Percent change

You can save a lot of time by learning the quickhand ways to solve for percentage changes. But before we get into a few tricks, let’s first make sure we understand percentage changes well. You probably intuitively know that 50% of 100 is 50, but how about:

What is 50% more than 100?

It’s important to know how to go the other way too:

What is 50% less than 100?

How about with numbers that are a bit harder?

What is 40% more than 990?

And the other way:

What is 40% less than 990?

There’s a quicker way to solve these operations. Since percentages add up to 100%, 40% less of something is equivalent to 60% of something. Try it out with 990:

$$990\times .6=594$$

This matches the result we found earlier. This is because the 40% is subtracted from the original 100%, leaving us with 100% - 40% = 60% of the original.

Similarly, 40% more of something equals 100% + 40% = 140% of something.

$$990\times 1.4=1386$$

Whether you use the 100% trick or not is up to you! What’s most important is that you can express the question in a proper math equation, since from there, solving it is straightforward.

Example percent change GRE question

Let’s try a GRE question using these techniques.

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.

First, let’s review some of the basics. Decreasing something by 10% is the same as multiplying it by $0.9$. Increasing something by 25% is the same as multiplying it by $1.25$, and increasing something by 50% is the same as multiplying it by $1.50$. We can use these facts to create the following equations:

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

Since the left side of these two equations are the same ($0.9y=0.9y$), we can set the right sides equal to each other. This will leave us with only the $x$ and $z$ variables, and we can simplify the math.

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

The question asks us:

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

We have $z$ expressed in terms of $x$, but we have it in terms of $1.35x$, and we need to find the value for $1.0x$, i.e. just $x$. We can do this easily by dividing both sides of the equation by $1.35$.

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

Rounded to the nearest percent, our answer is that $x$ is approximately 93% of $z$.

Bringing it all together: question walkthrough video

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