Simplify and compare

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

The goal of the strategy is to make Quantity A (QA) and Quantity B (QB) themselves simpler so the comparison also gets simpler. This might involve simplifying either quantity or using math to transform the equations so they’re more similar to each other, making it easier to see the differences between QA and QB.

Change one quantity

Often you’ll only need to simplify one quantity so it becomes similar to the other. Let’s try an example.

$x<0$

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

The quantities both contain quadratics, but they’re written in different formats. Use FOIL to transform Quantity A into the same format as Quantity B, and see if you can find the answer.

FOILing Quantity A results in $x^2+x-12$. Let’s compare them again:

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

They’re almost the same, except Quantity A adds an additional $x$. However, remember that the question gave us the condition that $x<0$. Adding a negative number will decrease Quantity A, so Quantity B will always be greater.

Change both quantities

If changing one quantity doesn’t simplify things enough, we can change both quantities. When you use this method, treat the question like an equation. Try to make QA and QB as simple as possible by adding, subtracting, multiplying, or dividing both quantities by the same values.

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 have $+3$ as part of their equation, so let’s start by subtracting $3$:

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

Since $x>2$, we know it will always be positive, which means we can divide both by $x$ to simplify further:

• Quantity A: $x$
• Quantity B: $2$

We’re left with a very simple comparison. Since $x>2$, Quantity A will always be greater.

Compare unique parts of the quantities

Essentially what we’ve been doing is deleting shared values from both quantities, and then comparing the remaining unique parts. We can apply this to equations as we did above, and we can use the same technique for word problems too.

Here is a quick example problem that drives home the point.

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

Think about the unique information of each quantity to simplify and compare them.

You could solve this by adding up the lists of numbers, but that isn’t the point. Instead, let’s eliminate the overlap between QA and QB to highlight the unique parts. Anything that is shared between both quantities will have the same impact on both, so it’s ok to remove it.

Both quantities include the numbers 8 through 18. With this removed, the quantities become:

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

This leaves us with simple math:

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

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