Percent Change - Definition & Examples

Calculating Percent Change

You're probably more familiar with the idea of percent change than you realize. Every time you see a TV commercial, a billboard, or a storefront sign advertising a sale, you're actually looking at a calculation of it!

Instead of saying, "Our TVs are now $347.78$ instead of $869.43$," it's a lot easier (and a lot catchier) to say "Our TVs are now $40\%$ off." Whenever you see a "$\%$ off" or "save ____$\%$" sign, someone's done the calculation for you!

So, let's look at a few examples to learn how to do these calculations ourselves. But first, let's take a look at a pretty simple formula to follow.

To use this formula, there are two values you need to know to calculate the percentage change: the initial (starting) value and the final value.

There are a few things to keep in mind:

• It's okay to have values over $100\%$
• A Positive number means that the overall value increased (% increase)
• A Negative number means that the overall value decreased (% decrease)

Your percent change can either increase or decrease.

The formula is multiplied by $100$ in order to convert the decimal into a percent.

Let's plug in what we know to our equation:

Once we plug and chug, we can solve:

Let's plug what we know into our equation:

(Video) Percent Change Formula

by The Organic Chemistry Tutor

This video by The Organic Chemistry Tutor demonstrates the percent change formula with an example.

$\large{\text{Summary}}$

The formula: $$\%\text{ change} = \frac{\text{new - original}}{\text{original}} \times 100 \%$$

Interpreting the result:

If your % change is...

• Positive, that means there is an increase from the original.

• Negative, that means there is a decrease from the original.

Example

Suppose we have something that changes from $30$ to $42$. What is the percent change?

The new value will be $42$ and the old value is $30$. Plugging these in and solving, we get: $$\begin{align} \%\text{ change} &= \frac{\text{new - original}}{\text{original}} \times 100 \%\\ &=\frac{42-30}{30}\times100\%\\ &=\frac{12}{30}\times100\%\\ &=\frac{2}{5}\times100\%\\ &=40\% \end{align}$$

Example: Increase in club member

In percent change problems, we are always comparing the change to the original. This makes sense, because we are interested in how we have progressed from the past. The formula is: $$\frac{\text{(total - start)}}{\text{start}}\times 100=\frac{\text{change}}{\text{start}}\times 100$$

Don't forget to multiply by $100$! Why? Because we are looking at a percentage, not a decimal.

We can use this concept when calculating lots of different things in the real world, like calculating a tip at a restaurant, looking at changes in populations, and calculating a change in your math test score! There are a lot of real world applications, so this is a fundamentally important concept to learn.

Let's take a look at a real world example.

Last year, the rowing team started with $18$ people. The team successfully recruited $8$ new members. At the end of the year, $6$ people graduated. This year, the team recruited $14$ people. Compare the amount of people on the team prior to recruitment this year and after. What is the percent change?

We want to compare the team before and after this year's recruitment. At the beginning of this year there are: $$18+8-6=20$$ We can see this because $8$ people joined last year, but $6$ people graduated.

The start in this case is $20$. In the question, we are not given the current total. However, we are given the change.

It is tricky when assigning a sign to change—is it negative or positive? In this case, since the team gained people, the change is positive, it is $+14$.

The calculation:

There is a $70\%$ increase in club member after recruitment! The population increased by a lot! Let's hope the team retains all the new members!

Keep in mind: you can have negative numbers when doing these problems, so just make sure you watch out for your rules for subtracting negative numbers!

If we want to take this concept a step further, we can explore percent error, which is a really important concept in science. This would be another real-world application!