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# 3.14: Percent Equations

Difficulty Level: At Grade Created by: CK-12
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What if you knew that 25% of a number was equal to 24? How could you find that number? After completing this Concept, you'll be able to use the percent equation to solve problems like this one.

### Guidance

The percent equation is often used to solve problems. It goes like this:

\begin{align*}& \text{Rate} \times \text{Total} = \text{Part}\\ & \qquad \qquad \text{or}\\ & R\% \ \text{of Total is Part}\end{align*}

Rate is the ratio that the percent represents (\begin{align*}R\%\end{align*} in the second version).

Total is often called the base unit.

Part is the amount we are comparing with the base unit.

#### Example A

Find 25% of $80. Solution We are looking for the part. The total is$80. ‘of’ means multiply. \begin{align*}R\%\end{align*} is 25%, so we can use the second form of the equation: 25% of 80 is Part, or \begin{align*}0.25 \times 80 = \text{Part}\end{align*}. \begin{align*}0.25 \times 80 = 20\end{align*}, so the Part we are looking for is20.

#### Example B

Express $90 as a percentage of$160.

Solution

This time we are looking for the rate. We are given the part ($90) and the total ($160). Using the rate equation, we get \begin{align*}\text{Rate} \times 160 = 90\end{align*}. Dividing both sides by 160 tells us that the rate is 0.5625, or 56.25%.

#### Example C

$50 is 15% of what total sum? Solution This time we are looking for the total. We are given the part ($50) and the rate (15%, or 0.15). Using the rate equation, we get \begin{align*}0.15 \times \text{Total} = \50\end{align*}. Dividing both sides by 0.15, we get \begin{align*}\text{Total} = \frac{50}{0.15} \approx 333.33\end{align*}. So $50 is 15% of$333.33.

Watch this video for help with the Examples above.

### Vocabulary

• A percent is simply a ratio with a base unit of 100—for example, \begin{align*}13\% = \frac{13}{100}\end{align*}.
• The percent equation is \begin{align*}\text{Rate} \times \text{Total} = \text{Part}\end{align*}, or R% of Total is Part.
• The percent change equation is \begin{align*}\text{Percent change} = \frac{\text{final amount - original amount}}{\text{original amount}} \times 100\%.\end{align*} A positive percent change means the value increased, while a negative percent change means the value decreased.

### Guided Practice

$96 is 12% of what total sum? Solution: This time we are looking for the total. We are given the part ($96) and the rate (12%, or 0.12). Using the rate equation, we get \begin{align*}0.12 \times \text{Total} = \96\end{align*}. Dividing both sides by 0.15, we get \begin{align*}\text{Total} = \frac{96}{0.12}=800\end{align*}. So $96 is 12% of$800.

### Practice

Find the following.

1. 30% of 90
2. 27% of 19
3. 16.7% of 199
4. 11.5% of 10.01
5. 0.003% of 1,217.46
6. 250% of 67
7. 34.5% of y
8. 17.02% of y
9. x% of 280
10. a% of 0.332
11. \begin{align*}y\%\end{align*} of \begin{align*}3x\end{align*}

### Texas Instruments Resources

In the CK-12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9613.

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### Vocabulary Language: English

Percent Equation

The percent equation can be stated as: "Rate times Total equals Part," or "R% of Total is Part."

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Date Created:
Aug 13, 2012
Last Modified:
Apr 11, 2016
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