just reference

2
A. BASIC ARITHMETIC
• Foundation of modern day life.
• Simplest form of mathematics.
Four Basic Operations :
• Addition plus sign
• Subtraction minus sign
• Multiplication multiplication sign
• Division division sign
x
Equal or Even Values equal sign

3
1. Beginning Terminology
Arabic number system - 0,1,2,3,4,5,6,7,8,9
• Digits - Name given to place or position of each numeral.
Number Sequence
2. Kinds of numbers
• Whole Numbers - Complete units , no fractional parts. (43)
May be written in form of words. (forty-three)
• Fraction - Part of a whole unit or quantity. (1/2)
• Numbers - Symbol or word used to express value or quantity.Numbers
Digits
Whole Numbers
Fraction

4
2. Kinds of numbers (con’t)
• Decimal Numbers - Fraction written on one line as whole no.
Position of period determines power of decimal.
Decimal Numbers

5
• Number Line - Shows numerals in order of value
• Adding on the Number Line (2 + 3 = 5)
• Adding with pictures
B. WHOLE NUMBERS
1. Addition
Number Line
Adding on the Number Line
Adding with pictures

6
1. Addition (con’t)
• Adding in columns - Uses no equal sign
5
+ 5
10
897
+ 368
1265
Simple Complex
Answer is called “sum”.
Table of Digits
Adding in columns

7
ADDITION PRACTICE EXERCISES
1. a. 222
+ 222
b. 318
+ 421
c. 611
+ 116
d. 1021
+ 1210
2. a. 813
+ 267
b. 924
+ 429
c. 618
+ 861
d. 411
+ 946
3. a. 813
222
+ 318
b. 1021
611
+ 421
c. 611
96
+ 861
d. 1021
1621
+ 6211
444 739 727 2231
1080 1353 1479 1357
1353 2053 1568 8853

8
2. Subtraction
• Number Line - Can show subtraction.
Number Line Subtraction with pictures
Position larger numbers above smaller numbers.
If subtracting larger digits from smaller digits, borrow from
next column.
5 3 8
- 3 9 7
1
4 1
41
Number Line

9
SUBTRACTION PRACTICE EXERCISES
1. a. 6
- 3
b. 8
- 4
c. 5
- 2
d. 9
- 5
2. a. 11
- 6
b. 12
- 4
c. 28
- 9
d. 33
- 7
3. a. 27
- 19
b. 23
- 14
c. 86
- 57
d. 99
- 33
3 4 3 4
5 8 19 26
8 9 29 66
e. 7
- 3
e. 41
- 8
e. 72
- 65
4
33
7

10
SUBTRACTION PRACTICE EXERCISES (con’t)
4. a. 387
- 241
b. 399
- 299
c. 847
- 659
d. 732
- 687
5. a. 3472
- 495
b. 312
- 186
c. 419
- 210
d. 3268
- 3168
6. a. 47
- 38
b. 63
- 8
c. 47
- 32
d. 59
- 48
146 100 188 45
2977 126 209 100
9 55 15 11
7. a. 372
- 192
b. 385
- 246
c. 219
- 191
d. 368
- 29
180 139 28 339

11
3. Checking Addition and Subtraction
2
+ 8
10
- 8
2
5
+ 3
8
- 3
5
73
+ 48
121
- 48
73
Result should produce other added number.
• Check Addition - Subtract one of added numbers from sum.Check Addition
927
318
426
183
927
ToAdd
ToCheck
Check Three or more #s• Check Three or more #s - Add from bottom to top.
5
- 4
1
+ 4
5
62
- 37
25
+ 37
62
103
- 87
16
+ 87
103
• Check SubtractionCheck Subtraction - Add subtracted number back.Check Subtraction

12
CHECKING ADDITION & SUBTRACTION PRACTICE EXERCISES
1. a. 6
+ 8
b. 9
+ 5
c. 18
+ 18
d. 109
+ 236
2. a. 87
- 87
b. 291
- 192
c. 367
- 212
d. 28
- 5
3. a. 34
+ 12
b. 87
13
81
+ 14
d. 21
- 83
13 14 26 335
1 99 55 24
46
195 746
104
4. a. 28
- 16
b. 361
- 361
c. 2793142
- 1361101
22 0 1432141
c. 87
13
81
+ 14
Check these answers using the method discussed.

13
CHECKING ADDITION & SUBTRACTION PRACTICE EXERCISES
1. a. 6
+ 8
13
- 8
5
b. 9
+ 5
14
- 5
9
c. 18
+ 18
26
- 18
8
d. 109
+ 236
335
- 236
99
2. a. 87
- 87
1
+ 87
88
b. 291
- 192
99
+ 192
291
c. 367
- 212
55
+ 212
267
d. 28
- 5
24
+ 5
29
3. a. 34
+ 12
46
- 12
34
b. 195
87
13
81
+ 14
195
d. 21
+ 83
104
- 83
21
4. a. 28
- 16
22
+ 16
38
b. 361
- 361
0
+ 361
361
c. 2793142
- 1361101
1432141
+ 1361101
2793242
c. 949
103
212
439
+ 195
746
# = Right
# = Wrong

14
4. Multiplication
• In Arithmetic - Indicated by “times” sign (x).
Learn “Times” Table
6 x 8 = 48
In Arithmetic

• Complex Multiplication - Carry result to next column.
15
Complex Multiplication
4. Multiplication (con’t)
Problem: 48 x 23
48
X 23
4
+ 2
48
X 23
144
+ 2
48
X 23
144
+ 1
6
48
X 23
144
+ 1
960
1104
Same process is used when multiplying
three or four-digit problems.

16
MULTIPLICATION PRACTICE EXERCISES
1. a. 21
x 4
b. 81
x 9
c. 64
x 5
d. 36
x 3
2. a. 87
x 7
b. 43
x 2
c. 56
x 0
d. 99
x 6
3. a. 24
x 13
b. 53
x 15
c. 49
x 26
d. 55
x 37
84 729 320 108
609 86 0 594
312 795 1274 2035

17
MULTIPLICATION PRACTICE EXERCISES (con’t)
4. a. 94
x 73
b. 99
x 27
c. 34
x 32
d. 83
x 69
5. a. 347
x 21
b. 843
x 34
c. 966
x 46
6. a. 360
x 37
b. 884
x 63
c. 111
x 19
6862 2673 1088 5727
7287 28,662 44,436
13,320 55,692 2109
7. a. 493
x 216
b. 568
x 432
c. 987
x 654
106,488 245,376 645,498

18
Finding out how many times a divider “goes into” a
whole number.
• Finding out how many times a divider “goes into” a
whole number.
5. Division
15 5 = 3 15 3 = 5

19
5. Division (con’t)
Shown by using a straight bar “ “ or “ “ sign.• Shown by using a straight bar “ “ or “ “ sign.
48 5040
1 48 “goes into” 50 one time.
48 1 times 48 = 48
2 50 minus 48 = 2 & bring down the 4
4
0
48 goes into 24 zero times.
0
Bring down other 0.
48 goes into 240, five times
5
240
0
5 times 48 = 240
240 minus 240 = 0 remainder
So, 5040 divided by 48 = 105 w/no remainder.
Or it can be stated:
48 “goes into” 5040, “105 times”

20
DIVISION PRACTICE EXERCISES
1. a. b. c.
2. a. b. c.
3. a. b.
211 62 92
13 310 101
256 687
4. a. b.
98 67
48 5040 7 434 9 828
9 117 12 3720 10 1010
23 5888 56 38472
98 9604 13 871
5. a. b.
50 123
50 2500 789 97047

21
DIVISION PRACTICE EXERCISES (con’t)
6. a. b.
7. a. b.
8. a. b.
7 9000
61 101
67 r 19 858 r 13
9. a. b.
12 r 955 22 r 329
21 147 3 27000
32 1952 88 8888
87 5848 15 12883
994 12883 352 8073

22
1. Changing whole numbers to fractions.
A smaller part of a whole number.C. FRACTIONS - A smaller part of a whole number.
Written with one number over the other, divided by a line.
3
8
11
16
or 3
8
11
16
Any number smaller than 1, must be a fraction.
Multiply the whole number times the number of parts being
considered.
Changing the whole number 4 to “sixths”:
4 = 4 x 6
6
=
24
6
or 24
6
Try thinking of the fraction as “so many of a specified number of parts”.
For example: Think of 3/8 as “three of eight parts” or...
Think of 11/16 as “eleven of sixteen parts”.

23
CHANGING WHOLE NUMBERS TO FRACTIONS EXERCISES
1. 49 to sevenths
2. 40 to eighths
3. 54 to ninths
4. 27 to thirds
5. 12 to fourths
6. 130 to fifths
49 x 7
7
= 343
7
or 343
7
=
40 x 8
8
= 320
8
or 320
8
=
54 x 9
9
= 486
9
or 486
9
=
27 x 3
3
= 81
3
or 81
3
=
12 x 4
4
= 48
4
or 48
4
=
130 x 5
5
= 650
5
or 650
5
=

24
3. Mixed numbers.
Combination of a whole number and a proper fraction.
4. Changing mixed numbers to fractions.
Change 3 7/8 into an improper fraction.
• Add both fractions together.
=24
8
7
8
+ 31
8
• Change whole number (3) to match fraction (eighths).
3 x 8
8
24
8
or
24
83 = =
2. Proper and improper fractions.
Proper Fraction - Numerator is smaller number than denominator.
Improper Fraction - Numerator is greater than or equal to denominator.
3/4
15/9

25
CHANGING MIXED NUMBERS TO FRACTIONS EXERCISES
1. 4 1/2
3. 19 7/16
5. 6 9/14
2. 8 3/4
4. 7 11/12
6. 5 1/64
1
2
4 x 2
2
8
2
= + = 9
2
19 x 16
16
304
16
= + 7
16
= 311
16
7 x 12
12
84
12
= + 11
12
= 95
12
6 x 14
14
84
14
= + 9
14
= 93
14
5 x 64
64
320
64
= + 1
64
= 321
64
8 x 4
4
24
4
= + 3
4
= 27
4
=
=
=
=
=
=

26
5. Changing improper fractions to whole/mixed
numbers.
Change 19/3 into whole/mixed number..
19/3 = 19 3 = 6, remainder 1 = 6 1/3 (a mixed number)
= 37 7 = 5, remainder 2 = 5 2/7 (a mixed number)
= 44 4 = 11, no remainder = 11 (a whole number)
= 240 8 = 30, no remainder = 30 (a whole number)
CHANGING IMPROPER FRACTIONS TO WHOLE/MIXED NUMBERS EXERCISES
1. 37/7 =
2. 44/4 =
3. 23/5 =
4. 43/9 =
5. 240/8 =
6. 191/6 =

27
6. Reducing Fractions
Terms - The name for numerator and denominator of a fraction.
Reducing - Changing to different terms.
Reducing does not change value of original fraction.
7. Reducing to Lower Terms
Divide both numerator and denominator by same number.
Example:
.
3 3 = 1
9 3 = 3
.
..
3
9 = 3
9 & 1
3 Have same value.
8. Reducing to Lowest Terms
Lowest Terms - 1 is only number which evenly divides both numerator
and denominator.
Example: 16
32 =
.
16 2 = 8
32 2 = 16
.
..
a.
.
8 2 = 4
16 2 = 8
.
..
b.
.
4 2 = 2
8 2 = 4
.
..
c.
.
2 2 = 1
4 2 = 2
.
..
d.

28
REDUCING TO LOWER/LOWEST TERMS EXERCISES
.15 5 = 3
20 5 = 4
.
..
1. Reduce the following fractions to LOWER terms:
15
20 =a. to 4ths
• Divide the original denominator (20) by the desired denominator (4) = 5..
• Then divide both parts of original fraction by that number (5).
36
40 =b. to 10ths
24
36 =c. to 6ths
12
36 =d. to 9ths
16
76 =f. to 19ths
30
45 =e. to 15ths
.36 4 = 9
40 4 = 10
.
..
.
24 6 = 4
36 6 = 6
.
..
.12 4 = 3
36 4 = 9
.
..
.30 3 = 10
45 3 = 15
.
..
.16 4 = 4
76 4 = 19
.
..

29
REDUCING TO LOWER/LOWEST TERMS EXERCISES (con’t)
2. Reduce the following fractions to LOWEST terms:
6
10a.
3
9 =b.
6
64 =c.
13
32 =d.
16
76 =f.
32
48 =e.
= 6 2 = 3
10 2 = 5
..a. ..
3 3 = 1
9 3 = 3
..a. ..
6 2 = 3
64 2 = 32
..a. ..
Cannot be reduced.
32 2 = 16
64 2 = 32
..a. ..
16 2 = 8
32 2 = 16
..b. ..
8 8 = 1
16 8 = 2
..c. ..
16 2 = 8
76 2 = 38
..a. ..
8 2 = 4
38 2 = 19
..b. ..

30
9. Common Denominator
Two or more fractions with the same denominator.
1
8
2
8
6
8
7
8
When denominators are not the same, a common denominator is
found by multiplying each denominator together.
1
6
3
8
2
9
5
12
5
18
7
24
1
36
6 x 8 x 9 x 12 x 18 x 24 x 36 = 80,621,568
80,621,568 is only one possible common denominator ...
but certainly not the best, or easiest to work with.
10. Least Common Denominator (LCD)
Smallest number into which denominators of a group of two or
more fractions will divide evenly.

31
The most number of times any single factors appears in a set is
multiplied by the most number of time any other factor appears.
10. Least Common Denominator (LCD) con’t.
To find the LCD, find the “lowest prime factors” of each denominator.
2 x 3 2 x 2 x 2 3 x 3 2 x 3 x 2 2 x 3 x 3 3 x 2 x 2 x 2 2 x 2 x 3 x 3
(2 x 2 x 2) x (3 x 3) = 72
Remember: If a denominator is a “prime number”, it can’t be
factored except by itself and 1.
LCD Exercises (Find the LCD’s)
3
8
2
9
5
12
5
18
7
24
1
361
6
1
6
1
8
1
12
2 x 3 2 x 2 x 2 2 x 3 x 2
2 x 2 x 2 x 3 = 24
1
12
1
16
1
24
2 x 2 x 2 x 2 x 3 = 48
3
10
4
15
7
20
2 x 5 3 x 5 2 x 2 x 5
2 x 2 x 3 x 5 = 60
2 x 2 x3 2 x 2 x 2 x 2 3 x 2 x 2 x 2

32
Divide the LCD by each of the other denominators, then multiply both
the numerator and denominator of the fraction by that result.
11. Reducing to LCD
Reducing to LCD can only be done after the LCD itself is known.
3
8
2
9
5
12
5
18
7
24
1
361
6
2 x 3 2 x 2 x 2 3 x 3 2 x 3 x 2 2 x 3 x 3 3 x 2 x 2 x 2 2 x 2 x 3 x 3
LCD = 72
1
6
72 6 = 12..
1 x 12 = 12
6 x 12 = 72
3
8
72 8 = 9..
3 x 9 = 27
8 x 9 = 72
2
9
72 9 = 8..
2 x 8 = 16
9 x 8 = 72
5
12
72 12 = 6..
5 x 6 = 30
12 x 6 = 72
Remaining fractions are handled in same way.

33
Reducing to LCD Exercises
Reduce each set of fractions to their LCD.
1
6
1
8
1
12
2 x 3 2 x 2 x 2 2 x 3 x 2
2 x 2 x 2 x 3 = 24
1
12
1
16
1
24
3
10
4
15
7
20
2 x 5 3 x 5 2 x 2 x 5
2 x 2 x 3 x 5 = 602 x 2 x 2 x 2 x 3 = 48
2 x 2 x3 2 x 2 x 2 x 2 3 x 2 x 2 x 2
1
6
24 6 = 4
.
.
1 x 4 = 4
6 x 4 = 24
1
8
24 8 = 3
.
.
1 x 3 = 3
8 x 3 = 24
1
12
24 12 = 2
.
.
1 x 2 = 2
12 x 2 = 24
1
12
48 12 = 4
.
.
1 x 4 = 4
12 x 4 = 48
1
16
48 16 = 3
.
.
1 x 3 = 3
16 x 3 = 48
1
24
48 24 = 2
.
.
1 x 2 = 2
24 x 2 = 48
3
10
60 10 = 6
.
.
3 x 6 = 18
10 x 6 = 60
4
15
60 15 = 4
.
.
4 x 4 = 16
15 x 4 = 60
7
20
60 20 = 3
.
.
7 x 3 = 21
20 x 3 = 60

34
12. Addition of Fractions
All fractions must have same denominator.
Determine common denominator according to previous process.
Then add fractions.
1
4
2
4
3
4
= 6
4+ + = 1
21
Always reduce to lowest terms.
13. Addition of Mixed Numbers
Mixed number consists of a whole number and a fraction. (3 1/3)
• Whole numbers are added together first.
• Then determine LCD for fractions.
• Reduce fractions to their LCD.
• Add numerators together and reduce answer to lowest terms.
• Add sum of fractions to the sum of whole numbers.

35
Adding Fractions and Mixed Numbers Exercises
Add the following fractions and mixed numbers, reducing answers to lowest terms.
1. =
3
4+
3
4
2. 7
10+
2
5
=
3. 15
16+
9
32 = 14. 3
4+
2
55 =
= 1
2
6
4 1 4
10
7
10+
1
10
=
1
11
10
=
39
32
30
32+
9
32 =
7
321=
5 + 1 = 6
8
20
15
20+ =
23
20
3
201= + 6
3
207=

36
14. Subtraction of Fractions
Similar to adding, in that a common denominator must be found first.
Then subtract one numerator from the other.
20
24
14
24
- =
6
24
To subtract fractions with different denominators: ( )
5
16
1
4
-
• Find the LCD...
5
16
1
4
-
2 x 2 x 2 x 2 2 x 2
2 x 2 x 2 x 2 = 16
• Change the fractions to the LCD...
5
16
4
16
-
• Subtract the numerators...
5
16
4
16
- =
1
16

37
15. Subtraction of Mixed Numbers
• Subtract the fractions first. (Determine LCD)
1
2
2
310 - 4
3 x 2 = 6 (LCD)
• Divide the LCD by denominator of each fraction.
6 3 = 2 6 2 = 3
.. ..
2
2
• Multiply numerator and denominator by their respective numbers.
2
3 x = 4
6
3
3
1
2 x = 3
6
• Subtract the fractions.
3
6
4
6 - = 1
6
• Subtract the whole numbers.
10 - 4 = 6
• Add whole number and fraction together to form complete answer.
6 +
1
6 = 6
1
6

38
15. Subtraction of Mixed Numbers (con’t)
• Subtract the fractions.
6
16
17
16
- = 11
16
• Subtract the whole numbers.
4 - 3 = 1
• Add whole number and fraction together to form complete answer.
1 +
11
16 = 1
11
16
Borrowing
• Subtract the fractions first. (Determine LCD)
3
8
1
165 - 3
(LCD) = 16
becomes 6
16
1
165 - 3
• Six-sixteenths cannot be subtracted from one-sixteenth, so
1 unit ( ) is borrowed from the 5 units, leaving 4.16
16
• Add to and problem becomes:16
16
1
16
6
16
17
164 - 3

39
Subtracting Fractions and Mixed Numbers Exercises
Subtract the following fractions and mixed numbers, reducing answers to lowest
terms.
1. =
1
3-
2
5
1
15
- 5
15
6
15
=
4. =
2
5-
1
333 15
14
15
=
6
15-
5
1533 15
=
6
15-
20
1532 15 17
2. =
3
12-
5
8
9
24
- 6
24
15
24
= = 3
8
3. =
1
3-2
547 28
1
15=
5
15-
6
1547 28 19
5. =
15
16-1
4101 57
5
16
=
15
16-
4
16101 57
=
15
16-
20
16100 57 43
6. =5
12-3
414 10
=
5
12-
9
1214 10
4
124 1
34=

40
16. MULTIPLYING FRACTIONS
• Common denominator not required for multiplication.
4
16
3
4 X
1. First, multiply the numerators.
2. Then, multiply the denominators.
3. Reduce answer to its lowest terms.
4
16
3
4 X = 12
=
4
16
3
4 X = 12
=64
4
4
12
64 = 3
16
..

41
17. Multiplying Fractions & Whole/Mixed Numbers
• Change to an improper fraction before multiplication.
1. First, the whole number (4) is changed to improper fraction.
2. Then, multiply the numerators and denominators.
3. Reduce answer to its lowest terms.
4
1
4
1
3
4 X = 12
4
3
4 X 4
4
4
12
4 = 3
1
.. = 3

42
18. Cancellation
• Makes multiplying fractions easier.
• If numerator of one of fractions and denominator of other
fraction can be evenly divided by the same number, they can be
reduced, or cancelled.
Example:
5
16
8
3 X = 5
16
8
3 X =
1
2
5
2
1
3 X = 5
6
Cancellation can be done on both parts of a fraction.
3
24
12
21
X =
1
7 2
1
1
2
1
7 X =
1
14

43
Multiply the following fraction, whole & mixed numbers.
Reduce to lowest terms.
Multiplying Fractions and Mixed Numbers Exercises
1. 2.
3. 4.
5. 6.
7. 8.
9.
1
26X =
4
5 X = 2
3
9
5 X =
4
16
3
4 X =
4
35
35
4 X =
7
12
1
6 X =
3
5
9
10 X =
5
11
2
3 X =
77
15X =
26
3
5
3
16
1
2 2
5 1 1
5
1 27
50
7
72
10
33
25 2
3

44
19. Division of Fractions
• Actually done by multiplication, by inverting divisors.
• The sign “ “ means “divided by” and the fraction to the
right of the sign is always the divisor.
Example:
1
5
3
4 becomes 5
1
3
4 X =
15
4 = 3 3
4
20. Division of Fractions and Whole/Mixed Numbers
• Whole and mixed numbers must be changed to improper fractions.
Example:
1
8
becomes3 3
16 2 16 3 + 3
16
=
51
16 and
2 X 8 + 1
8
=
17
8
X
X
51
16
17
8 Inverts to 51
16
8
17 = X
51
16
8
17
1
2 1
3
= 1
1
3
2 X
1
1
3
2 X = 3
2 = 1
2
1 Double
Cancellation

45
Divide the following fraction, whole & mixed numbers. Reduce
to lowest terms.
Dividing Fractions,Whole/Mixed Numbers Exercises
1. 2.
3. 4.
5.
3
8
=
= =
3
6
5
8 =
7
4
14
3 =
18 144
51
16
1
8 15 7
12
1
41
1
28
5
725
2
32

46
D. DECIMAL NUMBERS
• System of numbers based on ten (10).
• Decimal fraction has a denominator of 10, 100, 1000, etc.
Written on one line as a whole number, with a period (decimal point) in
front.
5
10 =
.5
5
100 =
.05
5
1000 =
.005
3 digits
.999 is the same as 999
1000
( 1 + same number of zeros
as digits in numerator)
1. Decimal System

47
2. Reading and Writing Decimals
7
105 is written 5.7
Whole Number Decimal Fraction (Tenths)
7
10055 is written 55.07
Whole Number Decimal Fraction (Hundredths)
Decimal Fraction (Tenths)
77
1000555 is written 555.077
Whole Number
Decimal Fraction (Tenths)
Decimal Fraction (Hundredths)
Decimal Fraction (Thousandths)

48
2. Reading and Writing Decimals (con’t)
• Decimals are read to the right of the decimal point.
.63 is read as “sixty-three hundredths.”
.136 is read as “one hundred thirty-six thousandths.”
.5625 is read as “five thousand six hundred twenty-five
ten-thousandths.”
3.5 is read “three and five tenths.”
• Whole numbers and decimals are abbreviated.
6.625 is spoken as “six, point six two five.”
One place .0 tenths
Two places .00 hundredths
Three places .000 thousandths
Four places .0000 ten-thousandths
Five places .00000 hundred-thousandths

49
3. Addition of Decimals
• Addition of decimals is same as addition of whole
numbers except for the location of the decimal point.
Add .865 + 1.3 + 375.006 + 71.1357 + 735
• Align numbers so all decimal points are in a vertical column.
• Add each column same as regular addition of whole numbers.
• Place decimal point in same column as it appears with each number.
.865
1.3
375.006
71.1357
+ 735.
“Add zeros to help eliminate errors.”000
0000
0
0
“Then, add each column.”
1183.3067

50
4. Subtraction of Decimals
• Subtraction of decimals is same as subtraction of whole
numbers except for the location of the decimal point.
Solve: 62.1251 - 24.102
• Write the numbers so the decimal points are under each other.
• Subtract each column same as regular subtraction of whole numbers.
• Place decimal point in same column as it appears with each number.
62.1251
- 24.102
“Add zeros to help eliminate errors.”
0
“Then, subtract each column.”
38.0231

51
5. Multiplication of Decimals
• Multiply the same as whole numbers.
• Count the number of decimal places to the right of the decimal
point in both numbers.
• Position the decimal point in the answer by starting at the
extreme right digit and counting as many places to the left as
there are in the total number of decimal places found in both numbers.
Solve: 38.639 X 2.08
3 8 .6 3 9
x 2.0 8
“Add zeros to help eliminate errors.”0
“Then, add the numbers.”
3 0 6 9 5 2
Rules For Multiplying Decimals
7 7 2 7 8 0
8 0 3 4 7 5 2
Decimal point 3 places over.
Decimal point 2 places over.
.
Place decimal point 5 places over from right.

52
6. Division of Decimals
• Place number to be divided (dividend) inside the division box.
• Place divisor outside.
• Move decimal point in divisor to extreme right. (Becomes whole number)
• Move decimal point same number of places in dividend. (NOTE: zeros
are added in dividend if it has fewer digits than divisor).
• Mark position of decimal point in answer (quotient) directly above decimal
point in dividend.
• Divide as whole numbers - place each figure in quotient directly above
digit involved in dividend.
• Add zeros after the decimal point in the dividend if it cannot be divided
evenly by the divisor.
• Continue division until quotient has as many places as required for the
answer.
Rules For Dividing Decimals
Solve: 123.573 137.4

53
6. Division of Decimals
137 4 1 2 3 5 7 3. .. .
.8
1 0 9 9 2
1 3 6 5 3
9
1 2 3 6 6
1 2 8 7
0
0
9
1 2 3 6 6
5 0 4
0
0
4 1 2 2
3
9 1 8 remainder

54
Decimal Number Practice Exercises
1. Add the following decimals.
a. .6 + 1.3 + 2.8 =
b. 72.8 + 164.02 + 174.01 =
c. 185.7 + 83.02 + 9.013 =
d. 0.93006 + 0.00850 + 3315.06 + 2.0875 =
2. Subtract the following decimals.
a. 2.0666 - 1.3981 =
b. 18.16 - 9.104 =
c. 1.0224 - .9428 =
d. 1.22 - 1.01 =
e. 0.6 - .124 =
f. 18.4 - 18.1 =
g. 1347.008 - 108.134 =
h. 111.010 - 12.163 =
i. 64.7 - 24.0 =
4.7
410.83
277.733
3318.08606
0.6685
9.056
0.0796
0.21
0.467
0.3
1238.874
98.847
40.7
“WORK ALL 4 SECTIONS (+, , X, )

55
3. Multiply the following decimals.
a. 3.01
x 6.20
b. 21.3
x 1.2
c. 1.6
x 1.6
d. 83.061
x 2.4
e. 1.64
x 1.2
f. 44.02
x 6.01
g. 63.12
x 1.12
h. 183.1
x .23
i. 68.14
x 23.6
18.662 25.56 2.56
199.3464 1.968 264.5602
70.6944 42.113 1608.104

56
4. Divide the following decimals.
3 0.5
a. 1.4 4 2.7 0 b. .8 4.6 3000
c. 1.2 6 2 0.4 d. 6 6.6 7 8 6
e. 1.1 110.0
5.7875
5 1 7 1.1 1 3 1
10 0

57
E. CHANGING FRACTIONS TO DECIMALS
A fraction can be changed to a decimal by dividing the
numerator by the denominator.
Change to a decimal.
3
4 4 3.0
.75
Write the following fractions and mixed numbers as decimals.
a. 6
10 b. 3
5 c. 4
5 d. 1
5 e. 1
2
f. 8
20 g. 7
20 h. 15
20 i. 7
25 j. 12
25
k. 17
20
l. 49
50
m.
9
10 n. o.1
1
251
15
256
.6 .6 .8 .2 .5
.4 .35 .75 .28 .48
.85 .98 1.9 1.04 6.6

58
F. PERCENTAGES
1. Percents
• Used to show how many parts of a total are taken out.
• Short way of saying “by the hundred or hundredths part of the whole”.
• The symbol % is used to indicate percent.
• Often displayed as diagrams.
100 Equal Squares = 100%
25% or 25/100
or
1/4
1/4
1/4
1/4
4/4 = 100%
25/100 = 25%
To change a decimal to a %, move decimal point two places to
right and write percent sign.
.15 = 15%
.55 = 55%
.853 = 85.3%
1.02 = 102%
“Zeros may be needed to hold place”.
.8 = 80%

59
Percents Practice Exercises
Write as a decimal.
1. 35% = _________
2. 14% = _________
3. 58.5% = _________
4. 17.45% = __________
5. 5% = _________
Write as a percent.
6. .75 = ______%
7. 0.40 = _____%
8. 0.4 =_______%
9. .4 = _______%
.35
.14
.585
.1745
.05
75
40
40
40

60
Rules For Any Equivalent
To convert a number to its decimal equivalent, multiply by 0.01
Change 6 1/4% to its decimal equivalent.
• Change the mixed number to an improper fraction, then divide the
numerator by the denominator.
6 1/4 = 25/4 = 6.25
• Now multiply the answer (6.25) times 0.01
6 .25 x 0.01 = 0.0625
Rules For Finding Any Percent of Any Number
• Convert the percent into its decimal equivalent.
• Multiply the given number by this equivalent.
• Point off the same number of spaces in answer as in both numbers multiplied.
• Label answer with appropriate unit measure if applicable.
Find 16% of 1028 square inches.
16 x .01 = .16
1028 x 0.16 = 164.48
Label answer: 164.48 square inches

61
2. Percentage
• Refers to value of any percent of a given number.
• First number is called “base”.
• Second number called “rate”... Refers to percent taken from base.
• Third number called “percentage”.
Rule: The product of the base, times the rate, equals the percentage.
Percentage = Base x Rate or P=BxR
NOTE: Rate must always be in decimal form.
To find the formula for a desired quantity, cover it and the
remaining factors indicate the correct operation.
R
P B
Only three types of percent problems exist.
1. Find the amount or rate. R=PxB
2. Find the percentage. P= R
B
3. Find the base. B= R
P

62
Percents Practice Exercises
1. Determine the rate or amount for each problem A through E for the
values given.
BASE
PERCENT-
AGE
2400 lbs
1875
gallons
148 feet
3268.5
Square
inches
$ 875.00
A. B. C. D. E.
80% 45% 15% 4 1/2% 19.5%
A. B. C. D. E.
2. The labor and material for renovating a building totaled $25,475. Of this amount,
70% went for labor and the balance for materials. Determine: (a) the labor cost,
and (b) the material cost.
1920 lbs. 843.75 Gal. 22.2 feet 147.08 sq.in. $170.63
a. $17,832.50 (labor) b. $ 7642.50 (materials)
3. 35% of 82 = 4. 14% of 28 =
5. Sales tax is 9%. Your purchase is $4.50. How much do you owe?
6. You have 165 seconds to finish your task. At what point are you 70%
finished?
7. You make $14.00 per hour. You receive a 5% cost of living raise. How
much raise per hour did you get? How much per hour are you making
now?
28.7 4.32
$4.91
115.5 seconds
$.70 /hr raise Making $14.70 /hr

63
G. APPLYING MATH TO THE REAL WORLD
1. 18 x 12 = 216
2. 240 x 8 = 30
3. 3.5 + 8.5 + 12 + 2.5 + 15 = 41.5
55 - 41.5 = 13.5 gallons more
4. 1.5 x 0.8 = 1.2 mm
5. 5 x .20 = 1 inch
6. 2400 divided by 6 = 400 per person
400 divided by 5 days = 80 per day per person
7. 6 x 200 = 1200 sq. ft. divided by 400 = 3 cans of dye
8. 2mm x .97 = 1.94 min 2mm x 1.03 = 2.06 max

64
H. METRICS
1. Metrication
• Denotes process of changing from English weights and measures
to the Metric system.
• U.S. is only major country not using metrics as standard system.
• Many industries use metrics and others are changing.
Metric Prefixes:
Most commonly used prefixes are Kilo, centi, and milli.
Kilo = 1000 units
Hecto = 100 units
Deka = 10 units
deci = 0.1 unit (one-tenth of the unit)
centi = 0.01 (one-hundredth of the unit)
milli = 0.001 (one thousandth of the unit)

65
A. Advantages of Metric System
• Based on decimal system.
• No fractions or mixed numbers
• Easier to teach.
Example 1:
Using three pieces of masking tape of the following English measurement lengths:
4 1/8 inches, 7 6/16 inches, and 2 3/4 inches, determine the total length of the tape.
Step 1: Find the least common
denominator (16). This
is done because unequal
fractions can’t be added.
Step 2: Convert all fractions to the
least common denominator.
Step 3: Add to find the sum.
Step 4: Change sum to nearest
whole number.
14 7/16
“Now, compare with Example 2 using Metrics”.
13 23/16
4 1/8 = 4 2/16
7 9/16 = 7 9/16
2 3/4 = 2 12/16

66
b. Advantages of Metric System
Example 2:
Using three pieces of masking tape of the following lengths: 85 mm, 19.4 cm, and
57 mm, determine the total length of the tape.
Step 1: Millimeters and centimeters
cannot be added, so convert
to all mm or cm.
85mm = 85mm
19.4cm = 194mm
57mm = 57mm
Step 2: Add to find the sum.
336 mm
or
85mm = 8.5cm
19.4cm = 19.4cm
57mm = 5.7cm
33.6 cm
“MUCH EASIER”

67
2. Metric Abbreviations
• Drawings must contain dimensions.
• Words like “inches, feet, millimeters, & centimeters take too much space.
• Abbreviations are necessary.
Metric Abbreviations:
mm = millimeter = one-thousandth of a meter
cm = centimeter = one-hundredth of a meter
Km = Kilometer = one thousand meters
76mm 25mm
30mm
Dimensioned Drawing
SLIDE BLOCK
12mm
76 25
30
Dimensioned Drawing with
Note for Standard Units
SLIDE BLOCK
NOTE: All dimensions are in millimeters.
12

68
3. The Metric Scale
• Based on decimal system. Easy to read.
• Graduated in millimeters and centimeters.
Metric Scales
• Both scales graduated the same... Numbering is different.
• Always look for the abbreviation when using metric scales.
• Always place “0” at the starting point and read to end point.
8.35cm or 83.5mm
110mm or 11.0cm

69
Metric Measurement Practice Exercises
Using a metric scale, measure the lines and record their length.
a. _______ mm
b. _______ mm
c. _______ cm
d. _______ mm
e. _______ cm
f. _______ mm
g. _______ cm
h. _______ mm
i. _______ mm
j. _______ cm
109
81.5
3.1
103
6.3
80.5
10.85
23
91.5
4.25

70
4. Comparisons and Conversions
• Manufacturing is global business.
• Metrics are everywhere.
• Useful to be able to convert.
Compare the following:
One Yard: About the length between your nose and the end
of your right hand with your arm extended.
One Meter: About the length between your left ear and the
end of your right hand with your arm extended.
One Centimeter: About the width of the fingernail on your pinky
finger.
One Inch: About the length between the knuckle and the
end of your index finger.

71
U.S. Customary and Metric Comparisons
Length:
A Kilometer is a little over 1/2 mile - .62 miles to be more precise.
Mile
Kilometer
A centimeter is about 3/8 inch.
Weight:
A paper clip weighs about one gram.
A nickel weighs about five grams.
A Kilogram is 2.2 pounds. - Two packs
of butter plus about 1 stick.

72
U.S. Customary and Metric Comparisons
Capacity:
One liter and one quart are approximately the same.
1 liter
There are about 5 milliliters in a teaspoon.
Pressure is measured in newton meters instead of foot pounds.
Equivalent Units:
KiloThousands
HectoHundreds
DekaTens
baseunitOnes
deciTenths
centiHundredths
milliThousandths
Place Value
Prefix
To change to a smaller unit,
move decimal to right.
To change to a larger unit,
move decimal to left.

73
Changing to a Smaller Unit
KiloThousands
HectoHundreds
DekaTens
baseunitOnes
deciTenths
centiHundredths
milliThousandths
15 liters = ________ milliliters (ml)
• Count the number of places from the base unit
to “milli”. There are 3 places.
• Move the decimal 3 places to the right.
15 liters = 15.000 liters = 15000ml
Changing to a Larger Unit
150 grams (g) = _____ Kilograms (Kg)
• Count the number of places from the base unit
to “Kilo”. There are 3 places.
• Move the decimal 3 places to the left.
150 grams = 150.00 grams = 0.150 Kg
15000
.150

74
1. 1 liter = _______ ml
2. 6000 ml = _______ liters
3. 10 cm = _______ mm
4. 500 cm = _______ m
5. 4 Kg = _______ g
6. 55 ml = _______ liters
7. 8.5 Km = _______ m
8. 6.2 cm = _______ mm
9. 0.562 mm = _______ cm
10. 75 cm = _______ mm
1000
6
100
5.0
4000
.055
8500
62
.0562
750
Comparison and Conversion Practice Exercises

75
5. Conversion Factors
Conversion Table for Length
Conversion Table for Area

76
5. Conversion Factors
Conversion of Volume
• Volume measures the total space occupied by three-dimensional
objects or substances.
• Volume of six-sided spaces is calculated as “length x width x height”.
• Volume of spheres and cylinders is more complicated.
• Term “cubic” is used because it is a math function involving 3 factors.
2ft x 4ft x 3ft = 24 Cubic Feet
English
1 cubic inch = 1 cubic inch
1 cubic foot = 1728 cubic inches (12 x 12 x 12)
1 cubic yard = 27 cubic feet (3 x 3 x 3)
Metric
1 cubic meter = 1,000,000 cubic centimeters (100 x 100 x 100)
1 foot = .305 meters
and
1 meter = 3.28 feet
Factors can be converted before or after initial calculation.

77
5. Conversion Factors (con’t)
Conversion Table for Pressure
Conversion Table for Weight

78
5. Conversion Factors (con’t)
Conversion Table for Temperature
To convert between Celsius and Fahrenheit:
Fahrenheit to Celsius . . . . (o
F-32) x 5/9 = o
C
Celsius to Fahrenheit . . . . (o
C x 9/5) + 32 = o
F

79
Metric System Practice Exercises
1. Which one of the following is not a metric measurement?
a. millimeter
b. centimeter
c. square feet
d. cm
2. Milli - is the prefix for which one of the following?
a. 100 ones
b. 0.001 unit
c. 0.0001 unit
d. 0.00001 unit
3. How long are lines A and B in this figure? A
B
4. How long is the line below? (Express in metric units).
5. Convert the following:
a. 1 meter = __________millimeters
b. 5 cm = ____________millimeters
c. 12 mm = ___________centimeters
d. 7m = _____________centimeters
A = 53 mm, or 5.3 cm
B = 38 mm, or 3.8 cm
69 mm
1000
50
1.2
700

80
H. THE CALCULATOR
• Functions vary from one manufacturer to the next.
• Most have same basic functions.
• More advanced scientific models have complicated
applications.
• Solar models powered by sunlight or normal indoor
light.
1. Basic Keys:
On/Off Key: Turns calculator on or off. Solar unit will not have “off” key..
C/AC: Press once ( C ) to clear last entry - Press twice (AC) to clear all functions.
Key: Controls the division function.
X Key: Controls the multiplication function.
- Key: Controls the subtraction function.
+ Key: Controls the addition function.
Key: Controls the square root function.
M+ Key: Adds a number or function to the memory register, to be recalled later.
M- Key: Subtracts number or function from memory register.
MR Key: Memory Recall recalls function stored in register.
MC Key: Memory Clear clears or erases all contents from memory.
% Key: Controls the percentage functions

81
2. Calculator Functions:
• Cannot give correct answer if given the wrong information or command.
• Decimals must be placed properly when entering numbers.
• Wrong entries can be cleared by using the C/AC button.
• Calculators usually provide a running total.
ADDITION
Add 3, 8, 9, and 14.
Step 1: Press “3” key - number 3 appears on screen..
Step 2: Press “+” key - number 3 remains on screen.
Step 3: Press “8” key - number 8 appears on screen.
Step 4: Press “+” key - running total of “11” appears on screen.
Step 5: Press the “9” key - number 9 appears on screen.
Step 6: Press “+” key - running total of “20” appears on screen.
Step 7: Press “1 & 4” keys - number 14 appears on screen.
Step 8: Press the = key - number 34 appears. This is the answer.
In step 8, pressing the + key would have displayed the total. Pressing the
= key stops the running total function and ends the overall calculation.

82
Calculator Addition Exercise
Use the calculator to add the following.
1. .06783
.49160
.76841
.02134
+ .87013
2. 154758
3906
4123
5434
+ 76
3. 12.54 + 932.67 + 13.4
2.21931 168297
= 958.61

83
Calculator Subtraction Exercise
Use the calculator to subtract the following.
1. .0543
- .0532
2. .0578
- .0463
3. 179853 - 4327
0.0011 0.0115
= 175526
SUBTRACTION
SUBTRACT 25 FROM 187.
Step 1: Press 1, 8, and 7 keys - number 187 appears on screen..
Step 2: Press “-” key - number 187 remains on screen.
Step 3: Press 2 & 5 keys- number 25 appears on screen.
Step 4: Press “=” key - number 162 appears on screen. This is the answer.
In step 4, pressing the - key would have displayed the total.

84
Calculator Multiplication Exercise
Use the calculator to multiply the following.
1. 2.45
x 16
2. 60.8
x 19
3. 12.8976 x 43.7 x 12.01
40.64 1155.2
= 6769.1376912
MULTIPLICATION
MULIPLY 342 BY 174.
Step 1: Press 3, 4, and 2 keys - number 342 appears on screen..
Step 2: Press “X” key - number 342 remains on screen.
Step 3: Press 1, 7 & 4 keys- number 174 appears on screen.
Step 4: Press “=” key - number 59508 appears on screen. This is the answer.

85
= 0.05922 = 1.22232 = 0.353
DIVISION
DIVIDE 66 BY 12.3
Step 1: Press the 6 key twice - number 66 appears on screen..
Step 2: Press “ ” key - number 66 remains on screen.
Step 3: Press 1, 2,. (decimal), & 3 keys- number 12.3 appears on screen.
Step 4: Press “=” key - number 5.3659 appears on screen. This is the answer.
Calculator Division Exercise
Use the calculator to divide the following.
1. .2961 5 2. 13.5678 11.1 3. .1765 .5

86
PERCENTAGES
FIND 1.3% OF 50
Step 1: Press the 5 and 0 keys - number 50 appears on screen..
Step 2: Press “ x ” key - number 50 remains on screen.
Step 3: Press 1, . (decimal), & 3 keys- number 1.3 appears on screen.
Step 4: Press “%” key - number .065 appears on screen. This is the answer.
Calculator Percentages Exercise
Use the calculator to find the following percentages.
1. Find 5% of:
a. 150
b. 675
c. 100
2. Find 10% of:
a. 1250
b. 871
c. 202
3. Find 26% of
a. 260
b. 212
c. 1817
= 7.5 = 125 = 67.6
= 33.75
= 5
= 87.1
= 20.2
= 55.12
= 472.42

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