Simplify and compare

Another quantitative comparison strategy is called the simplify and compare method.

The goal is to rewrite Quantity A (QA) and Quantity B (QB) in simpler forms so the comparison is easier. You might simplify one quantity, or you might transform both quantities so they look more alike. Once the expressions are in comparable forms, it’s usually much easier to see which one is larger.

Change one quantity

Often, you only need to simplify one quantity so it matches the form of the other. Here’s an example.

$x<0$

Quantity A: $(x-3)(x+4)$
Quantity B: $x^2-12$

Both quantities involve quadratic expressions, but they’re written differently. Use FOIL to rewrite Quantity A in the same form as Quantity B, then compare.

FOILing Quantity A gives:

• $(x-3)(x+4)=x^2+x-12$

Now compare:

• Quantity A: $x^2+x-12$
• Quantity B: $x^2-12$

The only difference is the extra $+x$ in Quantity A. Since the problem tells you $x<0$, the value of $x$ is negative. Adding a negative number makes Quantity A smaller, so Quantity B will always be greater.

Change both quantities

If simplifying one quantity doesn’t make the comparison clear enough, simplify both. When you use this method, treat the comparison like an equation: you can add, subtract, multiply, or divide both quantities by the same value to remove shared parts and make the remaining comparison simpler.

Let’s try an example:

$x>2$

Quantity A: $x^2+3$
Quantity B: $2x+3$

Simplify both quantities to make them easier to compare, and solve the question.

Both quantities include $+3$, so start by subtracting $3$ from each:

• Quantity A: $x^2$
• Quantity B: $2x$

Since $x>2$, $x$ is always positive. That means you can safely divide both quantities by $x$ (dividing by a positive number doesn’t change the direction of an inequality):

• Quantity A: $x$
• Quantity B: $2$

Now the comparison is straightforward. Because $x>2$, Quantity A will always be greater.

Compare unique parts of the quantities

What you’ve been doing is removing parts that are shared by both quantities, then comparing what’s left. You can use this idea with algebraic expressions (as above) and with word problems.

Here’s a quick example that shows the idea.

Quantity A: The sum of all integers from 4 to 18
Quantity B: The sum of all integers from 8 to 19

Focus on what’s unique to each quantity to simplify and compare them.

You could add each list of numbers, but you don’t need to. Instead, remove the overlap. Any numbers that appear in both sums contribute equally to both quantities, so they don’t affect which one is larger.

Both quantities include the integers 8 through 18. Remove that shared part:

• Quantity A: The sum of all integers from 4 to 7
• Quantity B: 19

Now compute:

• Quantity A: $4+5+6+7=22$
• Quantity B: $19$

Since $22>19$, Quantity A is always greater.