Hard GMAT Math Question - Absolute Value

Algebra – Absolute Value
Problem Solving – Question 10

Question
If a, b, and c are not equal to zero, what is the difference between the maximum and
minimum value of S?
S=1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
A. 12
B. 14
C. 22
D. 20
E. 18

What is the difference between the maximum and minimum
value of S?
1. When maximum?
Identify when will the
value of the expression
be maximum and
compute it.
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
; a, b, and c are not zero

value of S?
1. When maximum?
be maximum and
compute it.
2. When minimum?
be minimum and
compute it.
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c

value of S?
1. When maximum?
be maximum and
compute it.
2. When minimum?
be minimum and
compute it.
3. Compute the difference
Compute the difference
to find the answer.
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c

value of S?
1. When maximum?
be maximum and
compute it.
2. When minimum?
be minimum and
compute it.
3. Compute the difference
Compute the difference
to find the answer.
4. Traps to watch out
Where could one have
gone wrong?
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c

The value of
x
x
will either be 1 or -1.
x
x
= 1when x is positive.
x
x
= -1 when x is negative.
01 When will the value be maximum? What is it?
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
value of S?

The value of
x
x
x
x
x
x
The value of the expression is maximized if
we can make all the terms of the expression
positive.
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
value of S?

The value of
x
x
x
x
x
x
positive.
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
value of S?
a
a
,
b
b
,
ab
ab
will each be 1 when a > 0 and b > 0.
When c < 0,
c
c
= -1 and -4
c
c
will be maximized.

The value of
x
x
x
x
x
x
positive.
Therefore, the maximum value of the expression is 1 + 1 + 2 + 3 – (-4) = 11
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
value of S?
a
a
,
b
b
,
ab
ab
will each be 1 when a > 0 and b > 0.
When c < 0,
c
c
= -1 and -4
c
c
will be maximized.

02 When will the value be minimum? What is it?
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
value of S?
The value of the expression is minimized if
we can make as many terms negative as
possible.
Further, higher the magnitude of the terms
becoming negative, lower the value of the
expression.

c has to be positive for the expression to be
minimum.
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
value of S?
possible.
expression.

minimum.
If a > 0 and b < 0, the value is 1 + 1 -2 – 3 – 4 = -7
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
value of S?
possible.
expression.
The value of the expression reduces when ab is
negative.
If ab has to be negative, one of a or b has to be
negative and the other has to be positive.

minimum.
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
value of S?
possible.
expression.
negative.
If a < 0 and b > 0, the value is 1 - 1 + 2 – 3 – 4 = -5

minimum.
Therefore, the minimum value of S is -7 when a > 0, b < 0 and c > 0
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
value of S?
possible.
expression.
negative.
If a < 0 and b > 0, the value is 1 - 1 + 2 – 3 – 4 = -5

Maximum value of S = 11
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
03 Last step: Compute the difference
Minimum value of S = -7

Maximum value of S = 11
If A starts they finish the task in exactly 10 days. If B starts, they take half a day more.
How long does it take to complete the task if they both work
together?
The difference is 11 – (-7) = 18
03 Last step: Compute the difference
Choice E is the correct answer
Minimum value of S = -7

◴What should one have watched out for?

While computing the minimum value, we would have estimated that if all the terms other than the first term
‘1’ is negative, the value of the expression will be minimum.
04 Choice D is the trap
·
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
value of S?

·
However, if ab is negative ONLY ONE of a or b can be negative. The other HAS to be positive.·
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
value of S?

·
However, if ab is negative ONLY ONE of a or b can be negative. The other HAS to be positive.·
If you had made all the terms negative, then the minimum value would have been 1 – 1 – 2 – 3 - 4 = -9·
And the answer would have been 11 – (-9) = 20. That is INCORRECT.·
S = 1+
a
a
+2
b
b
+3
ab
ab
-4
c
c
value of S?

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Hard GMAT Math Question - Absolute Value

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