Textbook

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.

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.

(spoiler)

Answer: Quantity B is greater

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.

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.

(spoiler)

Answer: Quantity A is greater

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.

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.

(spoiler)

Answer: Quantity A is greater

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.

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