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# Percent Equations

## Equations to solve for rates, totals, and parts.

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Practice Percent Equations
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Estimated9 minsto complete
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Percent Equations

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:

Rate×Total=PartorR% of Total is Part\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 (R%\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. R%\begin{align*}R\%\end{align*} is 25%, so we can use the second form of the equation: 25% of 80 is Part, or 0.25×80=Part\begin{align*}0.25 \times 80 = \text{Part}\end{align*}. 0.25×80=20\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 Rate×160=90\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%.

$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 0.15×Total=50\begin{align*}0.15 \times \text{Total} = \50\end{align*}. Dividing both sides by 0.15, we get Total=500.15333.33\begin{align*}\text{Total} = \frac{50}{0.15} \approx 333.33\end{align*}. So50 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, 13%=13100\begin{align*}13\% = \frac{13}{100}\end{align*}. • The percent equation is Rate×Total=Part\begin{align*}\text{Rate} \times \text{Total} = \text{Part}\end{align*}, or R% of Total is Part. • The percent change equation is Percent change=final amount - original amountoriginal amount×100%.\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 Practice96 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 0.12×Total=$96\begin{align*}0.12 \times \text{Total} = \96\end{align*}. Dividing both sides by 0.15, we get Total=960.12=800\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. y%\begin{align*}y\%\end{align*} of 3x\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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