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Section2.5One-Step Equations With Percentages

With the skills we have learned for solving one-step linear equations, we can solve a variety of percent-related problems that arise in our everyday life.

Figure2.5.1Alternative Video Lesson

In many situations when translating from English to math, the word “of” translates as multiplication. For example:

\begin{align*} \text{One third }\amp\text{of thirty is ten.}\\ \frac{1}{3}\amp\cdot30=10 \end{align*}

It's also helpful to note that when we translate English into math, the word “is” (and many similar words related to “to be”) translates to an equals sign.

Here is another example, this time involving a percentage. We know that “\(2\) is \(50\%\) of \(4\text{,}\)” so we have:

\begin{align*} 2\amp\text{ is } 50\% \text{ of } 4\\ 2\amp= 0.5\cdot4 \end{align*}
Example2.5.2

Translate each statement involving percents below into an equation. Define any variables used. (Solving these equations is an exercise).

  1. How much is \(30\%\) of \(\$24.00\text{?}\)

  2. \(\$7.20\) is what percent of \(\$24.00\text{?}\)

  3. \(\$7.20\) is \(30\%\) of how much money?

Solution

Each question can be translated from English into a math equation by reading it slowly and looking for the right signals.

  1. The word “is” means about the same thing as the \(=\) sign. “How much” is a question phrase, and we can let \(x\) be the unknown amount (in dollars). The word “of” translates to multiplication, as discussed earlier. So we have:

    \begin{equation*} \underbrace{\strut x}_{\text{How much}}\underbrace{\strut =}_{\text{is}}\underbrace{\strut 0.30}_{\text{30}\%}\underbrace{\strut\cdot}_{\text{of}}\underbrace{\strut 24}_{\text{\$24}} \end{equation*}
  2. Let \(P\) be the unknown value. We have:

    \begin{equation*} \underbrace{\strut 7.2}_{\text{\$7.20}}\underbrace{\strut =}_{\text{is}}\underbrace{\strut P}_{\text{what percent}}\underbrace{\strut\cdot}_{\text{of}}\underbrace{\strut 24}_{\text{\$24}} \end{equation*}

    With this setup, \(P\) is going to be a decimal value (\(0.30\)) that you would translate into a percentage (\(30\%\)).

  3. Let \(x\) be the unknown amount (in dollars). We have:

    \begin{equation*} \underbrace{\strut 7.2}_{\text{\$7.20}}\underbrace{\strut =}_{\text{is}}\underbrace{\strut 0.30}_{\text{30}\%}\underbrace{\strut\cdot}_{\text{of}}\underbrace{\strut x}_{\text{how much money}} \end{equation*}
Checkpoint2.5.3

Solve each equation from Example 2.5.2.

Subsection2.5.1Modeling and Solving Percent Equations

An important skill for solving percent-related problems is to boil down a complicated word problem into a simple form like “\(2\) is \(50\%\) of \(4\text{.}\)” Let's look at some further examples.

Remark2.5.4

When solving problems that are not applications, we state the solution set. This communicates the set of all solution(s) to that equation or inequality. However, when solving an equation or inequality that arises in an application problem, it makes more sense to summarize our result with a sentence, using the context of the application. This allows us to communicate the full result, including appropriate units.

Example2.5.5

As of Fall 2016, Portland Community College had \(89{,}900\) enrolled students. According to the chart below, how many black students were enrolled at PCC as of Fall 2016 term?

a pie chart that indicates white students are 68%; Hispanic students are 11%; Asian students are 8%; black students are 6%; and students of other ethnicities make up 7%
Figure2.5.6Racial breakdown of PCC students in Fall 2016
Solution

After reading this word problem and the chart, we can translate the problem into “what is \(6\%\) of \(89{,}900\text{?}\)” Let \(x\) be the number of black students enrolled at PCC in Fall 2016. We can set up and solve the equation:

\begin{align*} x\amp=0.06\cdot89900\\ x\amp=5394 \end{align*}

(There was not much “solving” to do, since the variable we wanted to isolate was already isolated.)

As of Fall 2016, Portland Community College had \(5394\) black students. (Note: this is not likely to be perfectly accurate, because the information we started with (\(89{,}900\) enrolled students and \(6\%\)) appear to be rounded.)

Example2.5.7

According to the bar graph, what percentage of the school's student population is freshman?

a bar graph indicating there are 134 freshmen, 96 sophomores, 86 juniors, and 103 seniors
Figure2.5.8Number of students at a high school by class
Solution

The school's total number of students is:

\begin{equation*} 134+96+86+103=419 \end{equation*}

Now, we can translate the main question:

“What percentage of the school's student population is freshman?”

into:

“What percent of \(419\) is \(134\text{?}\)”

Using \(P\) to represent the unknown quantity, we write and solve the equation:

\begin{align*} \overbrace{\strut P}^{\text{what percent}}\overbrace{\strut \cdot}^{\text{of}} \overbrace{\strut 419}^{\text{419}}\amp\overbrace{\strut =}^{\text{is}}\overbrace{\strut 134}^{\text{134}}\\ \divideunder{P\cdot 419}{419}\amp=\divideunder{134}{419}\\ P\amp=0.319809\ldots\\ P\amp\approx31.98\% \end{align*}

Approximately \(31.98\%\) of the school's student population is freshman.

Example2.5.9

Kim just received her monthly paycheck. Her gross pay (the amount before taxes and related things are deducted) was \(\$2{,}346.19\text{,}\) and her total tax and other deductions was \(\$350.21\text{.}\) The rest was deposited directly into her checking account. What percent of her gross pay went into her checking account?

Solution

Train yourself to read the word problem and not try to pick out numbers to substitute into formulas. You may find it helps to read the problem over to yourself three or more times before you attempt to solve it. There are three dollar amounts to discuss in this problem, and many students fall into a trap of using the wrong values in the wrong places. There is the gross pay, the amount that was deducted, and the amount that was deposited.

Even though there are three dollar amounts to think about, only two have been explicitly written down. We need to compute a subtraction to find the dollar amount that was deposited:

\begin{equation*} 2346.19-350.21=1995.98 \end{equation*}

Now, we can translate the main question:

“What percent of her gross pay went into her checking account?”

into:

“What percent of \(2346.19\) is \(1995.98\text{?}\)”

Using \(P\) to represent the unknown quantity, we write and solve the equation:

\begin{align*} \overbrace{\strut P}^{\text{what percent}}\overbrace{\strut \cdot}^{\text{of}} \overbrace{\strut 2346.19}^{\text{\$2346.19}}\amp\overbrace{\strut =}^{\text{is}}\overbrace{\strut 1995.98}^{\text{\$1995.98}}\\ \divideunder{P\cdot 2346.19}{2346.19}\amp=\divideunder{1995.98}{2346.19}\\ P\amp=0.85073\ldots\\ P\amp\approx85.07\% \end{align*}

Approximately \(85.07\%\) of her gross pay went into her checking account.

Example2.5.10

Kandace sells cars for a living, and earns \(28\%\) of the dealership's sales profit as commission. In a certain month, she plans to earn \(\$2200\) in commissions. How much total sales profit does she need to bring in for the dealership?

Solution

Be careful that you do not calculate \(28\%\) of \(\$2200\text{.}\) That might be what a student would do who fails to read the question repeatedly. If you have ever trained yourself to quickly find numbers in word problems and substitute them into formulas, you must unlearn this. The issue is that \(\$2200\) is not the dealership's sales profit, and if you mistakenly multiply \(0.28\cdot2200=616\text{,}\) then \(\$616\) makes no sense as an answer to this question. How could Kandace bring in only \(\$616\) of sales profit, and be rewarded with \(\$2200\) in commission?

We can translate the problem into “\(2200\) is \(28\%\) of what?” Letting \(x\) be the sales profit for the dealership (in dollars), we can write and solve the equation:

\begin{align*} 2200\amp=0.28\cdot x\\ \divideunder{2200}{0.28}\amp=\divideunder{0.28x}{0.28}\\ 7857.142\ldots\amp=x\\ x\amp\approx7857.14 \end{align*}

To earn \(\$2200\) in commission, Kandace needs to bring in approximately \(\$7857.14\) of sales profit for the dealership.

In the following price increase/decrease problems, the key is to identify the original price, which represents \(100\%\text{.}\) For these problems, it is important to note that “a percent of” and “a percent off” are two very different things. To find \(10\%\) of \(\$50\text{,}\) simply multiply the percentage with the number: \(0.10\cdot\$50=\$5\text{.}\) So, \(\$5\) is \(10\%\) of \(\$50\text{.}\) To find \(10\%\) off from an original value of \(\$50\text{,}\) consider how if we took \(10\%\) off, we would only need to pay \(90\%\) of the original value. This means to find \(10\%\) off, multiply the \(90\%\) with the number: \(0.90\cdot\$50=\$45\text{.}\) So, \(\$45\) is \(10\%\) off from \(\$50\text{.}\)

Example2.5.11

A shirt is on sale with \(15\%\) off. The current price is \(\$51.00\text{.}\) What was the original price?

Solution

If the shirt was \(15\%\) off, then what you pay for it is \(100\%-15\%\text{,}\) or \(85\%\text{,}\) of its original price. So we can translate this problem into “\(51\) is \(85\%\) of what?” It's very important to note that we are using \(85\%\text{,}\) not \(15\%\text{.}\) Let \(x\) represent the shirt's original price, in dollars. We can set up and solve the equation:

\begin{align*} 51\amp=0.85\cdot x\\ \divideunder{51}{0.85}\amp=\divideunder{0.85x}{0.85}\\ 60\amp=x \end{align*}

The shirt's original price was \(\$60.00\text{.}\)

Example2.5.12

According to e-Literate, the average cost of a new college textbook has been increasing. Find the percentage of increase from 2009 to 2013.

a plot over time indicating average textbook price was $62.00 in 2009; $65.11 in 2010; $68.87 in 2011; $72.11 in 2012; and $79.00 in 2013
Figure2.5.13Average New Textbook Price from 2009 to 2013
Solution

The actual amount of increase from 2009 to 2013 was \(79-62=17\text{,}\) dollars. We need to answer the question “\(17\) is what percent of \(62\text{?}\)” Note that we are comparing the \(17\) to \(62\text{,}\) not to \(79\text{.}\) In these situations where one amount is the earlier amount, the earlier original amount is the one that represents \(100\%\text{.}\) Let \(x\) represent the percent of increase. We can set up and solve the equation:

\begin{align*} 17\amp=x\cdot62\\ 17\amp=62x\\ \divideunder{17}{62}\amp=\divideunder{62x}{62}\\ 0.274193\ldots\amp=x\\ x\amp\approx27.42\% \end{align*}

From 2009 to 2013, the average cost of a new textbook increased by approximately \(27.42\%\text{.}\)

Checkpoint2.5.14
Example2.5.15

Enrollment at your neighborhood's elementary school two years ago was \(417\) children. After a \(15\%\) increase last year and a \(15\%\) decrease this year, what's the new enrollment?

Solution

It is tempting to think that increasing by \(15\%\) and then decreasing by \(15\%\) would bring the enrollment right back to where it started. But the \(15\%\) decrease applies to the enrollment after it had already increased. So that \(15\%\) decrease is going to translate to more students lost than were gained.

Using \(100\%\) as corresponding to the enrollment from two years ago, the enrollment last year was \(100\%+15\%=115\%\) of that. But then using \(100\%\) as corresponding to the enrollment from last year, the enrollment this year was \(100\%-15\%=85\%\) of that. So we can set up and solve the equation

\begin{align*} \overbrace{\strut x}^{\text{this year's enrollment}}\amp\overbrace{\strut =}^{\text{is}}\overbrace{\strut 0.85}^{85\%}\overbrace{\strut \cdot}^{\text{of}}\overbrace{\strut 1.15}^{115\%}\overbrace{\strut \cdot}^{\text{of}}\overbrace{\strut 417}^{\text{enrollment two years ago}}\\ x\amp=0.85\cdot1.15\cdot417\\ x\amp=407.6175 \end{align*}

We would round and report that enrollment is now \(408\) students. (The percentage rise and fall of \(15\%\) were probably rounded in the first place, which is why we did not end up with a whole number of children.)

SubsectionExercises

Basic Percentage Problems

1

\(3\%\) of \(670\) is .

2

\(8\%\) of \(770\) is .

3

\(50\%\) of \(870\) is .

4

\(30\%\) of \(970\) is .

5

\(750\%\) of \(180\) is .

6

\(500\%\) of \(280\) is .

7

Fill in the blank with a percent:

\(72\) is of \(180\text{.}\)

8

Fill in the blank with a percent:

\(380\) is of \(760\text{.}\)

9

Fill in the blank with a percent:

\(90\) is of \(45\text{.}\)

10

Fill in the blank with a percent:

\(34.5\) is of \(15\text{.}\)

11

Fill in the blank with a percent. Round to the hundredth place, like \(12.34\%\text{.}\)

\(16\) is approximately of \(74\text{.}\)

12

Fill in the blank with a percent. Round to the hundredth place, like \(12.34\%\text{.}\)

\(18\) is approximately of \(47\text{.}\)

13

\(11\%\) of is \(106.7\text{.}\)

14

\(69\%\) of is \(117.3\text{.}\)

15

\(4\%\) of is \(10.8\text{.}\)

16

\(9\%\) of is \(33.3\text{.}\)

17

\(400\%\) of is \(1880\text{.}\)

18

\(230\%\) of is \(1311\text{.}\)

Percentage Application Problems

19

A town has \(3500\) registered residents. Among them, \(39\%\) were Democrats, \(21\%\) were Republicans. The rest were Independents. How many registered Independents live in this town?

There are registered Independent residents in this town.

20

A town has \(4000\) registered residents. Among them, \(36\%\) were Democrats, \(29\%\) were Republicans. The rest were Independents. How many registered Independents live in this town?

There are registered Independent residents in this town.

21

Holli is paying a dinner bill of \({\$46.00}\text{.}\) Holli plans to pay \(12\%\) in tips. How much in total (including bill and tip) will Holli pay?

Holli will pay in total (including bill and tip).

22

Benjamin is paying a dinner bill of \({\$49.00}\text{.}\) Benjamin plans to pay \(19\%\) in tips. How much in total (including bill and tip) will Benjamin pay?

Benjamin will pay in total (including bill and tip).

23

In the past few seasons’ basketball games, Ronda attempted \(310\) free throws, and made \(62\) of them. What percent of free throws did Ronda make?

Fill in the blank with a percent.

Ronda made of free throws in the past few seasons.

24

In the past few seasons’ basketball games, Jeff attempted \(170\) free throws, and made \(51\) of them. What percent of free throws did Jeff make?

Fill in the blank with a percent.

Jeff made of free throws in the past few seasons.

25

A painting is on sale at \({\$350.00}\text{.}\) Its original price was \({\$500.00}\text{.}\) What percentage is this off its original price?

Fill in the blank with a percent.

The painting was off its original price.

26

A painting is on sale at \({\$440.00}\text{.}\) Its original price was \({\$550.00}\text{.}\) What percentage is this off its original price?

Fill in the blank with a percent.

The painting was off its original price.

27

The pie chart represents a collector’s collection of signatures from various artists.

If the collector has a total of \(1150\) signatures, there are signatures by Sting.

28

The pie chart represents a collector’s collection of signatures from various artists.

If the collector has a total of \(1350\) signatures, there are signatures by Sting.

29

Gustav sells cars for a living. Each month, he earns \({\$1{,}100.00}\) of base pay, plus a certain percentage of commission from his sales.

One month, Gustav made \({\$69{,}600.00}\) in sales, and earned a total of \({\$3{,}271.52}\) in that month (including base pay and commission). What percent commission did Gustav earn?

Fill in the blank with a percent.

Gustav earned in commission.

30

Kristen sells cars for a living. Each month, she earns \({\$1{,}000.00}\) of base pay, plus a certain percentage of commission from her sales.

One month, Kristen made \({\$74{,}000.00}\) in sales, and earned a total of \({\$2{,}909.20}\) in that month (including base pay and commission). What percent commission did Kristen earn?

Fill in the blank with a percent.

Kristen earned in commission.

31

A community college conducted a survey about the number of students riding each bus line available. The following bar graph is the result of the survey.

What percent of students ride Bus No. 1?

Fill in the blank with a percent. Round your percent to whole numbers, like \(11\%\text{.}\)

Approximately of students ride Bus No. 1.

32

A community college conducted a survey about the number of students riding each bus line available. The following bar graph is the result of the survey.

What percent of students ride Bus No. 1?

Fill in the blank with a percent. Round your percent to whole numbers, like \(11\%\text{.}\)

Approximately of students ride Bus No. 1.

33

In the last election, \(32\%\) of a county’s residents, or \(5536\) people, turned out to vote. How many residents live in this county?

This county has residents.

34

In the last election, \(58\%\) of a county’s residents, or \(12586\) people, turned out to vote. How many residents live in this county?

This county has residents.

35

Aleric earned \({\$96.57}\) of interest from a mutual fund, which was \(0.37\%\) of his total investment. How much money did Aleric invest into this mutual fund?

Aleric invested in this mutual fund.

36

Aaron earned \({\$3.06}\) of interest from a mutual fund, which was \(0.01\%\) of his total investment. How much money did Aaron invest into this mutual fund?

Aaron invested in this mutual fund.

37

Sydney paid a dinner and left \(17\%\text{,}\) or \({\$6.63}\text{,}\) in tips. How much was the original bill (without counting the tip)?

The original bill (not including the tip) was .

38

Morah paid a dinner and left \(13\%\text{,}\) or \({\$5.46}\text{,}\) in tips. How much was the original bill (without counting the tip)?

The original bill (not including the tip) was .

39

The following is a nutrition fact label from a certain macaroni and cheese box.

The highlighted row means each serving of macaroni and cheese in this box contains \({15\ {\rm g}}\) of fat, which is \(20\%\) of an average person’s daily intake of fat. What’s the recommended daily intake of fat for an average person?

The recommended daily intake of fat for an average person is .

Use g for grams.

40

The following is a nutrition fact label from a certain macaroni and cheese box.

The highlighted row means each serving of macaroni and cheese in this box contains \({12.8\ {\rm g}}\) of fat, which is \(16\%\) of an average person’s daily intake of fat. What’s the recommended daily intake of fat for an average person?

The recommended daily intake of fat for an average person is .

Use g for grams.

Percent of Increase/Decrease Problems

41

The population of cats in a shelter decreased from \(20\) to \(17\text{.}\) What is the percentage decrease of the shelter’s cat population?

The percentage decrease is .

42

The population of cats in a shelter decreased from \(40\) to \(22\text{.}\) What is the percentage decrease of the shelter’s cat population?

The percentage decrease is .

43

The population of cats in a shelter increased from \(30\) to \(44\text{.}\) What is the percentage increase of the shelter’s cat population?

Fill in the blank with percent. Round your answer to \(2\) decimal places, like \(12.34\%\text{.}\)

The percentage increase is approximately .

44

The population of cats in a shelter increased from \(38\) to \(46\text{.}\) What is the percentage increase of the shelter’s cat population?

Fill in the blank with percent. Round your answer to \(2\) decimal places, like \(12.34\%\text{.}\)

The percentage increase is approximately .

45

Last year, a small town had \(650\) population. This year, the population decreased to \(649\text{.}\) What is the percentage decrease of the town’s population?

Fill in blank with a percent. Round your answer to two decimal places, like \(1.23\%\text{.}\)

The percentage decrease of the town’s population was approximately .

46

Last year, a small town had \(690\) population. This year, the population decreased to \(686\text{.}\) What is the percentage decrease of the town’s population?

Fill in blank with a percent. Round your answer to two decimal places, like \(1.23\%\text{.}\)

The percentage decrease of the town’s population was approximately .

47

Your salary used to be \({\$33{,}000}\) per year.

You had to take a \(4\%\) pay cut. After the cut, your salary was per year.

Then, you earned a \(4\%\) raise. After the raise, your salary was per year.

48

Your salary used to be \({\$47{,}000}\) per year.

You had to take a \(5\%\) pay cut. After the cut, your salary was per year.

Then, you earned a \(5\%\) raise. After the raise, your salary was per year.

49

This line graph shows a certain stock's price change over a few days.

From Nov. 1 to Nov. 5, what is the stock price’s percentage change?

Fill in the blank with a percent. Round your percent to two decimal places, like \(12.34\%\text{.}\)

From Nov. 1 to Nov. 5, the stock price’s percentage change was approximately .

50

This line graph shows a certain stock's price change over a few days.

From Nov. 1 to Nov. 5, what is the stock price’s percentage change?

Fill in the blank with a percent. Round your percent to two decimal places, like \(12.34\%\text{.}\)

From Nov. 1 to Nov. 5, the stock price’s percentage change was approximately .

51

A house was bought two years ago at the price of \({\$420{,}000}\text{.}\) Each year, the house’s value decreased by \(3\%\text{.}\) What’s the house’s value this year?

The house’s value this year is .

52

A house was bought two years ago at the price of \({\$280{,}000}\text{.}\) Each year, the house’s value decreased by \(4\%\text{.}\) What’s the house’s value this year?

The house’s value this year is .

53

After a \(15\%\) increase, a town has \(460\) people. What was the population before the increase?

Before the increase, the town’s population was .

54

After a \(45\%\) increase, a town has \(725\) people. What was the population before the increase?

Before the increase, the town’s population was .