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# Percent Equation to Find the Base b

## Solve "x is y percent of what number?"

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Use the Percent Equation to Find the Base, b

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, b\begin{align*}b\end{align*}.

### Finding the Base b

You can use the proportion ab=p100\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. 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, b\begin{align*}b\end{align*}.

When you solve the proportion ab=p100\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 a\begin{align*}a\end{align*}.

ab100a100a100a====p100pbpb100pb100\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 a=pb100\begin{align*}a = \frac{pb}{100}\end{align*}is equal to a=0.01pb\begin{align*}a = 0.01pb\end{align*}. 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 65%\begin{align*}65\%\end{align*} is the same as 65100\begin{align*}\frac{65}{100}\end{align*}.

78=65100×b\begin{align*}78 = \frac{65}{100} \times b\end{align*}

or

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

Next, divide both sides of the equation by 0.65 to solve for b\begin{align*}b\end{align*}.

78780.65b===0.65×b0.65×b0.65120\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*}

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 77100\begin{align*}\frac{77}{100}\end{align*}.

11=77100×b\begin{align*}11 = \frac{77}{100} \times b\end{align*}

or

11=0.77×b\begin{align*}11 = 0.77 \times b\end{align*}

Next, divide both sides of the equation by 0.77 to solve for b\begin{align*}b\end{align*}.

11110.77b===0.77×b0.77×b0.7714.29\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*}

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 35100\begin{align*}\frac{35}{100}\end{align*}.
850=35100×b\begin{align*}850 = \frac{35}{100} \times b\end{align*}

or

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

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

8508500.35b===0.35×b0.35×b0.352428.57\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*}

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 60100\begin{align*}\frac{60}{100}\end{align*}.

25=60100×b\begin{align*}25 = \frac{60}{100} \times b\end{align*}

or

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

Next, divide both sides of the equation by 0.60 to solve for b\begin{align*}b\end{align*}.

25250.60b===0.60×b0.60×b0.6041.67\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 50100\begin{align*}\frac{50}{100}\end{align*}.

10=50100×b\begin{align*}10 = \frac{50}{100} \times b\end{align*}

or

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

Next, divide both sides of the equation by 0.50 to solve for b\begin{align*}b\end{align*}.

10100.50b===0.50×b0.50×b0.5020\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*}

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 20100\begin{align*}\frac{20}{100}\end{align*}.

45=20100×b\begin{align*}45 = \frac{20}{100} \times b\end{align*}

or

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

Next, divide both sides of the equation by 0.20 to solve for b\begin{align*}b\end{align*}.

45450.20b===0.20×b0.20×b0.20225\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*}

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 40100\begin{align*}\frac{40}{100}\end{align*}.

68=40100×b\begin{align*}68 = \frac{40}{100} \times b\end{align*}

or

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

Next, divide both sides of the equation by 0.40 to solve for b\begin{align*}b\end{align*}.

68680.40b===0.40×b0.40×b0.40170\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.

1. 23 is 9% of what number?
2. 10 is 35% of what number?
3. 580 is 82% of what number?
4. 58 is 8% of what number?
5. 58 is 80% of what number?
6. 11 is 82% of what number?
7. 33 is 2% of what number?
8. 14 is 9% of what number?
9. 50 is 67% of what number?
10. 33 is 45% of what number?
11. 40 is 80% of what number?
12. 68 is 99% of what number?
13. 78 is 55% of what number?
14. 16 is 12% of what number?
15. 1450 is 80% of what number?

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### Vocabulary Language: English

TermDefinition
Decimal In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).
Equation An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.