Alice lives in New Hampshire and by mid-September 35% of the trees lose their leaves. Alice considers the trees in her grandparents’ grove, and sees that approximately 850 trees have lost their leaves. How can Alice know how many trees are there in all?

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

### Finding the Base b

You can use the proportion 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. Sometimes, you will know the percent and a part of the ratio, or part a, but you will need to find the whole or the base, .

to solve for a percent. You can also solveWhen you solve the proportion \begin{align*}\frac{a}{b} = \frac{p}{100}\end{align*}, you cross multiply to find the missing variable. You can rearrange this formula so that you are solving for just the variable .

\begin{align*}\begin{array}{rcl} \frac{a}{b} &=& \frac{p}{100} \\ 100a &=& pb \\ \frac{100a}{100} &=& \frac{pb}{100} \\ a &=& \frac{pb}{100} \end{array}\end{align*}You could also say that is equal to . As well, you could convert your percent directly into a decimal and therefore the formula becomes even simpler.

Let’s look at a problem.

78 is 65% of what number?

First, write the equation. Remember that \begin{align*}\frac{65}{100}\end{align*}.

is the same as\begin{align*}78 = \frac{65}{100} \times b\end{align*}

or

\begin{align*}78 = 0.65 \times b\end{align*}

Next, divide both sides of the equation by 0.65 to solve for

.\begin{align*}\begin{array}{rcl} 78 &=& 0.65 \times b \\ \frac{78}{0.65} &=& \frac{0.65 \times b}{0.65} \\ b &=& 120 \end{array}\end{align*}

The answer is 120.

Therefore, 78 is 65% of 120.

Let’s try another example.

11 is 77% of what number?

First, write the equation. Remember that 77% is the same as \begin{align*}\frac{77}{100}\end{align*}.

\begin{align*}11 = \frac{77}{100} \times b\end{align*}

or

Next, divide both sides of the equation by 0.77 to solve for

.\begin{align*}\begin{array}{rcl} 11 &=& 0.77 \times b \\ \frac{11}{0.77} &=& \frac{0.77 \times b}{0.77} \\ b &=& 14.29 \end{array}\end{align*}

The answer is 14.29.

Therefore, 11 is 77% of 14.29.

In this problem, you could round to the nearest hundredths place as you did here. Sometimes, you may be asked to round to the nearest tenths place. In that case, the answer would have been 14.3.

### Examples

#### Example 1

Earlier, you were given a problem about Alice’s family grove.

Alice knows that 850 trees in the grove lost their leaves, but doesn’t know the total number of trees in the grove. She also knows that the 850 represents 35% of the total grove.

First, write the equation. Remember that 35% is the same as \begin{align*}\frac{35}{100}\end{align*}.

or

\begin{align*}850 = 0.35 \times b\end{align*}

Next, divide both sides of the equation by 0.35 to solve for \begin{align*}b\end{align*}.

\begin{align*}\begin{array}{rcl} 850 &=& 0.35 \times b \\ \frac{850}{0.35} &=& \frac{0.35 \times b}{0.35} \\ b &=& 2428.57 \end{array}\end{align*}

The answer is 2428.

Therefore, 850 is approximately 35% of 2428.

#### Example 2

25 is 60% of what number?

First, write the equation. Remember that 60% is the same as \begin{align*}\frac{60}{100}\end{align*}.

or

\begin{align*}25 = 0.60 \times b\end{align*}

Next, divide both sides of the equation by 0.60 to solve for

.\begin{align*}\begin{array}{rcl} 25 &=& 0.60 \times b \\ \frac{25}{0.60} &=& \frac{0.60 \times b}{0.60} \\ b &=& 41.67 \end{array}\end{align*}The answer is 41.67.

Therefore, 25 is 60% of 41.67.

#### Example 3

10 is 50% of what number?

First, write the equation. Remember that 50% is the same as \begin{align*}\frac{50}{100}\end{align*}.

\begin{align*}10 = \frac{50}{100} \times b\end{align*}

or

\begin{align*}10 = 0.50 \times b\end{align*}

Next, divide both sides of the equation by 0.50 to solve for

.\begin{align*}\begin{array}{rcl} 10 &=& 0.50 \times b \\ \frac{10}{0.50} &=& \frac{0.50 \times b}{0.50} \\ b &=& 20 \end{array}\end{align*}

The answer is 20.

Therefore, 10 is 50% of 20.

#### Example 4

45 is 20% of what number?

First, write the equation. Remember that 20% is the same as \begin{align*}\frac{20}{100}\end{align*}.

\begin{align*}45 = \frac{20}{100} \times b\end{align*}

or

\begin{align*}45 = 0.20 \times b\end{align*}

Next, divide both sides of the equation by 0.20 to solve for

.\begin{align*}\begin{array}{rcl} 45 &=& 0.20 \times b \\ \frac{45}{0.20} &=& \frac{0.20 \times b}{0.20} \\ b &=& 225 \end{array}\end{align*}

The answer is 225.

Therefore, 45 is 20% of 225.

#### Example 5

68 is 40% of what number?

First, write the equation. Remember that 40% is the same as \begin{align*}\frac{40}{100}\end{align*}.

\begin{align*}68 = \frac{40}{100} \times b\end{align*}

or

\begin{align*}68 = 0.40 \times b\end{align*}

Next, divide both sides of the equation by 0.40 to solve for

.\begin{align*}\begin{array}{rcl} 68 &=& 0.40 \times b \\ \frac{68}{0.40} &=& \frac{0.40 \times b}{0.40} \\ b &=& 170 \end{array}\end{align*}The answer is 170.

Therefore, 68 is 40% of 170.

### Review

Solve each percent problem. You may round your answers to the nearest tenth when necessary.

- 23 is 9% of what number?
- 10 is 35% of what number?
- 580 is 82% of what number?
- 58 is 8% of what number?
- 58 is 80% of what number?
- 11 is 82% of what number?
- 33 is 2% of what number?
- 14 is 9% of what number?
- 50 is 67% of what number?
- 33 is 45% of what number?
- 40 is 80% of what number?
- 68 is 99% of what number?
- 78 is 55% of what number?
- 16 is 12% of what number?
- 1450 is 80% of what number?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.9.