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Section A.4 Percentages

Percent-related problems arise in everyday life. This section reviews some basic calculations that can be made with percentages.
Figure A.4.1. Alternative Video Lesson

Subsection A.4.1 Converting Percents, Decimals, and English

In many situations when translating from English to math, the word β€œof” translates as multiplication. Also the word β€œis” (and many similar words related to β€œto be”) translates to an equal sign. For example:
One third of thirty is ten.13β‹…30=10
Here is another example, this time involving a percentage. We know that β€œ2 is 50% of 4”, so we can say:
2 is 50% of 42=0.5β‹…4

Example A.4.2.

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?
  2. $7.20 is what percent of $24.00?
  3. $7.20 is 30% of how much money?
Explanation.
Each question can be translated from English into a math equation by reading it slowly and looking for the right signals.
  1. The word β€œis” means about the same thing as the equal sign. β€œHow much” is a question phrase, and we can let x be the unknown amount (in dollars). The word β€œof” translates to multiplication, as discussed earlier. So we have:
    x| how much =| is 0.30| 30% β‹…| of 24| $24 
  2. Let P be the unknown value. We have:
    7.2| $7.20 =| is P| what percent β‹…| of 24| $24 
    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:
    7.2| $7.20 =| is 0.30| 30% β‹…| of x| how much 

Subsection A.4.2 Setting up 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 β€œ2 is 50% of 4”. Let’s look at some further examples.

Example A.4.4.

In Fall 2016, Portland Community College had 89,900 enrolled students. According to Figure 5, how many Black students were enrolled at PCC in Fall 2016?
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%
Figure A.4.5. Racial breakdown of PCC students in Fall 2016
Explanation.
After reading this word problem and the chart, we can translate the problem into β€œwhat is 6% of 89,900?” Let x be the number of Black students enrolled at PCC in Fall 2016. We can set up and solve the equation:
x| what =| is 0.06| 6% β‹…| of 89900| 89,900 x=5394
There was not much β€œsolving” 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 numbers we started with (89,900 enrolled students and 6%) appear to be rounded.

Example A.4.6.

The bar graph in Figure 7 displays how many students are in each class at a local high school. According to the bar graph, what percentage of the school’s student population is freshman?
a bar graph indicating there are 134 freshmen, 103 sophomores, 96 juniors, and 86 seniors
Figure A.4.7. Number of students at a high school by class
Explanation.
The school’s total number of students is:
134+103+96+86=419
With that calculated, we can translate the main question:
β€œWhat percentage of the school’s student population is freshman?”
into:
β€œWhat percent of 419 is 134?”
Using P to represent the unknown quantity, we write and solve the equation:
P| what percent β‹…| of 419| 419 =| is 134| 134 Pβ‹…419419=134419Pβ‰ˆ0.3198Pβ‰ˆ31.98%
Approximately 31.98% of the school’s student population is freshman.

Remark A.4.8.

When solving equations that do not have context we state the solution set. However, when solving an equation or inequality that arises in an application problem (such as the context of the high school in Example 6), 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.

Example A.4.9.

Carlos just received his monthly paycheck. His gross pay (the amount before taxes and related things are deducted) was $2,346.19, and his total tax and other deductions was $350.21. The rest was deposited directly into his checking account. What percent of his gross pay went into his checking account?
Explanation.
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. Only two of these have been explicitly written down. We need to use subtraction to find the dollar amount that was deposited:
2346.19βˆ’350.21=1995.98
Now, we can translate the main question:
β€œWhat percent of his gross pay went into his checking account?”
into:
β€œWhat percent of $2346.19 is $1995.98?”
Using P to represent the unknown quantity, we write and solve the equation:
P| what percent β‹…| of 2346.19| $2346.19 =| is 1995.98| $1995.98 Pβ‹…2346.192346.19=1995.982346.19Pβ‰ˆ0.8507Pβ‰ˆ85.07%
Approximately 85.07% of his gross pay went into his checking account.

Checkpoint A.4.10.

Alexis 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?
Alexis needs to bring in in sales profit.
Explanation.
Be careful that you do not calculate 28% of $2200. That might be what a student would do who doesn’t thoroughly read the question. 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 0.28β‹…2200=616, then $616 makes no sense as an answer to this question. How could Alexis bring in only $616 of sales profit, and earn $2200 in commission?
We can translate the problem into β€œ$2200 is 28% of what?” Letting x be the sales profit for the dealership (in dollars), we can write and solve the equation:
2200| $2200 =| is 0.28| 28% β‹…| of x| what 22000.28=0.28x0.287857.14β‰ˆxxβ‰ˆ7857.14
To earn $2200 in commission, Alexis needs to bring in approximately $7857.14 of sales profit for the dealership.

Example A.4.11.

According to e-Literate
 1 
mfeldstein.com/how-much-do-college-students-actually-pay-for-textbooks
, 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
Figure A.4.12. Average New Textbook Price from 2009 to 2013
Explanation.
The actual amount of increase from 2009 to 2013 was 79βˆ’62=17, dollars. We need to answer the question β€œ$17 is what percent of $62?” Note that we are comparing the 17 to 62, not to 79. In these situations where one amount is the earlier amount, the earlier original amount is the one that represents 100%. Let P represent the percent of increase. We can set up and solve the equation:
17| $17 =| is P| what percent β‹…| of 62| $62 17=62P1762=62P620.2742β‰ˆP
From 2009 to 2013, the average cost of a new textbook increased by approximately 27.42%.

Checkpoint A.4.13.

Last month, a full tank of gas for a car you drive cost you $40.00. You hear on the news that gas prices have risen by 12%. By how much, in dollars, has the cost of a full tank gone up?
A full tank of gas now costs more than it did last month.
Explanation.
Let x represent the amount of increase. We can set up and solve the equation:
0.12| 12% β‹…| of 40| old cost =| is x| how much 4.8=x
A full tank now costs $4.80 more than it did last month.

Example A.4.14.

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?
Explanation.
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%βˆ’15%=85% of that. So we can set up and solve the equation
x| this year's enrollment =| is 0.85| 85% β‹…| of 1.15| 115% β‹…| of 417| enrollment two years ago x=0.85β‹…1.15β‹…417x=407.6175
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.)

Exercises A.4.3 Exercises

Review and Warmup.

Exercise Group.

Basic Percentage Calculation.

Exercise Group.

Applications.

31.

A town has 4200 registered residents. Among them, 31% were Democrats, 27% were Republicans. The rest were Independents. How many registered Independents live in this town?
There are registered Independent residents in this town.

32.

A town has 4600 registered residents. Among them, 36% were Democrats, 35% were Republicans. The rest were Independents. How many registered Independents live in this town?
There are registered Independent residents in this town.

33.

Joshua is paying a dinner bill of $20.00. Joshua plans to pay 13% in tips. How much tip will Joshua pay?
Joshua will pay in tip.

34.

Fabrienne is paying a dinner bill of $23.00. Fabrienne plans to pay 20% in tips. How much tip will Fabrienne pay?
Fabrienne will pay in tip.

35.

Alyson is paying a dinner bill of $27.00. Alyson plans to pay 16% in tips. How much in total (including bill and tip) will Alyson pay?
Alyson will pay in total (including bill and tip).

36.

Parnell is paying a dinner bill of $30.00. Parnell plans to pay 12% in tips. How much in total (including bill and tip) will Parnell pay?
Parnell will pay in total (including bill and tip).

37.

A watch’s wholesale price was $340.00. The retailer marked up the price by 40%. What’s the watch’s new price (markup price)?
The watch’s markup price is .

38.

A watch’s wholesale price was $380.00. The retailer marked up the price by 30%. What’s the watch’s new price (markup price)?
The watch’s markup price is .

39.

In the past few seasons’ basketball games, Sydney attempted 190 free throws, and made 133 of them. What percent of free throws did Sydney make?
Sydney made of free throws in the past few seasons.

40.

In the past few seasons’ basketball games, Kurt attempted 450 free throws, and made 360 of them. What percent of free throws did Kurt make?
Kurt made of free throws in the past few seasons.

41.

A painting is on sale at $640.00. Its original price was $800.00. What percentage is this off its original price?
The painting was off its original price.

42.

A painting is on sale at $360.00. Its original price was $400.00. What percentage is this off its original price?
The painting was off its original price.

43.

The pie chart represents a collector’s collection of signatures from various artists.
If the collector has a total of 450 signatures, there are signatures by Sting.

44.

The pie chart represents a collector’s collection of signatures from various artists.
If the collector has a total of 650 signatures, there are signatures by Sting.

45.

In the last election, 36% of a county’s residents, or 8460 people, turned out to vote. How many residents live in this county?
This county has residents.

46.

In the last election, 61% of a county’s residents, or 17080 people, turned out to vote. How many residents live in this county?
This county has residents.

47.

61.69 grams of pure alcohol was used to produce a bottle of 19.9% alcohol solution. What is the weight of the solution in grams?
The alcohol solution weighs .

48.

38.76 grams of pure alcohol was used to produce a bottle of 11.4% alcohol solution. What is the weight of the solution in grams?
The alcohol solution weighs .

49.

Carmen paid a dinner and left 18%, or $7.92, in tips. How much was the original bill (without counting the tip)?
The original bill (not including the tip) was .

50.

Sean paid a dinner and left 14%, or $6.58, in tips. How much was the original bill (without counting the tip)?
The original bill (not including the tip) was .

51.

Olivia sells cars for a living. Each month, she earns $1,800.00 of base pay, plus a certain percentage of commission from her sales.
One month, Olivia made $40,100.00 in sales, and earned a total of $2,866.66 in that month (including base pay and commission). What percent commission did Olivia earn?
Olivia earned in commission.

52.

Phil sells cars for a living. Each month, he earns $1,800.00 of base pay, plus a certain percentage of commission from his sales.
One month, Phil made $44,500.00 in sales, and earned a total of $3,593.35 in that month (including base pay and commission). What percent commission did Phil earn?
Phil earned in commission.

53.

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 7.7 g of fat, which is 14% 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 .

54.

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 6 g of fat, which is 10% 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 .

55.

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 #1?
Approximately of students ride Bus #1.

56.

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 #1?
Approximately of students ride Bus #1.

57.

Nina earned $337.64 of interest from a mutual fund, which was 0.92% of his total investment. How much money did Nina invest into this mutual fund?
Nina invested in this mutual fund.

58.

Nenia earned $271.92 of interest from a mutual fund, which was 0.66% of his total investment. How much money did Nenia invest into this mutual fund?
Nenia invested in this mutual fund.

59.

A town has 4600 registered residents. Among them, there are 1518 Democrats and 1840 Republicans. The rest are Independents. What percentage of registered voters in this town are Independents?
In this town, of all registered voters are Independents.

60.

A town has 5000 registered residents. Among them, there are 1950 Democrats and 1400 Republicans. The rest are Independents. What percentage of registered voters in this town are Independents?
In this town, of all registered voters are Independents.

Percent Increase/Decrease.

61.

The population of cats in a shelter decreased from 40 to 28. What is the percentage decrease of the shelter’s cat population?
The percentage decrease is .

62.

The population of cats in a shelter decreased from 60 to 51. What is the percentage decrease of the shelter’s cat population?
The percentage decrease is .

63.

The population of cats in a shelter increased from 33 to 52. What is the percentage increase of the shelter’s cat population?
The percentage increase is approximately .

64.

The population of cats in a shelter increased from 41 to 54. What is the percentage increase of the shelter’s cat population?
The percentage increase is approximately .

65.

Last year, a small town’s population was 670. This year, the population decreased to 664. What is the percentage decrease?
The percentage decrease of the town’s population was approximately .

66.

Last year, a small town’s population was 700. This year, the population decreased to 698. What is the percentage decrease?
The percentage decrease of the town’s population was approximately .

67.

Your salary used to be $40,000 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.

68.

Your salary used to be $32,000 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.

69.

A house was bought two years ago at the price of $420,000. Each year, the house’s value decreased by 8%. What’s the house’s value this year?
The house’s value this year is .

70.

A house was bought two years ago at the price of $280,000. Each year, the house’s value decreased by 2%. What’s the house’s value this year?
The house’s value this year is .

71.

This line graph shows a certain stock’s price change over a few days.
From 11/1 to 11/5, what is the stock price’s percentage change?
From 11/1 to 11/5, the stock price’s percentage change was approximately .

72.

This line graph shows a certain stock’s price change over a few days.
From 11/1 to 11/5, what is the stock price’s percentage change?
From 11/1 to 11/5, the stock price’s percentage change was approximately .
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