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General Principles for Solving Quantitative Comparisons

The following principles can greatly simplify quantitative comparison problems.

 

Strategy: You Can Add or Subtract the Same Term (Number) from Both Sides of a Quantitative Comparison Problem.

 

Strategy: You Can Multiply or Divide Both Sides of a Quantitative Comparison Problem by the Same Positive Term (Number).
(Caution: This cannot be done if the term can ever be negative or zero.)

 

You can think of a quantitative comparison problem as an inequality/equation. Your job is to determine whether the correct symbol with which to compare the columns is <, =, >, or that it cannot be determined. Therefore, all the rules that apply to solving inequalities apply to quantitative comparisons. That is, you can always add or subtract the same term to both columns of the problem. If the term is always positive, then you can multiply or divide both columns by it. (The term cannot be negative because it would then invert the inequality. And, of course, it cannot be zero if you are dividing.)
 
Example-1
Column A Column B
 
Don’t solve this problem by adding the fractions in each column; that would be too time consuming—the LCD is 120! Instead, merely subtract 1/5 and 1/8 from both columns:
 
 
Column A Column B
1/3 1/4
 


Now 1/3 is larger than 1/4, so Column A is larger than Column B.
 

 

If there are only numbers (i.e., no variables) in a quantitative comparison problem, then “not-enough-information” cannot be the answer. Hence (D), not-enough-information, cannot be the answer to the example above.
 
Example-2
y > 0
Column A
Column B
y3 + y4
y4 – 2y2
 
First cancel y4 from both columns:
y > 0
 
Column A
Column B
y3
–2y2
 
Since y > 0, we can divide both columns by y2:
 
y > 0
 
Column A
Column B
y
–2
 
 
Now, we are given that y > 0. Hence, Column A is greater. The answer is (A).
 
 
Example-3
x > 1
Column A Column B
1/x
 
Since x > 1, x – 1 > 0. Hence, we can multiply both columns by x(x – 1) to clear fractions. This yields
x > 1
 
Column A Column B
x – 1 x
 
 
Subtracting x from both columns yields
x > 1
 
 
Column A Column B
– 1 0
 
 

In this form, it is clear that Column B is larger. The answer is (B).
 

 
Example-4
n is a positive integer and 0 < x < 1
Column A Column B
n2

Since we are given that n is positive, we may multiply both columns by :
 
Column A Column B
 
 
Reducing yields
 
 
Column A Column B
1/x 1
 
 
We are also given that 0 < x < 1.  So we may multiply both columns by x to get
 
 
 
Column A Column B
1 x
 
 

But again, we know that 0 < x < 1. Hence, Column A is larger.
 

 

Watch Out: You Must Be Certain That the Quantity You Are Multiplying or Dividing by Can Never Be Zero or Negative. (There are no restrictions on adding or subtracting.)

 

The following example illustrates the false results that can occur if you don’t guarantee that the number you are multiplying or dividing by is positive.
 
0 ≤ x < 1

Column A

Column B

 x3

x2


Solution (Invalid): Dividing both columns by x2 yields

 

Column A

Column B

x

1


We are given that
x < 1, so Column B is larger. But this is a false result because when x = 0, the two original columns are equal:

 

Column A

Column B

03 = 0

02 = 0

 

Hence, the answer is actually (D), not-enough-information to decide.

 

Watch Out: Don’t Cancel Willy-Nilly.


​Some people are tempted to cancel the
x2s from both columns of the following problem:

 

Column A

Column B

x2 + 4x – 6

6 + 4xx2

 

You cannot cancel the x2s from both columns because they do not have the same sign. In Column A, x2 is positive. Whereas in Column B, it is negative.

 

Strategy: You Can Square Both Sides of a Quantitative Comparison Problem to Eliminate Square Roots.

 

Example-5
Column A Column B
 
Squaring both columns yields
 
Column A Column B
or
 
Column A Column B
8
 
Reducing gives
 
Column A Column B
8
 

Now, clearly Column A is larger.
 

 

Example-6
Column A Column B
2/5
 
Multiplying both columns by 15 to clear fractions yields
 
Column A Column B
6
 
Squaring both columns yields
 
Column A Column B
25 * 2 36
 
Performing the multiplication in Column A yields
 
Column A Column B
50 36
 

Hence, Column A is larger, and the answer is (A).
 





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