248.9884
= 1.0116
= 1.01 cm3
(to 3 s.f.)
(iv) Mass of each coin
= density × volume
= 6.5 × 1.696 68
= 11.028 42 g
= 11.0 g (to 3 s.f.)
177

1
33. (i) Convert 3780 litres to m3
.
3780 l = (3780 × 1000) ÷ 100 ÷ 100 ÷ 100
= 3.78 m3
Let the depth of the liquid in the tank be d m.
3.78 = 4.2 × 1.8 × d
∴ d = 0.5
The depth of the liquid in the tank is 0.5 m or
50 cm.
(ii) Volume of increase in liquid level
= 420 × 180 × (1.6)
= 120 960 cm3
Volume of one solid brick
=
120960
380
= 318.315 789 5
= 318 cm3
(to 3 s.f.)
(iii)Mass of bricks
= 1.8 × 318.315 789 5 × 380
= 217 728 g
Mass of liquid
= 1.2 × 3 780 000
= 4 536 000 g
Total mass
= 4 536 000 + 217 728
= 4 753 728 g
= 4753.728 kg
= 4753.73 kg (to 2 d.p.)
34. (i) Volume of open rectangular tank
= 110 × 60 × 40
= 264 000 cm3
Amount of liquid required to fill up the tank
=
3
8
× 264 000
= 99 000 cm3
= 99 litres
(ii) Amount of time needed, in minutes, to fill up
the tank
=
99
5.5
= 18 minutes
(iii)Volume of liquid in tank, in m3
= 264 400 ÷ 100 ÷ 100 ÷ 100
= 0.264 m3
Mass of liquid in the whole tank
= density × volume of liquid in tank
= 800 × 0.264
= 211.2 kg
35. (i) Volume of closed container
= volume of cuboid + volume of half – cylinder
= (28 × 60 × 40) +
1
2
× 3.142 ×
28
2






2
× 60








= 67 200 + 18 474.96
= 85 674.96 cm3
= (85 674.96 ÷ 1000) litres
= 85.7 litres (to 3 s.f.)
(ii) Total surface area of the container
= surface area of the cuboid (without the
top surface)
+ surface area of the half cylinder
= 2 × 28 × 40 +
1
2
× 3.142 × 142






+ 2(60 × 40) + (28 × 60)
+
1
2
× 2 × 3.142 × 14 × 60






= 2[1120 + 307.916] + 4800 + 1680 + 2639.28
= 2855.832 + 4800 + 1680 + 2639.28
= 11 975.112 cm2
= 1.197 511 2 m2
= 1.20 m2
(to 3 s.f.)
36. Volume of two cylindrical discs
= 2 π ×
120
2
⎛
⎝
⎜
⎞
⎠
⎟
2
× 12
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
= 86 400p cm3
Volume of the connecting cylinder of diameter 40 cm
= p ×
40
2






2
× (94 – 12 – 12)
= 28 000p cm3
Total volume of drum (before the cylinder of diameter
of 16 cm is removed)
= (86 400 + 28 000)p
= 114 400p cm3
Volume of cylinder, of diameter 16 cm, removed from
the drum
= p ×
16
2






2
× 94
= 6016p cm3
Volume of wood used to make the drum
= (114 400 – 6016)p
= 108 384p cm3
= 108 000p cm3
(to 3 s.f.)
37. (i) Volume of cylindrical container
= 3.142 × (14)2
× 40
= 24 633.28
= 24 600 cm3
(to 3 s.f.)
178

1
(ii) Surface area of one cylindrical container
= (2 × 3.142 × 14 × 40) + (2 × 3.142 × 142
)
= 4750.704 cm2
Surface area of 450 cylinders
= 450 × 4750.704
= 2 137 816.8 cm2
4200 cm2
surface requires 0.24 litres of paint.
2 137 816.8 cm2
requires 122.160 96 litres of paint.
123 litres of paint must be purchased to paint all
450 containers.
Cost to paint the containers
= $8.70 × 123
= $1070.10
38. (i) Volume of the rectangular block
= 12 × 18 × 10
= 2160 cm3
Volume of cylinder removed from the block
= 3.142 ×
7
2






2
× 18
= 692.811 cm3
Volume of remaining solid
= 2160 – 692.811
= 1467.189
= 1470 cm3
(to 3 s.f.)
(ii) Total surface area of the remaining solid
= 2 12 × 10 − 3.142 ×
7
2






2
















+ 2[(18 × 12) + (18 × 10)]
= 2[120 – 38.4895] + 2[396]
= 955.021
= 955 cm2
(to 3 s.f.)
Advanced
39. (i) Let the radius of cylinder A be r cm and the height
be h cm.
Volume of cylinder A
= pr2
h
= 343 cm3
Then the radius of cylinder B will be
1
3
r cm and
the height will be h cm.
Volume of cylinder B
= p ×
1
3
r






2
× h
= p ×
1
9
× r2
× h
=
1
9
× pr2
h
=
1
9
× 343
= 38.111
= 38.1 cm3
(to 3 s.f.)
(ii) Number of cubes formed
=
38.111
23
= 4.763…
The maximum number of cubes formed is 4.
40. Total thickness of the paper towel after it is being rolled
= (1 ÷ 10) × 90
= 9 cm
Total radius of the paper towel and roll
= 9 + 2.5
= 11.5 cm
Base area of the paper towel, in the form of a cylinder
=
22
7
× [(11.5)2
– (2.5)2
]
= 396 cm2
Volume of paper towel
= base area × width of towel
= 396 × 14
= 5544
= 5540 cm3
(to 3 s.f.)
New Trend
41. Surface area of cross-section
= π
1.3
2






2
+
1
2
× 2.6 × 2.25 − π
0.5
2






2
= 0.4225π + 2.925 − 0.0625π
= (0.36π + 2.925) cm2
Volume of platinum = 0.2(0.36π + 2.925)
= (0.072π + 0.585) cm3
Price of platinum used = 21.5(0.072π + 0.585) × $43.48
= $758.3210 (to 4 d.p.)
Total value of pendant = $4200 + $758.3210
42. Convert 100 litres to cm3
.
100 l = 100 × 1000 = 100 000 cm3
Volume of tank = cross-sectional area × height
Height = volume of tank ÷ cross-sectional area
= 100 000 ÷ π(30)2
= 35.4 cm (to 3 s.f.)
179

1
Chapter 15 Statistical Data Handling
Basic
1. (a)
Daily Earnings of ABC Pte Ltd for the Week
Monday
Sunday –
Each circle represents $1000.
Tuesday
Thursday
Saturday
Friday
Wednesday
(b) Total earnings for the week
= 4500 + 6000 + 6500 + 7000 + 12 000 + 8000
= $44 000
Percentage of Friday’s earning to the total earnings
for the week
=
12 000
44 000
× 100%
= 27
3
11
%
2. (i)
Number
of
Students
Prsent
0
36
40
34
38
42
33
37
41
35
39
43
Monday Wednesday
Tuesday Thursday Friday
Students Present for the Week
Number
of students
present
42 40
36 36 38
Day
(ii) All students were present on Monday.
(iii) Number of absentees on Friday = 42 – 38
= 4
Percentage of absentees on Friday
=
4
42
× 100%
= 9.52% (to 3 s.f.)
(iv) Ethan is right to say that because on Monday, everyone
is present. So, if student A is absent from Tuesday to
Friday, he is still present at least once in that week
and not absent for the whole week.
3. (a) Total number of foreign countries
= 9 + 6 + 8 + 12 + 5 = 40
Number of
foreign countries
Angle of sector
A
9
40
× 360° = 81°
B
6
40
× 360° = 54°
C
8
40 × 360° = 72°
D
12
40
× 360° = 108°
E
5
40
× 360° = 45°
Number of Foreign Countries
45°
B
A
C
D
E
81°
54°
72°
108°
(b) Total number of students surveyed
= 40 + 64 + 10 + 24 + 102 = 240
Mode of Transport Angle of sector
Bus
40
240
× 360° = 60°
Car
64
240
× 360° = 96°
Bicycle
10
240
× 360° = 15°
Foot
24
240
× 360° = 36°
MRT
102
240
× 360° = 153°
180

1
60°
Bus
Bicycle
Mode of Transport
Car
Foot
MRT
96°
15°
36°
153°
(c)
Sports Angle of sector
Badminton
70
600
× 360° = 42°
Basketball
90
600
× 360° = 54°
Athletics
105
600
× 360° = 63°
Soccer
205
600
× 360° = 123°
Tennis
130
600
× 360° = 78°
Badminton
Favourite Sports
54°
42°
63°
123°
78°
Athletics
Basketball
Soccer
Tennis
4. Total number of students in the school
= 30 + 20 + 10 + 20
= 80
Angle of the smallest sector
=
10
80
× 360°
= 45°
It represents the number of Secondary 3 students in a
school for the year 2013.
5. (i)
Mass
(kg)
0.00
2.00
6.00
4.00
8.00
1.00
3.00
7.00
5.00
9.00
0.50
2.50
6.50
4.50
8.50
1.50
3.50
7.50
5.50
9.50
10.00
2 4 6 9
1 3 5 7 8 10
Mass of a Baby from Birth to 10 months
Month
(ii) From the line graph, the increase in the mass
of the baby is the largest between the 5th
and
6th
months.
(iii) From the line graph, the first decrease in the mass
is on the 8th
month.
(iv) Total mass of the baby from birth to 10 months
= 3.7 + 3.8 + 4.7 + 5.4 + 6.6 + 8.2 + 8.3 + 8.2
+ 8.7 + 9.4
= 67 kg
Average mass of the baby
=
67
10
= 6.7 kg
6. (i)
Nunber
of
Hours
(in
1000
hours)
0
80
240
160
40
120
200
20
100
180
60
140
220
2003 2007
2005 2009 2011
2004 2008
2006 2010 2012
Time Spent on Work
Year
(ii) The years in which there was a decrease in the
number of hours the workers spent in work are
2004, 2008 and 2009.
(iii) The years in which there was an increase in the
number of hours the workers spent in work are
2005, 2006, 2007, 2010, 2011 and 2012.
(iv) From the line graph, the year in which the increase is
the largest is 2011 and the year in which the increase
is the least is 2007.
181

1
(v) The possible years in which the workers spent more
than 172 000 hours in work are 2003, 2005, 2006,
2007 and 2012.
Intermediate
7. (i) (a) Number of cars produced on Tuesday
= 6.5 × 20
= 130
(b) Number of cars produced on Thursday
= 5 × 20
= 100
(c) Number of cars produced on Saturday
= 0 × 20 = 0
(ii) The greatest number of cars produced was on Tuesday.
(iii) Production line has stopped for half a day on
Wednesday. One possible indication is that the
number of cars produced is low as compared to
the other days. The number of cars produced on
Wednesday is approximately half the number of cars
produced on Monday and on Thursday.
(iv) Increase in production of cars from Monday to
Tuesday
= 130 – 80
= 50
Percentage increase
=
50
80
× 100%
= 62.5%
(v) One possible explanation may be the workers are
resting on weekends. The other reason may be there
may not be orders on weekends and the number of
cars produced on Friday may be sufficient to meet
the demands for the coming week.
(vi) Total number of cars produced
= 80 + 130 + 50 + 100 + 120
= 480
8. (a) (i) Number of students in the class
= 6 + 7 + 10 + 8 + 4 + 3 + 3
= 41
(ii) Most students have 2 coins.
(iii) Total number of coins
= 7 × 1 + 10 × 2 + 8 × 3 + 4 × 4 + 3 × 5 + 3 × 6
= 7 + 20 + 24 + 16 + 15 + 18
= 100
Average number of coins
=
100
41
= 2.44 (to 3 s.f.)
(iv) Number of students having 4 or more coins
= 4 + 3 + 3
= 10
Percentage of students having 4 or more coins
=
10
41
× 100%
= 24.4% (to 3 s.f.)
(b) Angle representing students having 0 coins
=
6
41
× 360°
= 52.7°
Angle representing students having 1 coin
=
7
41
× 360°
= 61.5°
Angle representing students having 2 coins
=
10
41
× 360°
= 87.8°
Angle representing students having 3 coins
=
8
41
× 360°
= 70.2°
Angle representing students having 4 or
more coins
=
10
41
× 360°
= 87.8°
9. (a) (i) February
(ii) June
(iii) August
(b) The month in which he is the heaviest is in the month
of November. His weight is about 54 kg.
(c) The months in which his weights were the same are
May, October and December.
(d) His largest weight = 54 kg
His smallest weight = 46 kg
Range of weight = 54 – 46
= 8 kg
(e) (i) On 1st June, he lost weight greatly after his weight
increased for the past 5 months. Therefore, he
was sick in May.
(ii) On 1st December, he lost weight slightly after his
weight increased for the past 5 months. Therefore,
he was controlling his diet in November.
(f) November
182

1
(g)
0
30
40
50
35
45
55
Jan May Sep
Mar Jul Nov
Feb Jun
Month
Michael’s Weight for the Year
Weight
(in
kg)
Oct
Apr Aug Dec
(h) Line graph is more suitable to represent and interpret
the above data as we can observe the trends of his
weight over the months easily.
We can observe the increase or decrease of his
weight easily from the line graph.
10. (a) (i) The value of sales in 2007 is 64 × $10 000
= $640 000.
(ii) The value of sales in 2009 is 110 × $10 000
= $1 100 000.
(iii) The value of sales in 2011 is 140 × $10 000
= $1 400 000.
(b) The value of sales is $1 000 000 in 2008.
(c) Between 2009 and 2010, the increase in the value
of sales is the greatest.
The maximum value of sales (from 2009 to 2010)
= (160 × $10 000) – $1 100 000
= $500 000
(d) Amount exceeded the sales target
= $1 600 000 – $1 300 000
= $300 000
Percentage of amount exceeded the target
=
300 000
1 300 000
× 100%
= 23
1
13
%
(e) Amount below the sales target
= $1 650 000 – $1 400 000
= $250 000
Percentage of amount below the target
=
250 000
1 650 000
× 100%
= 15
5
33
%
(f) Total value of sales over the past 6 years
= (64 + 100 + 110 + 160 + 140 + 50) × $10 000
= $6 240 000
(g) The sudden increase may be due to the increase in the
popularity of the product. Another reason may be the
population in the country has increased over the past
year and the demand for the product increases as it is
a necessity.
11. (i) 5x° + 2x° + 52° = 360°
7x° + 52° = 360°
7x° = 308°
x° = 44°
x = 44
(ii) 2 × 44° = 88° represents 66 vehicles
1° represents 0.75 vehicles
360° represents 0.75 × 360 = 270 vehicles
The total number of vehicles included in the
survey is 270.
12. (i) When it rained the whole day, the average temperature
should be the lowest among the 10 days. In this case,
the day in which it rained the whole day is Monday
during the 1st week and its temperature is 24°C.
(ii) Friday, the 1st week; the temperature in the classroom
on that day is 31°C.
(iii) The days when the temperature is below 29°C are 1st
week on Monday, Tuesday and Wednesday and 2nd
week on Friday.
(iv) Number of days in which the temperature is above
28°C
= 6
Percentage of days in which the temperature is above
28°C
=
6
10
× 100%
= 60%
(v) The sudden increase in temperature may be due to
a change in weather. Another reason may be the
monsoon season has ended and the temperature has
resumed to its initial temperature before the monsoon
season.
183

1
13. (i) The sale first exceeds the 50 000 mark in year 2010.
(ii) In year 2012, the sale was exactly 100 000.
(iii) Between 2011 and 2012, the sales in the soap powder
were the greatest.
(iv)
Year 2008 2009 2010 2011 2012
Number of
Packets (in
thousands)
40 40 60 70 100
(v) Increase in sales from 2010 to 2012
= 100 000 – 60 000
= 40 000
Percentage increase in sales from 2010 to 2012
=
40 000
60 000
× 100%
= 66
2
3
%
Advanced
14. Yes.
Suggested answer:
The increase in the size of the diagram does not represent
accurately that the sales have increased by 300%. What the
advertisement is trying to show is that there is an increase
in the sales but it is unable to represent the increase as
300%.
15. No.
Suggested answer:
The charts did show an increase in the radius of the circle
by two times. However, the actual figures of the sales are
not given. Therefore, it is not conclusive that the sales
have doubled from the year 2010 to 2012.
Suggestion: A better representation is a bar graph which
compares the sales in 2010 and 2012 using bars.
16. No. I do not agree with Amirah. The person who collected
the data did not mention whether taking more projects of
the same nature contributes to people involved in more
community work.
Reason 1: More people may have increased their
involvement from May to June by taking part in more
projects within the same organisation. Therefore, the
nature of the projects may not have changed but the number
of projects involved has increased.
Reason 2: There may be a higher chance of people
involving in community work due to demand for more
volunteers as part of the school’s holiday programmes.
New Trend
17. (a) Number of females who use public transport
= 55 − 20 − 2 − 9
= 24
(b) Angle representing students walking to school
=
16
120
× 360°
= 48°
(c) For males,
percentage who travel using other modes of transport
=
12
65
× 100%
= 18.462% (to 5 s.f.)
For females,
percentage who travel using other modes of transport
=
9
55
× 100%
= 16.364% (to 5 s.f.)
Difference in percentage = 18.462% − 16.364%
= 2.10%
A greater percentage travel using other modes of
transport in males as compared to females. The
percentage in males is 2.10% higher.
18. (a) Ratio of manufactured goods and the minerals
=
85
115
=
17
23
= 17 : 23
(b) Angle representing agricultural produce
= 360° – 10° – 85° – 115°
= 150°
Ratio of agricultural produce and the manufactured
goods
=
150
85
=
30
17
= 30 : 17
(c) 115° represent 23 million
1° represents 0.2 million
360° represent 72 million
The total value of exports of the country is 2012 is
72 million.
184

1
185
Revision Test D1
1. (i) Length of OA = 28 – 11.2 = 16.8 cm
Circumference of semicircle with diameter
28 cm
=
1
2
× 2 × 3.142 ×
28
2






= 43.988 cm
Circumference of semicircle with diameter
11.2 cm
=
1
2
× 2 × 3.142 ×
11.2
2






= 17.5952 cm
Perimeter of figure
= 43.988 + 17.5952 + 16.8
= 78.3832
= 78.4 cm (to 3 s.f.)
(ii) Area of semicircle with diameter 28 cm
=
1
2
× 3.142 ×
28
2






2
= 307.916 cm2
Area of semicircle with diameter 11.2 cm
=
1
2
× 3.142 ×
11.2
2






2
= 49.266 56 cm2
= 307.916 – 49.266 56
= 258.649 44
= 259 cm2
(to 3 s.f.)
2. Length of DC = 4.7 – 2.3 = 2.4 cm
Area of BDC =
1
2
× DC × BD
2.64 =
1
2
× 2.4 × BD
2.64 = 1.2 × BD
BD = 2.2 cm
Area of trapezium =
1
2
× (2.3 + 4.7) × 2.2
= 7.7 cm2
3. Base area = (140 × 90) – [(140 – 30 – 30) × 60]
= 12 600 – 4800
= 7800 cm2
Volume of solid
= base area × width
= 7800 × 50
= 390 000 cm3
Total surface area
= 2(7800) + 2(50 × 90) + 2(30 × 50) + (140 × 50)
+ 2(60 × 50) + (80 × 50)
= 15 600 + 9000 + 3000 + 7000 + 6000 + 4000
= 44 600 cm2
4. (i) Volume of water in the cylindrical tank
= 3.142 ×
140
2






2
× 60
= 923 748
= 924 000 cm3
(to 3 s.f.)
(ii) Let the height of water in the trough be h cm.
Volume of water in the cylindrical tank
= volume of water in trough
923 748 = 120 × 80 × h
h = 96.223 75
= 96.2 cm (to 3 s.f.)
5. (a) Volume of cylindrical tank
= 3.142 ×
350
2






2
× 260
= 25 018 175 cm3
= 25 018.175 l
Time taken for the pipe to fill the tank
=
25 018.175
25
= 1000.727
= 1001 seconds
(to the nearest second)
(b) Volume of pipe
= 3.142 ×
6
2






2
× 120








– 3.142 ×
4.8
2






2
× 120








= 3393.36 – 2171.7504
= 1221.6096 cm3
Mass of pipe
= density × volume
= 7.6 × 1221.6096
= 9284.232 96 g
= 9.28 kg (to 3 s.f.)
6. (a) 5x° + 2x° + 255° = 360°
7x° = 360° – 255°
7x° = 105°
x° = 15°
∴ x = 15
(b) 255° represent 153 cars.
1° represents 0.6 cars.
5 × 15° = 75° represent 0.6 × 75° = 45.
There are 45 motorcycles in the car park.

1 186
Revision Test D2
1. (a) Area of AKC
=
1
2
× 6.8 × 5.6
= 19.04 cm2
Area of AKB
=
1
2
× 6.8 × 6.4
= 21.76 cm2
= 19.04 + 21.76
= 40.8 cm2
(b) Area of semicircle with diameter 24 cm
=
1
2
× 3.142 ×
24
2






2
= 226.224 cm2
Area of ABC
=
1
2
× 24 × 8
= 96 cm2
= 226.224 – 96
= 130.224
= 130 cm2
(to 3 s.f.)
2. Let the length of the cube be l cm.
Volume of solid cube = l3
∴ l3
= 125
l = 5
Total surface area of cube
= 6l2
= 6 × 52
= 150 cm2
3. (i) Volume of the cake before it is being cut
= 3.142 × (14)2
× 8
= 4926.656 cm3
Volume of remaining cake
=
3
4
× 4926.656
= 3694.992
= 3690 cm3
(to 3 s.f.)
(ii) Total surface area of the cake after it is being
cut
= (2 ×
3
4
× 3.142 × (14)2
)
+ (2 ×
3
4
× 3.142 × 14 × 8) + 2(14 × 8)
= 923.748 + 527.856 + 224
= 1675.604
= 1680 cm2
(to 3 s.f.)
4. Volume of water in the cylindrical container
=
11
14
× 3.142 ×
28
2






2
× 35








= 16 935.38 cm3
Number of glasses of water
=
16 935.38
245
= 69.124
≈ 69
Volume of water = 69 × 245
= 16 905 cm3
Volume of water left in the container
= 16 935.38 – 16 905
= 30.38
= 30.4 cm3
(to 3 s.f.)
5. Since water is discharged through the pipe at a rate
of 8 km/h, the volume of water discharged is the
volume of water that fills the pipe to a length of
8 km or 8000 m.
= volume of pipe of length 8000 m
= πr2
h
= 3.142 × [2.4 ÷ 100]2
× 8000
= 14.478 336 m3
Volume of rectangular tank
= 4 × 3 × 2.8
= 33.6 m3
Amount of time needed to fill the tank
= 33.6 ÷ 14.478 336
= 2.3208 hours
= 139 minutes (to the nearest minute)

1
187
6. (i)
Number
of
Students
0
80
160
240
40
120
200
280
20
100
180
260
60
140
220
300
2006 2008 2010
2007 2009 2011 2012
Number of students
Year
(ii) Between 2009 and 2010, the school has the greatest
increase in the number of students taking additional
mathematics.
(iii)Total number of students
= 80 + 100 + 110 + 70 + 160 + 200 + 240
= 960
Angle of sector that represents the number of
students taking additional mathematics in 2009
=
70
960
× 360°
= 26.25
= 26.3° (to 3 s.f.)
(iv) One possible reason for the sudden increase may
be the school has increased the number of classes
taking additional mathematics from 2.5 classes to
about 4 classes. This may be due to a change in
the expectation by the Ministry of Education or an
increase in demand requested by parents.

1 188
End-of-Year Examination Specimen Paper A
Part I
1. (a) 37 850 = 38 000 (to 2 s.f.)
(b) 1.3249 = 1.32 (to 2 d.p.)
2. (a) Convert 47.56 cm to mm.
47.56 cm = 47.56 × 10 = 475.6 mm
475.6 mm = 476 mm (to the nearest mm)
(b) 75 489 cm2
= 75 500 cm2
(to the nearest 100 cm2
)
3. (a) {[(56 + 34) ÷ 5 – 7] × 3 – 17} ÷ 4
= {[90 ÷ 5 – 7] × 3 – 17} ÷ 4
= {[18 – 7] × 3 – 17} ÷ 4
= {11 × 3 – 17} ÷ 4
= {33 – 17} ÷ 4
= 16 ÷ 4
= 4
(b) (–2)2
– (7 – 8)2
– (11 – 15)3
=
4 – (−1)2
– (–4)3
= 4 – 1 – (–64)
= 4 – 1 + 64
= 67
4. (a) 3
1
4
+ 1
1
4
÷
3
8
= 3
1
4
+
5
4
÷
3
8
= 3
1
4
+
5
4
×
8
3
= 3
1
4
+ 3
1
3
= 6
7
12
(b) 5
5
8
–
2
3






3
÷
9
16
= 5
5
8
–
8
27
÷
3
4
= 5
5
8
–
8
27
×
4
3
= 5
5
8
–
32
81
= 5
149
648
5. 12 = 22
× 3
28 = 22
× 7
64 = 26
22
HCF of 12, 28 and 64 = 22
= 4
12 = 22
× 3
28 = 22
× 7
64 = 26
26
3 7
LCM of 12, 28 and 64 = 26
× 3 × 7 = 1344
The HCF and LCM of 12, 28 and 64 are 4 and 1344
respectively.
6. (a) 2x – 2[3(1 – x) – 5(x – 2y)]
= 2x – 2[3 – 3x – 5x + 10y]
= 2x – 2[3 – 8x + 10y]
= 2x – 6 + 16x – 20y
= 2x + 16x – 20y – 6
= 18x – 20y – 6
(b) (i) 15hx + 10hy – 5h
= 5h(3x + 2y – 1)
(ii) 3p(x + h) + p(2h – 1)
= p[3(x + h) + (2h – 1)]
= p(3x + 5h – 1)
7. (a) When a = 2, b = 3 and c = –1,
5(2) – (–1)
3 – (–1)2
+
+
3 (–1)
2 – 2(3)
3
3
=
+
10 1
2
+
3 – 1
8 – 6
= 5
1
2
+ 1
= 6
1
2
(b)
x
3 – 2
4
=
+
x 2
5
–
x
2 – 1
10
20 ×
x
3 – 2
4
=
x + 2
5
–
2x – 1
10





 × 20
5(3x – 2) = 4(x + 2) – 2(2x – 1)
15x – 10 = 4x + 8 – 4x + 2
15x – 10 = 4x – 4x + 8 + 2
15x = 8 + 2 + 10
15x = 20
x = 1
1
3

1
189
8. (a)
×
63.2 2.8
5.53
+
×
2.826
0.9 1.57
≈
×
63 3
6
+
×
3
1 2
= 31.5 + 1.5
= 33
= 30 (to 1 s.f.)
(b) Decrease = $38.50 − $30.80
= $7.70
Percentage decrease =
7.70
38.50
× 100%
= 20%
9. (a) Sum of interior angles in a n-sided polygon
= (n – 2) × 180°
153° + 144° + 100° + [(n – 3) × 149°]
= (n – 2) × 180°
397° + 149n° – 447° = 180n° – 360°
180n° – 149n° = 397° – 447° + 360°
31n° = 310°
n° = 10°
n = 10
(b) Area of circle = pr2
154 = 3.142 × r2
r = 7.00 cm (to 3 s.f.)
(c) Let the length of the cube be l cm.
Surface area of cube = 6l2
150 = 6l2
l = 5
Volume of a cube = l3
= 53
= 125 cm3
10. (a) Draw a line CD through F that is parallel to
AB and PQ.
A
G
C
P
B
H
D
F
E
x°
R
58°
52°
23°
Q
PFC = 23° (alt. s, CD // PQ)
CFE = 52 – 23 = 29°
Draw a line GH through E that is parallel to AB
and PQ.
GEF = 180° – 29° = 151° (int. s, GH // CD)
GER = 180° – 58° = 122° (int. s, GH // AB)
x° = 151° + 122°
= 273°
x = 273
(b) One semicircle and two quarters of diameter
14 cm are removed means a circle of diameter
14 cm is removed.
Area of square = 14 × 14
= 196 cm2
= 3.142 ×
14
2






2
= 153.958 cm2
= 196 – 153.958
= 42.042
= 42.0 cm2
(to 3 s.f.)
11. (i) 5x° + 81° + 2x° + 55° = 360°
5x + 2x = 360 – 81 – 55
7x = 224
x = 32 (shown)
(ii) Percentage of students who chose Science
=
81
360
× 100%
= 22.5%
(iii) 5 × 32 = 160° represent 48 students.
1° represents 0.3.
360° represent 0.3 × 360 = 108.
The total number of students in the group is 108.

1 190
Part II
Section A
1. (a) 5x > 16
x >
16
5
x > 3
1
5
(b) 1 km requires
1
12.4
=
5
62
litres.
300 km require
5
62
× 300 = 24
6
31
litres.
The minimum number of petrol required is
25 litres (to the nearest whole number).
(Note: The answer is not 24 litres as it requires
slightly more than 24 litres of petrol to run
300 km.)
2. (a) 7x – {3x – 4(2x – y) – [(5x – 3y) – (2y – 3x)]}
= 7x – {3x – 4(2x – y) – [5x – 3y – 2y + 3x]}
= 7x – {3x – 4(2x – y) – [5x + 3x – 3y – 2y]}
= 7x – {3x – 4(2x – y) – [8x – 5y]}
= 7x – {3x – 8x + 4y – 8x + 5y}
= 7x – {3x – 8x – 8x + 4y + 5y}
= 7x – {−13x + 9y}
= 7x + 13x – 9y
= 20x – 9y
(b)
x
1
4 –
+ 3 =
x
3
– 4
–
1
x – 4





 + 3 =
x
3
– 4
x
3
– 4
+
x
1
– 4
= 3
x
4
– 4
=
3
1
3(x – 4) = 4
3x – 12 = 4
3x = 4 + 12
3x = 16
x = 5
1
3
(c) Let the breadth of the rectangle be x cm.
Then the length of the rectangle will be 2x cm.
2(2x + x) = 24
2(3x) = 24
6x = 24
x = 4
Then the length of the rectangle is 4 cm and its
breadth is 8 cm.
= 32 cm2
3. (a) 0.086 = 0.086 × 100%
= 8.6%
(b) 42
1
8
% =
42
1
8
100
= 0.42 125
(c)
60
100
× 6.8 m = 4.08 m
4. (i) 8 + 9 + x + 7 + 3 + 1 = 40
x = 40 – 8 – 9 – 7 – 3 – 1
= 12
(ii)
Number
of
Students
0
4
8
12
2
6
10
14
1
5
9
13
3
7
11
0 2 4
1 3 5
Number of Siblings of 40 Students
Number of Siblings
(iii) Angle representing students with 1 sibling
=
9
40
× 360°
= 81°

1
191
Section B
5.
S
A
6.5 cm
9.4 cm
B
T
C
D
(ii)
(i)
(i)
75°
85°
(i) Length of AD = 7.1 cm
Length of CD = 6.7 cm
(ii) Length of ST = 8.5 cm
6. (i) x –2 0 2
y = 5 + 4x –3 5 13
y = –2x + 2 6 2 –2
A
B
x
y
–3
–2
–2 –1 1 2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
–1
0
y = 5 + 4x
2x + y = 2
C
(ii) From the graph, the coordinates of A, B and C are
(0, 5), (0, 2) and –
1
2
, 3





 .
Area of ABC
=
1
2
× (5 – 2) ×
1
2
=
3
4
square units

1 192
7. (a) Time taken for the first part of the journey
=
240
60
= 4 hours
Time taken for the second part of the journey
= 8
4
5
– 4
= 4
4
5
hours
Distance for the second part of the journey
= 600 – 240
= 360 km
Average speed for the second part of the journey
=
360
4
4
5
= 75 km/h
(b) (i)
1
5
–
1
6
=
1
30
i.e. 30 = 5 × 6
(ii) 182 = x(x + 1)
Since 13 × 13 = 169, we can try 13 × 14.
13 × 14 = 182
x = 13
8. (a) (i) Convert 1.2 mm to cm.
1.2 mm = 1.2 ÷ 10 = 0.12 cm
Convert 200 m to cm.
200 m = 200 × 100 = 20 000 cm
Volume of copper wire
= pr2
h
= 3.142 × (0.12 ÷ 2)2
× 20 000
= 226.224
= 226 cm3
(to 3 s.f.)
(ii) Mass of wire
= density × volume of wire
= 8.9 × 226.224
= 2013.3936
= 2010 g (to 3 s.f.)
(b) Area of trapezium =
1
2
[(2x – 3) + (3x + 4)] × 12
216 =
1
2
[2x – 3 + 3x + 4] × 12
216 =
1
2
[2x + 3x – 3 + 4] × 12
216 = 6[5x + 1]
30x + 6 = 216
30x = 216 – 6
30x = 210
x = 7
(c) Volume of water in cylinder B
= p × [(3x) ÷ 2]2
× 20
= 45px2
cm3
Let the height of water in cylinder A be h cm.
Volume of water in cylinder A
= p × [(5x) ÷ 2]2
× h
45px2
= 6.25px2
h
h =
45πx2
6.25πx2
= 7.2
Height of water in cylinder A is 7.2 cm.

1
193
End-of-Year Examination Specimen Paper B
Part I
1. (a) 4.002 56 = 4.00 (to 2 d.p.)
(b) 0.002 045 6 = 0.00 205 (to 3 s.f.)
(c) 10.0245 cm2
= 10.025 cm2
to the nearest
1
1000
cm2






2. (a) (11 – 7)2
– 72
– (28 – 33)3
= (4)2
– 49 – (–5)3
=
16 – 49 – (−125)
= 16 – 49 + 125
= 92
(b) 21 + (–65) ÷ 5 × {3 + [42 ÷ (–7)]}
= 21 + (–65) ÷ 5 × {3 + [–6]}
= 21 + (–65) ÷ 5 × {−3}
= 21 + 39
= 60
3. (a) 7x – 5(4x – 5) = 2(3x – 2) – 9
7x – 20x + 25 = 6x – 4 – 9
6x – 7x + 20x = 25 + 4 + 9
19x = 38
x = 2
(b)
+
x
x
– 2
4 1
= 0.5
+
x
x
– 2
4 1
=
1
2
2(x – 2) = 4x + 1
2x – 4 = 4x + 1
4x – 2x = –5
2x = –5
x = −2
1
2
4. Let the first odd number be n.
Then the next consecutive odd number will be n + 2.
4(n + 2) +
1
3
n = 73
4n + 8 +
1
3
n = 73
4
1
3
n = 73 – 8
4
1
3
n = 65
n = 15
The two consecutive odd numbers are 15 and
15 + 2 = 17.
5. (i) 2 4900
2 2450
5 1225
5 245
7 49
7 7
1
4900 = 22
× 52
× 72
3 9261
3 3087
3 1029
7 343
7 49
7 7
1
9261 = 33
× 73
(ii) 4900 = 22
× 52
× 72
= (2 × 5 × 7)2
9261 = (3 × 7)3
4900 = 2 × 5 × 7
9261
3
= 3 × 7
HCF of 4900 and 9261
3
= 7
LCM of 4900 and 9261
3
= 2 × 3 × 5 × 7
= 210
6. (i) 3(x – 2) – 12 + 5(3 – x)
= 3x – 6 – 12 + 15 – 5x
= 3x – 5x – 6 – 12 + 15
=
−2x – 3
(ii) 3(x – 2) – 12 + 5(3 – x) ø 5
3x – 6 – 12 + 15 – 5x ø 5

−2x – 3 ø 5
2x ø −3 – 5
2x > –8
x > –4

1 194
7. (a) Time taken for the first 120 km
=
120
40
= 3 hours
Distance for the remaining journey
= 200 – 120
= 80 km
Time taken for the remaining 80 km
=
80
60
= 1
1
3
hours
Total time taken for the whole journey
= 3 + 1
1
3
= 4
1
3
hours or 4 hours 20 minutes
(b) Let the distance of AB be d km.
d
16
+
d
24
= 5
+
d d
3 2
48
= 5
d
5
48
= 5
5d = 5 × 48
5d = 240
d = 48
The distance of AB is 48 km.
8. The ratio is 8 : 7 : 5.
8 + 7 = 15 parts represent $270.
1 part represents $18.
8 + 7 + 5 = 20 parts represent $18 × 20 = $360.
The sum of money to be divided among the three boys is
$360.
9. AGB = 49° (vert. opp. s)
55° + 49° + y° = 180° ( sum of )
y° = 180° – 55° – 49°
= 76°
GBD = 49° (alt. s, AE // BD)
x° + 76° + 49° + 128° = 360°
x° = 360° – 76° – 49° – 128°
= 107°
x = 107 and y = 76
10. (i) Arc length of quadrant
=
1
4
× (2 × p × 10)
= 5p cm
Perimeter of figure
= 5p + (28 – 10) + 3 + 5 + 5 + 5 + (12 – 5 – 3)
+ 28 + (12 – 10)
= (70 + 5p) cm
(ii) Area of quadrant
=
1
4
× p × (10)2
= 25p cm2
Area of figure
= area of rectangle – area of quadrant
– area of square
= (28 × 12) – 25p – (5 × 5)
= 336 – 25p – 25
= (311 – 25p) cm2
11. (a) Size of each interior angle of a 24-sided
regular polygon
=
(24 – 2) × 180°
24
= 165°
(b) An octagon has 8 sides.
Sum of angles in an octagon = (8 – 2) × 180°
= 1080°
Let one of the remaining interior angles be x°.
86° + (8 – 1)x° = 1080°
86° + 7x° = 1080°
7x° = 1080° – 86°
= 994°
x° = 142°
The size of one of the remaining interior angles
of the octagon is 142°.
(c) A hexagon has 6 sides.
Sum of angles in a hexagon = (6 – 2) × 180°
= 720°
3 + 3 + 3 + 3 + 4 + 4 = 20 parts represent 720°
1 part represents 36°.
3 parts represent 36° × 3 = 108°.
The smallest interior angle gives the largest
exterior angle.
Largest exterior angle
= 180° – 108°
= 72°

1
195
12. (a) (i)
Z
X
A
Y
(ii)
(iii)
6.9 cm
4.8 cm
5.8 cm
(ii) ∠XYZ = 43°
(iii) Length of AX = 2.6 cm
(b) (i) 2x° + 72° + x° + 90° = 360°
2x + x + 72 + 90 = 360
3x = 360 – 72 – 90
3x = 198
x = 66
(ii) Percentage of students who chose cinema as their
favourite form of entertainment
=
72
360
× 100%
= 20%
Part II
Section A
1. (a) Percentage of his income on savings
= 100% – 10% – 15% – 12% – 8% – 21%
= 34%
34% represent $1292.
1% represents $38.
100% represent 38 × 100 = $3800
His monthly income is $3800.
(b) Price of car after first year
= (100 – 15)% of $56 000
= $47 600
Price of car after second year
= (100 – 15)% of $47 600
= $40 460
Price of car after third year
= (100 – 15)% of $40 460
= $34 391
Price of car after fourth year
= (100 – 15)% of $34 391
= $29 232.35
= $29 200 (to the nearest 100 dollars)
2. (i)
C (6, –7)
A (1, 3)
B (–2, 0)
4
1
2
3
y
–7
–6
–5
–7
–6
–5
–3
–1
–4
–2
–2
–3
–4 –1 4
2 3 6
5
0
x
1
(ii) Area of ABC
=
1
2
× (2.5 + 2) × 3





 +
1
2
× (2.5 + 2) × 7






= 6.75 + 15.75
= 22.5 square units
3. (a) When s = 110, v = 36.5 and u = 2.56,
110 =
a
(36.5) – (2.56)
2
2 2
2a =
(36.5) – (2.56)
110
2 2
2a = 12.051 785 45
a = 6.025 892 727
= 6.03 (to 3 s.f.)
(b) 2xa – 8pa + 4ya – 6a
= 2a(x – 4p + 2y – 3)
(c)
x
4 – 1
4
–
x
5 –
2
=
x
5(7 – 2 )
6
+
11
12
12 ×
4x – 1
4
–
5 – x
2






=
5(7 – 2x)
6
+
11
12





 × 12
3(4x – 1) – 6(5 – x) = 10(7 – 2x) + 11
12x – 3 – 30 + 6x = 70 – 20x + 11
12x + 20x + 6x = 70 + 11 + 3 + 30
38x = 114
x = 3

1 196
4. We observe that 5 = 1 + 22
, 14 = 5 + 32
and so on.
(i) She counted 14 + 42
= 30 squares.
(ii) 1 =
+ +
1(1 1)(2 1)
6
, 5 =
+ +
2(2 1)(4 1)
6
,
14 =
+ +
3(3 1)(6 1)
6
, 30 =
+ +
4(4 1)(8 1)
6
The general formula for the nth
term of the
sequence is
+ +
n n n
( 1)(2 1)
6
.
(iii) When n = 51, the number of squares is
=
+ × +
51(51 1)(2 51 1)
6
=
51(92)(103)
6
= 45 526
Section B
5. (a) Let 1 part of the length of the sides of the quadrilateral
be n cm.
2n + 3n + 6n + 7n = 108
18n = 108
n = 6
The longest side is 7 × 6 = 42 cm.
The shortest side is 2 × 6 = 12 cm.
Difference between the length of the longest and
the shortest sides
= 42 – 12
= 30 cm
(b) Percentage of bulbs that are not defective
= 100% – 6%
= 94%
94% represents 611 bulbs.
1% represents 6.5 bulbs.
100% represents 6.5 × 100 = 650 bulbs.
He must produce 650 bulbs in order to obtain
611 bulbs which are not defective.
6. (i) Ratio of weight of pineapple, sugar and water
= 6 : 9 : 16
2
3
– 6 – 9






= 6 : 9 : 1
2
3
= 18 : 27 : 5
(ii) Total weight loss = 16
2
3
– 15 = 1
2
3
kg
Ratio of total weight loss and the original weight
of mixture
= 1
2
3
: 16
2
3
= 5 : 50
= 1 : 10
(iii) Total cost of producing 15 kg of pineapple jam
= cost of pineapples + cost of sugar +
cost of electricity
= [(0.80) × 6] + [$1.20 × 9] + [2 × (0.45)]
= 4.80 + 10.80 + 0.90
= $16.50
Cost of each kg of pineapple jam
=
16.50
15
= $1.10
7. (i) Area of cross-section ABCDEFG
= area of rectangle ABCG – area of semicircle of
diameter 40 cm DEF
= (40 × 60) –
1
2
× 3.142 ×
40
2






2








= 2400 – 628.4
= 1771.6
= 1770 cm2
(to 3 s.f.)
(ii) Volume of slab
= area of cross-section ABCDEFG × length of slab
= 1771.6 × 120
= 212 592
= 213 000 cm3
(to 3 s.f.)
(iii) Total surface area of the slab
= 2(1771.6) + 2(120 × 40) + (60 × 120)
+ 2(10 × 120) +
1
2
× 2 × 3.142 ×
40
2
× 120






= 3543.2 + 9600 + 7200 + 2400 + 7540.8
= 30 284
= 30 300 cm2
(to 3 s.f.)
(iv) Convert density to kg/cm3
.
2300 kg/m3
= × ×
2300 kg
(100 100 100) cm3
= 0.0023 kg/cm3
Mass of slab
= density × volume
= 0.0023 × 212 592
= 488.9616
= 490 kg (to 3 s.f.)

1
197
8. (i)

Number
of
intervals
0
4
8
12
15
18
21
2
6
10
14
17
20
1
5
9
13
16
19
22
3
7
11
0 2 4 6
1 3 5 7
Calls Received in a Minute Interval
Number of calls per minute
(ii) From the line graph, the most common number
of calls per minute is 3.
(iii) 18 intervals
(iv) Total number of intervals
= 9 + 14 + 18 + 21 + 17 + 10 + 6 + 5
= 100
Number of intervals with more than 3 calls per minute
= 17 + 10 + 6 + 5
= 38
Ratio of the number of intervals with more than
3 calls to total number of intervals
= 38 : 100
= 19 : 50
(v) Number of intervals with 3 or less calls per minute
= 100 – 38
= 62
Percentage of intervals with 3 or less calls
per minute =
62
100
× 100%
= 62%
(vi) Angle representing 0 calls per minute
=
9
100
× 360° = 32.4°
Angle representing 1 call per minute
=
14
100
× 360° = 50.4°
Angle representing 2 calls per minute
=
18
100
× 360° = 64.8°
=
21
100
× 360° = 75.6°
=
17
100
× 360° = 61.2°
=
10
100
× 360° = 36°
=
6
100
× 360° = 21.6°
=
5
100
× 360° = 18°
(vii)Answer varies. One possible reason: for calls of
more than 4, it may be because some callers hang
up the phone before even speaking to the operators.
Therefore, more calls are recorded.

workbook_full_solutions_1 (1).pdf

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