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# 5.9: Use the Percent Equation to Find the Base, b

Difficulty Level: At Grade Created by: CK-12
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Practice Percent Equation to Find the Base b

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Have you ever enjoyed autumn? Take a look at this dilemma involving trees and their leaves.

By mid-September, 50% of the trees lose their leaves. If 850 trees in a grove lost their leaves, how many trees are there in all?

Use the percent equation to solve this problem. You will learn all that you need to know in this Concept.

### Guidance

Did you know that you can use the percent equation to solve percent problems? Take a look at how you can move from the proportion to the percent equation.

ab=p100\begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*}

When we solve the proportion ab=p100\begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*}, we use cross products to find the missing variable. However, even if we leave it in terms of the variables, we can still use cross multiply.

ab100aaa=p100=pb=pb100=.01pb\begin{align*}\frac{a}{b} &= \frac{p}{100}\\ 100 a &= pb\\ a &= \frac{pb}{100}\\ a &= .01pb\end{align*}

If we change the percent to a decimal by moving the decimal point two places to the left, then there is no need to multiply p\begin{align*}p\end{align*} by .01 as we will have already accounted for the coefficient of .01 by moving the decimal point.

Okay, let’s go through it again. Look at what we just wrote.

We wrote the same thing we just didn’t include values. The variables stayed and we multiplied them.

The key is that if we change the percent to a decimal, then all we have to do is to multiply it by the base and we will be able to figure out the value of a\begin{align*}a\end{align*}.

This situation showed us how to use go from a proportion to the percent equation when solving for part a.

Sometimes, you will know the percent and a part of the ratio, or part a\begin{align*}a\end{align*}, but you will need to find the whole or the base, b\begin{align*}b\end{align*}. When this happens, you can use the same key words as before and simply figure out the base by using the percent equation. Let’s look at one like this.

78 is 65% of what number?

Here we know that the word “is” means equals. The numbers may be in a different location, but just pay attention to the key words and you will know what to do. Notice that we have been given the percent and we are missing the “of what number” that is the value of the base. Let’s write the equation.

78=65%b\begin{align*}78 = 65\% b\end{align*}

To work with the 65%, it makes sense to convert it to a decimal. We do this by dropping the percent sign and moving the decimal two places to the left.

78=.65b\begin{align*}78 = .65b\end{align*}

Now we can solve it for the value of b\begin{align*}b\end{align*}. Divide both sides of the equation by .65.

78.65120=.65b.65=b\begin{align*}\frac{78}{.65} &= \frac{.65b}{.65}\\ 120 &= b\end{align*}

This is the answer.

Here is another one.

11 is 77% of what number?

Once again, pay attention to the key words. You can see that we are once again going to be looking for the value of the base. Let’s write the equation.

11=77%b\begin{align*}11 = 77 \% b\end{align*}

Convert the percent to a decimal and solve.

1111.7714.28=.77b=b=b\begin{align*}11 &= .77b\\ \frac{11}{.77} &= b\\ 14.28 &= b\end{align*}

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

Solve each problem using the percent equation.

#### Example A

10 is 50% of what number?

Solution:20\begin{align*}20\end{align*}

#### Example B

45 is 20% of what number?

Solution:225\begin{align*}225\end{align*}

#### Example C

68 is 40% of what number?

Solution:170\begin{align*}170\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

Let’s start by breaking apart this problem. We have a percent, so we know that we won’t be looking for the percent. We know that 850 trees in a grove lost their leaves, but we don’t know the total number of trees in the grove. The total could be thought of as the whole and this is the base. We are going to be looking for the base.

Let’s write the equation.

850=.50b\begin{align*}850 = .50b\end{align*}

Now we solve by dividing both sides of the equation by .50.

850.501700=.50b.50=b\begin{align*}\frac{850}{.50} &= \frac{.50b}{.50}\\ 1700 &= b\end{align*}

There are 1700 trees in the grove.

### Vocabulary

Percent
a part of a whole out of 100.

### Guided Practice

Here is one for you to try on your own.

25 is 60% of what number?

Solution

First, let's write down the percent equation.

100a=pb\begin{align*}100a = pb\end{align*}

In this problem, we are solving for the base. Let's fill in the values that we know.

100(25)=60b\begin{align*}100(25) = 60b\end{align*}

2500=60b\begin{align*}2500 = 60b\end{align*}

Now we divide to solve for b\begin{align*}b\end{align*}.

250060=b\begin{align*}\frac{2500}{60} = b\end{align*}

41.6=42\begin{align*}41.6 = 42\end{align*}

Our answer is 41.6\begin{align*}41.6\end{align*} or 42\begin{align*}42\end{align*}.

### Practice

Directions: 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

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.

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