## Percent Equation

#### Common Questions

#### What is a percent?

Good question! Like decimals, a percent is another way of expressing a fraction. The “cent” in percent means $100$ (like how **cent**ipedes have $\textbf{100}$ legs), so percent means “out of $100$.” For example, $39\%$ means $39$ out of $100$ and is equal to the fraction $\frac{39}{100}$.

Read below to learn more.

#### Common Questions

Good question! Like decimals, a percent is another way of expressing a fraction. The “cent” in percent means $100$ (like how Read below to learn more. | |

Great question! The percent equation is ${\color{#b5179e}\text{part}} = \frac{{\color{#4ac783}\text{percent}} \times {\color{#3A86FF}\text{whole}}}{100}$. If we know two of the unknowns, we can always divide or multiply to solve for the third. For example, take our previous example $45\%$ of $60$. We can rewrite this question into equation form using the percent equation: ${\color{#b5179e}\text{What}} \text{ is } {\color{#4ac783}45\%} \text{ of } {\color{#3A86FF}60}?$${\color{#b5179e}x} = \frac{{\color{#4ac783}45} \times {\color{#3A86FF}60}}{100}$Now, we just need to multiply and divide like before, and we should get ${\color{#b5179e}27}$ as our answer. Read below to learn more. |

### What is a percent?

Like fractions and decimals, percents are a ratio between two numbers. The “cent” in percent means $100$ (like how **cent**ipedes have $\textbf{100}$ legs), so “percent” means “per $100$” or “out of $100$.” A percent is always a ratio out of $100$.

Since percentages are just fractions out of $100$, we can use equivalent proportions to solve problems involving percentages. The equation below relates proportions and percents:

$\frac{{\color{#b5179e}\text{part}}}{{\color{#3A86FF}\text{whole}}} = \frac{{\color{#4ac783}\text{percent}}}{100}$Click the button below if you need to review proportions!

Let’s rearrange our equation a bit. If we multiply both sides of the equation by “whole,” we can change this equation into a different form.

$\frac{{\color{#b5179e}\text{part}}}{{\color{#3A86FF}\text{whole}}} = \frac{{\color{#4ac783}\text{percent}}}{100}$$\frac{{\color{#b5179e}\text{part }}\times{\sout\color{#3A86FF}\text{ whole}}}{{\sout\color{#3A86FF}\text{whole}}} = \frac{{\color{#4ac783}\text{percent}} \times {\color{#3A86FF}\text{whole}}}{100}$${\color{#b5179e}\text{part}} = \frac{{\color{#4ac783}\text{percent}} \times {\color{#3A86FF}\text{whole}}}{100}$This equation is called the percent equation. If we know two of the unknowns, we’ll always be able to solve for the third. As you’ll see later, writing the equation in this form makes it easier for us to solve problems involving percentages.

### The Percent Equation

Let’s look at a few examples involving the percent equation.

** Example 1: ** What is $60\%$ of $25$?

Here’s where the percent equation really comes in handy. It can help us transform word problems into equations. For this problem, our percent is $60\%$, and our whole is $25$. We are looking for the part, so we can replace it with the variable $x$:

${\color{#b5179e}\text{What }} \text{is } {\color{#4ac783}\text{60}}\% \text{ of } {\color{#3A86FF}\text{25}}?$${\color{#b5179e}\text{part}} = \frac{{\color{#4ac783}\text{percent}} \times {\color{#3A86FF}\text{whole}}}{100}$${\color{#b5179e}x} = \frac{{\color{#4ac783}\text{60}} \times {\color{#3A86FF}\text{25}}}{100}$Now that we've transformed the question into an equation, we can flex our algebra skills and solve for $x$!

${\color{#b5179e}x} = \frac{{\color{#4ac783}60} \times {\color{#3A86FF}25}}{100} = {\textbf{\color{#b5179e}15}}$Great! We should get $15$ as our answer. At first, you might not know how to answer a percent problem by just looking at it, but as you can see from this example, the percent equation is like a pair of glasses. Using it can make things a whole lot clearer!

### Exchanging Values 💱

From Example $1$, we know that ${\color{#4ac783}60}\%$ of ${\color{#3A86FF}25}$ is ${\color{#b5179e}15}$. Now let’s look at a similar problem:

What is $25\%$ of $60$?

Using the percent equation, we’ll get:

${\color{#b5179e}\text{What}} \text{ is } {\color{#4ac783}25}\% \text{ of } {\color{#3A86FF}60}?$${\color{#b5179e}\text{part}} = \frac{{\color{#4ac783}\text{percent}} \times {\color{#3A86FF}\text{whole}}}{100}$${\color{#b5179e}x} = \frac{{\color{#4ac783}25}\times {\color{#3A86FF}60}}{100}$${\color{#b5179e}x} = {\textbf{\color{#b5179e}15}}$Hmmm. It looks like $60\%$ of $25$ and $25\%$ of $60$ are the same value! This is because in the percent equation, we multiply percent times whole. Since order doesn’t matter for multiplication, swapping the order of our percent and whole values won’t change the value of our part.

Keep in mind that this exchanging trick only works for percent and whole! Knowing this trick can be useful, but to keep things consistent, we will use ${\color{#b5179e}\text{part}} = \frac{{\color{#4ac783}\text{percent}} \times {\color{#3A86FF}\text{whole}}}{100}$ for the rest of this lesson.

### Is and Of 🌈

Here are a few more tips to help you remember the percent equation! In addition to changing the part, percent, and whole, we can also change the following:

- is → =
- of → $\times$ (multiplication)
- % → /$100$

“Is” means the same as “equals to,” so when we change our problem to equation form, we can replace “is” with the equals sign. Similarly, “of” becomes a multiplication sign. And last but not least, percent means “out of $100$,” so when we see percentage signs, we know we have to divide by $100$.

Here’s an example of how we could use these tips:

$\text{What} \text{ is } 60\% \text{ of } 90?$$\text{What} {\color{#00bbff}\text{ is }} 60 {\color{#925cff}\%} {\color{#f92a82}\text{ of }} 90?$$\text{part} {\color{#00bbff}\text{ = }} \frac{\text{percent}} {\color{#925cff}100} {\color{#f92a82}\times} \text{whole}$$\text{part} {\color{#00bbff}\text{ = }} \frac{\text{percent} {\color{#f92a82}\times} \text{whole}}{{\color{#925cff}100}}$$x {\color{#00bbff}\text{ = }} \frac{60 {\color{#f92a82}\times} 90}{{\color{#925cff}100}}$$x = \textbf{54}$If you ever forget the percent equation, the “is” and “of” trick can help you remember where to multiply and divide.

Let’s look at another example.

** Example 2: ** $24$ is what percent of $64$?

100

Awesome job! Let’s isolate ${\color{#4ac783}x}$ by first multiplying both sides by $100$:

Then, we’ll divide both sides by ${\color{#3A86FF}\text{64}}$, so we have ${\color{#4ac783}x}$ alone on the right side

$\frac{{\color{#b5179e}\text{24}}\times100}{{\color{#3A86FF}64}} = {\color{#4ac783}x}$$\frac{{\color{#b5179e}\text{24}}\times100}{{\color{#3A86FF}64}} = {\textbf{\color{#4ac783}37.5}}$We get that ${\color{#b5179e}\text{24}}$ is ${\textbf{\color{#4ac783}\text{37.5}\%}}$ of ${\color{#3A86FF}\text{64}}$.

Let’s look at one more example together!

** Example 3: ** $60$ is $20\%$ of what number?

100

Awesome job! Let’s isolate ${\color{#4ac783}x}$ by first multiplying both sides by $100$:

Then, we’ll divide both sides by ${\color{#4ac783}\text{20}}$, so we have ${\color{#3A86FF}x}$ alone on the right side

$\frac{{\color{#b5179e}\text{60}}\times100}{{\color{#4ac783}20}} = {\color{#3A86FF}x}$$\frac{{\color{#b5179e}\text{60}}\times100}{{\color{#4ac783}20}} = {\textbf{\color{#3A86FF}300}}$We get that ${\color{#b5179e}\text{60}}$ is ${\color{#4ac783}\text{20}\%}$ of ${\textbf{\color{#3A86FF}\text{300}}}$.

If you want to see more examples of the percent equation in action, check out our **Percent Equation Calculator**. And when you’re ready, click the 🤩 button below to explore some fantas**Tik ** applications of percentages!

### Percent Equation Calculator

What is your unknown variable?

First, let’s identify our known and unknown variables for the problem you entered:

Known | Unknown |
---|---|

Next, let’s plug the values into the percent equation:

${\color{#b5179e}\text{part}}= \frac{{\color{#4ac783}\text{percent}}\times{\color{#3A86FF}\text{whole}}}{100}$To isolate $x$, we’ll multiply both sides by $100$ and then divide both sides by${\text{ }}$.

${\color{#b5179e}\text{ }} = \frac{{\color{undefined}x} \times {\color{undefined}\text{}}}{100}$${\color{#b5179e}\text{ }} \times 100 = {\color{undefined}x} \times {\color{undefined}\text{}}$$\frac{{\color{#b5179e}\text{}} \times 100}{{\color{undefined}\text{ }}} = {\color{undefined}x}$Click the 🏆 when you’re ready to see the answer!

### Let’s Tok About It

You may not know it, but influencers frequently use percentages to see how their posts are doing. On many platforms like YouTube and TikTok, creators have a statistic known as the CTR or Click-Through Rate. The CTR is equal to the number of views on a video divided by the number of thumbnail impressions, or how many times the thumbnail is seen. Most CTRs are between $2\%$ and $10\%$. They can tell creators how engaging their thumbnails are and help them estimate how much money they’ll make from a video.

Let’s look at some questions about the CTR and practice answering word problems using the percent equation.

** Application: ** YouTuber Nyma Tang’s new makeup review has $\text{536k}$ ($536$ thousand) views and a CTR of $6.7\%$. Approximately how many people saw the thumbnail of Nyma’s video?

We want to change this word problem into number form. To start off, we want to connect our YouTube lingo to the percent equation

${\color{#b5179e}\text{part}} = \frac{{\color{#4ac783}\text{percent}} \times {\color{#3A86FF}\text{whole}}}{100}$Match the number of views, number of impressions, and CTR with their counterparts in the percent equation:

Great work! Now that we’ve identified the components of our percent equation, we can plug our numbers in and solve:

${\color{#b5179e}\text{part}} = {\color{#b5179e}\text{number of views}} = {\color{#b5179e}\text{536k}}$${\color{#4ac783}\text{percent}} = {\color{#4ac783}\text{CTR}} = {\color{#4ac783}6.7}$${\color{#3A86FF}\text{whole}} = {\color{#3A86FF}\text{number of impressions}} = {\color{#3A86FF}x}$${\color{#b5179e}\text{part}} = \frac{{\color{#4ac783}\text{percent}} \times {\color{#3A86FF}\text{whole}}}{100}$${\color{#b5179e}\text{536k}} = \frac{{\color{#4ac783}6.7} \times {\color{#3A86FF}x}}{100}$We isolate $x$ by multiplying both sides by $100$ , then dividing both sides by $6.7$:

${\color{#b5179e}\text{536k}} \times 100 = {\color{#4ac783}6.7} \times {\color{#3A86FF}x}$$\frac{{\color{#b5179e}\text{536k}} \times 100}{{\color{#4ac783}6.7}} = {\color{#3A86FF}x}$$\frac{{\color{#b5179e}\text{536k}} \times 100}{{\color{#4ac783}6.7}} = {\color{#3A86FF}\text{8000k}} = {\textbf{\color{#3A86FF}\text{8M}}}$In total, about ${\color{#3A86FF}8}$ million people saw the thumbnail of Nyma’s video.

Great work! Now it's time to try a few practice questions by yourself.

In the practice below, select the correct dropdown to rewrite these word problems in the following equation form:

${\color{#b5179e}\text{part}} = \frac{{\color{#4ac783}\text{percent}} \times {\color{#3A86FF}\text{whole}}}{100}$These numbers might look a little trickier but don’t worry, we’ll do the calculations for you!

### ** Quick Practice 1 / 3 **

Tabitha Brown’s latest TikTok has a CTR of about $10\%$. If the TikTok thumbnail is shown to $15\text{M}$ people, approximately how many views will the TikTok have

100

Awesome! Let’s use algebra to solve for $x$:

We want to isolate $x$ by multiplying both sides by $100$, then dividing both sides by $21\text{M}$:

Click the 🏆 when you’re ready to reveal the answer.

If Tabitha Brown’s TikTok has a CTR of ${\color{#4ac783}10}\%$, and it was shown to ${\color{#3A86FF}15\text{M}}$ people, it would get about $\textbf{{\color{#b5179e}1.5\text{M}}}$ views.

Nice work! You completed all the questions! 👏🏿👏🏽👏🏻