GMAT Challenging Question 10

This GMAT 700 800 level problem solving practice question tests your understanding of absolute values of numbers. It also tests your ability to find values for which the expression will be positive and for those for which it will be negative and determine for what values will the expression be maximum and when it will be minimum.

Question 10: If a, b, and c are not equal to zero, what is the difference between the maximum and minimum value of S? S = 1 + \\frac{\text{|a|}}{\text{a}}) + \\frac{\text{2|b|}}{\text{b}}) + \\frac{\text{3|ab|}}{\text{ab}}) − \\frac{\text{4|c|}}{\text{c}})

1. 12
2. 14
3. 22
4. 20
5. 18

Video Explanation

Explanatory Answer | GMAT Algebra - Absolute values

Understanding Absolute Values

\\frac{\text{|a|}}{\text{a}}) = 1 When a is positive.
\\frac{\text{|a|}}{\text{a}}) = 1 When a is negative.

Step 1: Compute the Maximum Value of S

When will the value of the expression be maximum?

S = 1 + \\frac{\text{|a|}}{\text{a}}) + \\frac{\text{2|b|}}{\text{b}}) + \\frac{\text{3|ab|}}{\text{ab}}) − \\frac{\text{4|c|}}{\text{c}})
The value of the expression will be maximum when all of the terms become positive.
i.e., a, b, and ab should be positive and c should be negative.

When a is positive, \\frac{\text{|a|}}{\text{a}}) = 1
When b is positive, \\frac{\text{2|b|}}{\text{b}}) = 2
When a and b are positive, as required in the previous two steps, ab will be positive and the expression, \\frac{\text{3|ab|}}{\text{ab}}) = 3
When c is negative, \\frac{\text{4|c|}}{\text{c}}) = -4

Therefore, the maximum value = 1 + 1 + 2 + 3 -(-4) = 11

Step 2: Compute the Minimum Value of S

When will the value of the expression be minimum?

S = 1 + \\frac{\text{|a|}}{\text{a}}) + \\frac{\text{2|b|}}{\text{b}}) + \\frac{\text{3|ab|}}{\text{ab}}) − \\frac{\text{4|c|}}{\text{c}})
The value of the expression will be minimum if we make as many terms negative as possible.
Higher the magnitude of the terms made negative, lower the value of the expression.

c has to be positive for S to be minimum. The last term will then be -4.
If ab is negative, then \\frac{\text{3|ab|}}{\text{ab}}) = -3
If ab has to be negative, one of a or b has to be positive and the other has to be negative.

Possibility 1: If a > 0 and b < 0, \\frac{\text{|a|}}{\text{a}}) = 1 and \\frac{\text{2|b|}}{\text{b}}) = -2.
The value of the expression is 1 + 1 - 2 - 3 - 4 = -7.

Possibility 2: If a < 0 and b > 0, \\frac{\text{|a|}}{\text{a}}) = -1 and \\frac{\text{2|b|}}{\text{b}}) = 2.
The value of the expression is 1 - 1 + 2 - 3 - 4 = -5.

Therefore, the minimum value is -7

Step 3: Compute the difference

Maximum value of S = 11.
Minimum value of S = -7.
The difference is 18.

Choice E is the correct answer.