Donna is framing a picture with a blue border. Her picture has an area of 400 square inches. After framing with the border, Donna wants her final framed picture to be 1200 square inches. What percent of the final size is her original picture?

In this concept, you will use the percent equation to find a percent.

### Percent Equation

You can use the proportion \begin{align*}\frac{a}{b}= \frac{p}{100}\end{align*} to solve for a percent. You can also solve percent problems by using an equation. In this concept, you will use a proportion to create a different kind of equation that will help you solve percent problems.

To find percent, you know that the number being compared (\begin{align*}a\end{align*}) to the base ( ) is equal to the percent ( ).

Therefore the equation to use is:

\begin{align*}a = \frac{p}{100} \times b\end{align*}

Let’s look at a problem.

What percent of 32 is 18?

First, you know that you are looking for a percent. You want to set up the percent equation to solve for

.\begin{align*}18 = \frac{p}{100} \times 32\end{align*}

Next, solve for the value of \begin{align*}\frac{p}{100}\end{align*} by dividing both sides by 32.

\begin{align*}\begin{array}{rcl} 18 &=& \frac{32p}{100}\\ \frac{18}{32} &=& \frac{p}{100}\\ \frac{p}{100} &=& 0.5625 \end{array}\end{align*} Then, solve for \begin{align*}p\end{align*}, the percent by multiplying both sides by 100.

\begin{align*}\begin{array}{rcl} \frac{p}{100} &=& 0.5625\\ 100 \times \frac{p}{100} &=& 0.5625 \times 100\\ p &=& 56.25 \end{array}\end{align*} The answer is 56.25%.

Therefore, 18 is 56.25% of 32.

Let’s try another example.

10 is what percent of 12?

First, set up the percent equation to solve for \begin{align*}p\end{align*}.

\begin{align*}10 = \frac{p}{100} \times 12\end{align*}

Next, solve for the value of \begin{align*}\frac{p}{100}\end{align*} by dividing both sides by 12.

\begin{align*}\begin{array}{rcl} 10 &=& \frac{12p}{100}\\ \frac{10}{12} &=& \frac{p}{100}\\ \frac{p}{100} &=& 0.833 \end{array}\end{align*} Then, solve for \begin{align*}p\end{align*}, the percent by multiplying both sides by 100.

\begin{align*}\begin{array}{rcl} \frac{p}{100} &=& 0.833\\ 100 \times \frac{p}{100} &=& 0.833 \times 100\\ p &=& 83.3 \end{array}\end{align*} The answer is 83.3%.

Therefore, 10 is 83.3% of 12.

### Examples

#### Example 1

Earlier, you were given a problem about Donna and her frame.

Donna’s picture is 400 square inches and the final framed picture is 1200 square inches. She wants to know what percent of the final framed picture is her photo.

First, set up the percent equation to solve for

.\begin{align*}400 = \frac{p}{100} \times 1200\end{align*}

Next, solve for the value of by dividing both sides by 1200.

\begin{align*}\begin{array}{rcl} 400 &=& \frac{1200p}{100}\\ \frac{400}{1200} &=& \frac{p}{100}\\ \frac{p}{100} &=& 0.333 \end{array}\end{align*}

Then, solve for

, the percent by multiplying both sides by 100.\begin{align*}\begin{array}{rcl} \frac{p}{100} &=& 0.333\\ 100 \times \frac{p}{100} &=& 0.333 \times 100\\ p &=& 33.3 \end{array}\end{align*} The answer is 33.3%.

Therefore, Donna’s original picture represents 33.3% of the final framed picture.

#### Example 2

33 is what percent of 50?

First, set up the percent equation to solve for \begin{align*}p\end{align*}.

\begin{align*}33 = \frac{p}{100} \times 50\end{align*}

Next, solve for the value of by dividing both sides by 50.

\begin{align*}\begin{array}{rcl} 33 &=& \frac{50p}{100}\\ \frac{33}{50} &=& \frac{p}{100}\\ \frac{p}{100} &=& 0.66 \end{array}\end{align*}

Then, solve for

, the percent by multiplying both sides by 100.\begin{align*}\begin{array}{rcl} \frac{p}{100} &=& 0.66\\ 100 \times \frac{p}{100} &=& 0.66 \times 100\\ p &=& 66 \end{array}\end{align*} The answer is 66%.

Therefore, 33 is 66% of 50.

#### Example 3

18 is what percent of 20?

First, set up the percent equation to solve for \begin{align*}p\end{align*}.

\begin{align*}18 = \frac{p}{100} \times 20\end{align*}

Next, solve for the value of \begin{align*}\frac{p}{100}\end{align*} by dividing both sides by 20.

\begin{align*}\begin{array}{rcl} 18 &=& \frac{20p}{100}\\ \frac{18}{20} &=& \frac{p}{100}\\ \frac{p}{100} &=& 0.9 \end{array}\end{align*}Then, solve for , the percent by multiplying both sides by 100.

\begin{align*}\begin{array}{rcl} \frac{p}{100} &=& 0.9\\ 100 \times \frac{p}{100} &=& 0.9 \times 100\\ p &=& 90 \end{array}\end{align*}

The answer is 90%.

Therefore, 18 is 90% of 20.

#### Example 4

5 is what percent of 300?

First, set up the percent equation to solve for

.\begin{align*}5 = \frac{p}{100} \times 300\end{align*}

Next, solve for the value of by dividing both sides by 300.

\begin{align*}\begin{array}{rcl} 5 &=& \frac{300p}{100}\\ \frac{5}{300} &=& \frac{p}{100}\\ \frac{p}{100} &=& 0.0167 \end{array}\end{align*}

Then, solve for

, the percent by multiplying both sides by 100.\begin{align*}\begin{array}{rcl} \frac{p}{100} &=& 0.0167\\ 100 \times \frac{p}{100} &=& 0.0167 \times 100\\ p &=& 1.67 \end{array}\end{align*} The answer is 1.67%.

Therefore, 5 is 1.67% of 300.

#### Example 5

60 is what percent of 400?

First, set up the percent equation to solve for

.\begin{align*}60 = \frac{p}{100} \times 400\end{align*}

Next, solve for the value of by dividing both sides by 400.

\begin{align*}\begin{array}{rcl} 60 &=& \frac{400p}{100}\\ \frac{60}{400} &=& \frac{p}{100}\\\ \frac{p}{100} &=& 0.15 \end{array}\end{align*}

Then, solve for

, the percent by multiplying both sides by 100.\begin{align*}\begin{array}{rcl} \frac{p}{100} &=& 0.15\\ 100 \times \frac{p}{100} &=& 0.15 \times 100\\ p &=& 15 \end{array}\end{align*} The answer is 15%.

Therefore, 60 is 15% of 400.

### Review

Solve each percent problem by using the percent equation. You may round when necessary.

- What percent of 600 is 82?
- What percent of 18 is 17?
- 150 is what percent of 175?
- 200 is what percent of 450?
- 34 is what percent of 70?
- 12 is what percent of 88?
- 15 is what percent of 90?
- 230 is what percent of 600?
- 334 is what percent of 1000?
- 2 is what percent of 8?
- 55 is what percent of 1800?
- 61 is what percent of 80?
- 33 is what percent of 90?
- 78 is what percent of 156?
- 19 is what percent of 31?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.8.