Percent change

You can save a lot of time by learning quick ways to work with percent changes. Before you use any shortcuts, make sure you’re clear on what a percent change means.

You probably know that $50\%$ of $100$ is $50$, but consider this:

What is 50% more than 100?

It’s just as important to be able to go the other direction:

What is 50% less than 100?

Now try numbers that are a bit less convenient:

What is 40% more than 990?

And the other direction:

What is 40% less than 990?

There’s a quicker way to do these. Since the original amount is $100\%$, you can combine the percent change with $100\%$ first.

• 40% less means you keep $100\%-40\%=60\%$ of the original.

$$990\times .6=594$$

This matches the result from subtracting $40\%$ directly.

• 40% more means you have $100\%+40\%=140\%$ of the original.

$$990\times 1.4=1386$$

Whether you use this “100%” approach or the add/subtract approach is up to you. The key skill is translating the wording into a correct equation; once you do that, the arithmetic is straightforward.

Example percent change GRE question

Let’s try a GRE-style question using these ideas.

There are three variables: $x$, $y$, and $z$. If both $x$ and $y$ are decreased by 10%, $y$ becomes 25% greater than $z$, and 50% greater than the decreased value of $x$. What percent of $z$ is the original value of $x$? Round to the nearest percent.

Try it yourself, and then check your work.

Start by translating each percent change into a multiplier:

• Decreasing by 10% means multiplying by $0.9$.
• Increasing by 25% means multiplying by $1.25$.
• Increasing by 50% means multiplying by $1.50$.

Now write equations from the statements in the problem:

$$\begin{array}{rl}0.9y& =1.25z\\ 0.9y& =1.5(0.9x)\end{array}$$

Both equations have the same left-hand side ($0.9y$), so set the right-hand sides equal to each other. This eliminates $y$ and leaves an equation in $x$ and $z$:

$$\begin{array}{rl}1.25z& =1.5(0.9x)\\ 1.25z& =1.35x\end{array}$$

The question asks:

What percent of $z$ is the original value of $x$?

So you want $x$ written as a multiple of $z$. Solve for $x$ by dividing both sides by $1.35$:

$$\begin{array}{rl}1.25z& =1.35x\\ 1.25z/1.35& =1.35x/1.35\\ 0.92592592592z& \approx x\end{array}$$

So $x\approx 0.9259z$, meaning $x$ is about $92.59\%$ of $z$. Rounded to the nearest percent, $x$ is 93% of $z$.

Bringing it all together: question walkthrough video

If you’d like additional explanation, here’s a video walkthrough of a percent change problem: