3.4 Solve General Applications of Percent

Pooja Gupta

3.4 Solve General Applications of Percent

Learning Objectives

By the end of this section, you will be able to:

Translate and solve basic percent equations
Solve applications of percent
Find percent increase and percent decrease

Translate and Solve Basic Percent Equations

In the last section, we solved percent problems by setting them up as proportions. That is the best method available when you did not have the tools of algebra. Now, in this section we will translate word sentences into algebraic equations, and then solve the percent equations.

We’ll look at a common application of percent—tips to a server at a restaurant—to see how to set up a basic percent application.

When Kim and her friends went on a road trip to Vancouver, they ate lunch at Marta’s Cafe Tower. The bill came to $\text{\$80}$ . They wanted to leave a $\text{20\%}$ tip. What amount would the tip be?

To solve this, we want to find what amount is $\text{20\%}$ of $\text{\$80}$ . The $\text{\$80}$ is called the base. The amount of the tip would be $0.20\left(80\right)$ , or $\text{\$16}$ See (Figure 1). To find the amount of the tip, we multiplied the percent by the base.

A $\text{20\%}$ tip for an $\text{\$80}$ restaurant bill comes out to $\text{\$16}$ .

In the next examples, we will find the amount. We must be sure to change the given percent to a decimal when we translate the words into an equation.

EXAMPLE 1

What number is $\text{35\%}$ of $90?$

Solution

Translate into algebra. Let $n=\phantom{\rule{0.2em}{0ex}}$ the number.

${\underbrace{\text{What number }}_{\text{n}}} {\underbrace{\text{ is }}_{=}} {\underbrace{35\%}_{0.35}} {\underbrace{\text{ of }}_{\cdot}} {\underbrace{90?}_{90}}$

$n=0.35 \cdot 90$

Multiply.

$n=31.5$

$31.5$ is $35\%$ of $90$

TRY IT 1.1

What number is $\text{45\%}$ of $80?$

Answer

36

TRY IT 1.2

What number is $\text{55\%}$ of $60?$

Answer

33

EXAMPLE 2

$\text{125\%}$ of $28$ is what number?

Solution

Translate into algebra. Let $a\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}$ the number.

${\underbrace{125\%}_{1.25}} {\underbrace{\text{ of }}_{\cdot}} {\underbrace{28}_{28}} {\underbrace{\text{ is }}_{=}} {\underbrace{\text{ what number?}}_{a}}$

$1.25 \cdot 28=a$

Multiply.

$35=a$

$125\%$ of $28$ is $35$ .

Remember that a percent over $100$ is a number greater than $1$ . We found that $\text{125\%}$ of $28$ is $35$ , which is greater than $28$ .

TRY IT 2.1

$\text{150\%}$ of $78$ is what number?

Answer

117

TRY IT 2.2

$\text{175\%}$ of $72$ is what number?

Answer

126

In the next examples, we are asked to find the base.

EXAMPLE 3

Translate and solve: $36$ is $\text{75\%}$ of what number?

Solution

Translate. Let $b=$ the number.	${\underbrace{36}_{36}} {\underbrace{\text{ is }}_{=}} {\underbrace{75\%}_{0.75}} {\underbrace{\text{ of }}_{\cdot}} {\underbrace{\text{ what number?}}_{b}}$ $36=0.75 \cdot b$
Divide both sides by 0.75.	$\dfrac{36}{\color{red}0.75}=\dfrac{0.75 b}{\color{red}0.75}$
Simplify.	$48=b$
	$36$ is $75 \%$ of $48$

TRY IT 3.1

$17$ is $\text{25\%}$ of what number?

Answer

68

TRY IT 3.2

$40$ is $\text{62.5\%}$ of what number?

Answer

64

EXAMPLE 4

$\text{6.5\%}$ of what number is $\text{\$1.17}?$

Solution

Translate. Let $b=$ the number.	${\underbrace{6.5\%}_{0.065}} {\underbrace{\text{ of }}_{\cdot}} {\underbrace{\text{ what number }}_{\text{b}}} {\underbrace{\text{ is }}_{=}} {\underbrace{\$1.17?}_{1.17}}$ $0.065 \cdot b=1.17$
Divide both sides by 0.065.	$\dfrac{0.065 n}{\color{red}0.065}=\dfrac{1.17}{\color{red}0.065}$
Simplify.	$n=18$
	$6.5\%$ of $\$18$ is $\$1.17$

TRY IT 4.1

$\text{7.5\%}$ of what number is $\text{\$1.95}?$

Answer

$26

TRY IT 4.1

$\text{8.5\%}$ of what number is $\text{\$3.06}?$

Answer

$36

In the next examples, we will solve for the percent.

EXAMPLE 5

What percent of $36$ is $9?$

Solution

Translate into algebra. Let $p=$ the percent.	${\underbrace{\text{What percent }}_{\text{p}}} {\underbrace{\text{ of }}_{\cdot}} {\underbrace{36}_{36}} {\underbrace{\text{ is }}_{=}} {\underbrace{9?}_{9}}$ $p \cdot 36 =9$
Divide by 36.	$\dfrac{36 p}{\color{red}36}=\dfrac{9}{\color{red}36}$
Simplify.	$p=\dfrac{1}{4}$
Convert to decimal form.	$p=0.25$
Convert to percent.	$p = 25\%$
	$25\%$ of $36$ is $9$

TRY IT 5.1

What percent of $76$ is $57?$

Answer

75%

TRY IT 5.2

What percent of $120$ is $96?$

Answer

80%

EXAMPLE 6

$144$ is what percent of $96?$

Solution

Translate into algebra. Let $p=$ the percent.	${\underbrace{144}_{144}} {\underbrace{\text{ is }}_{=}} {\underbrace{\text{ what percent}}_{p}} {\underbrace{\text{ of }}_{\cdot}} {\underbrace{96?}_{96}}$ $144 = 9 \cdot 96$
Divide by 96.	$\dfrac{144}{\color{red}96}=\dfrac{96p}{\color{red}96}$
Simplify.	$1.5=p$
Convert to percent.	$150\%=p$
	$144$ is $150\%$ of 96.

TRY IT 6.1

$110$ is what percent of $88?$

Answer

125%

TRY IT 6.2

$126$ is what percent of $72?$

Answer

175%

Solve Applications of Percent

Many applications of percent occur in our daily lives, such as tips, sales tax, discount, and interest. To solve these applications we’ll translate to a basic percent equation, just like those we solved in the previous examples in this section. Once you translate the sentence into a percent equation, you know how to solve it.

We will update the strategy we used in our earlier applications to include equations now. Notice that we will translate a sentence into an equation.

HOW TO: Solve an Application

Identify what you are asked to find and choose a variable to represent it.
Write a sentence that gives the information to find it.
Translate the sentence into an equation.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Write a complete sentence that answers the question.

Now that we have the strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we’ll solve involve everyday situations, you can rely on your own experience.

EXAMPLE 7

Dezohn and his girlfriend enjoyed a dinner at a restaurant, and the bill was $\text{\$68.50}$ . They want to leave an $\text{18\%}$ tip. If the tip will be $\text{18\%}$ of the total bill, how much should the tip be?

Solution

What are you asked to find?	The amount of the tip
Choose a variable to represent it.	Let $t=$ amount of tip.
Write a sentence that give the information to find it.	The tip is 18% of the total bill.
Translate the sentence into an equation.	${\underbrace{\text{The tip }}_{\text{t}}} {\underbrace{\text{ is }}_{=}} {\underbrace{18\%}_{0.018}} {\underbrace{\text{ of }}_{\cdot}} {\underbrace{\$68.50?}_{68.50}}$ $t=0.18 \cdot 68.50$
Multiply.	$t=12.33$
Check. Is this answer reasonable?	If we approximate the bill to $70 and the percent to 20%, we would have a tip of $14. So a tip of $12.33 seems reasonable.
Write a complete sentence that answers the question.	The couple should leave a tip of $12.33.

TRY IT 7.1

Cierra and her sister enjoyed a special dinner in a restaurant, and the bill was $\text{\$81.50}$ . If she wants to leave $\text{18\%}$ of the total bill as her tip, how much should she leave?

Answer

$14.67

TRY IT 7.2

Kimngoc had lunch at her favorite restaurant. She wants to leave $\text{15\%}$ of the total bill as her tip. If her bill was $\text{\$14.40}$ , how much will she leave for the tip?

Answer

$2.16

EXAMPLE 8

The label on Masao’s breakfast cereal said that one serving of cereal provides $85$ milligrams (mg) of potassium, which is $\text{2\%}$ of the recommended daily amount. What is the total recommended daily amount of potassium?

The figures shows the nutrition facts for cereal.

Solution

What are you asked to find?	the total amount of potassium recommended
Choose a variable to represent it.	Let $a=$ total amount of potassium.
Write a sentence that gives the information to find it.	85 mg is 2% of the total amount.
Translate the sentence into an equation.	${\underbrace{85\text{ mg }}_{\text{85}}} {\underbrace{\text{ is }}_{=}} {\underbrace{2\%}_{0.02}} {\underbrace{\text{ of }}_{\cdot}} {\underbrace{a?}_{a}}$ $85=0.02 \cdot a$
Divide both sides by 0.02.	$\dfrac{85}{\color{red}0.02}=\dfrac{0.02a}{\color{red}0.02}$
Simplify.	$4,250=a$
Check: Is this answer reasonable?	Yes. 2% is a small percent and 85 is a small part of 4,250.
Write a complete sentence that answers the question.	The amount of potassium that is recommended is 4250 mg.

TRY IT 8.1

One serving of wheat square cereal has $7$ grams of fiber, which is $\text{29\%}$ of the recommended daily amount. What is the total recommended daily amount of fiber?

Answer

24.1 grams

TRY IT 8.2

One serving of rice cereal has $190$ mg of sodium, which is $\text{8\%}$ of the recommended daily amount. What is the total recommended daily amount of sodium?

Answer

2,375 mg

EXAMPLE 9

Mitzi received some gourmet brownies as a gift. The wrapper said each brownie was $480$ calories, and had $240$ calories of fat. What percent of the total calories in each brownie comes from fat?

Solution

What are you asked to find?	the percent of the total calories from fat
Choose a variable to represent it.	Let $p=$ percent from fat.
Write a sentence that gives the information to find it.	What percent of 480 is 240?
Translate the sentence into an equation.	${\underbrace{\text{What percent}}_{\text{p}}} {\underbrace{\text{ of }}_{\cdot}} {\underbrace{480}_{480}} {\underbrace{\text{ is }}_{=}} {\underbrace{240?}_{240}}$ $p \cdot 480=240$
Divide both sides by 480.	$\dfrac{480 p}{\color{red}480}=\dfrac{240}{\color{red}480}$
Simplify.	$p=0.5$
Convert to percent form.	$p=50\%$
Check. Is this answer reasonable?	Yes. 240 is half of 480, so 50% makes sense.
Write a complete sentence that answers the question.	Of the total calories in each brownie, 50% is fat.

TRY IT 9.1

Veronica is planning to make muffins from a mix. The package says each muffin will be $230$ calories and $60$ calories will be from fat. What percent of the total calories is from fat? (Round to the nearest whole percent.)

Answer

26%

Exercises

The brownie mix Ricardo plans to use says that each brownie will be $190$ calories, and $70$ calories are from fat. What percent of the total calories are from fat?

Answer

37%

Find Percent Increase and Percent Decrease

People in the media often talk about how much an amount has increased or decreased over a certain period of time. They usually express this increase or decrease as a percent.

To find the percent increase, first we find the amount of increase, which is the difference between the new amount and the original amount. Then we find what percent the amount of increase is of the original amount.

HOW TO: Find Percent Increase

Step 1. Find the amount of increase.

$\text{increase}=\text{new amount}-\text{original amount}$

Step 2. Find the percent increase as a percent of the original amount.

EXAMPLE 10

In $2017$ , university tuition fees in Canada for domestic students increased from $\text{\$26}$ per school year to $\text{\$36}$ per school year. Find the percent increase. (Round to the nearest tenth of a percent.)

Solution

What are you asked to find?	the percent increase
Choose a variable to represent it.	Let $p=$ percent.
Find the amount of increase.	$\text{increase}=\text{new amount}-\text{original amount}$ $=\$36-\$26=\$10$
Find the percent increase.	The increase is what percent of the original amount?
Translate to an equation.	${\underbrace{\text{10}}_{\text{10}}} {\underbrace{\text{ is }}_{=}} {\underbrace{\text{what percent}}_{p}} {\underbrace{\text{ of }}_{\cdot}} {\underbrace{26?}_{26}}$ $10=p \cdot 26$
Divide both sides by 26.	$\dfrac{10}{\color{red}26}=\dfrac{26p}{\color{red}26}$
Round to the nearest thousandth.	$0.384=p$
Convert to percent form.	$38.4\%=p$
Write a complete sentence.	The new fees represent a $38.4\%$ increase over the old fees.

TRY IT 10.1

In $2011$ , the IRS increased the deductible mileage cost to $55.5$ cents from $51$ cents. Find the percent increase. (Round to the nearest tenth of a percent.)

Answer

8.8%

TRY IT 10.2

In $1984$ , the standard bus fare in Vancouver was $\text{\$1.25}$ . In $2008$ , the standard bus fare was $\text{\$2.50}$ . Find the percent increase. (Round to the nearest tenth of a percent.)

Answer

100%

Finding the percent decrease is very similar to finding the percent increase, but now the amount of decrease is the difference between the original amount and the final amount. Then we find what percent the amount of decrease is of the original amount.

HOW TO: Find Percent Decrease

Find the amount of decrease.
- $\text{decrease}=\text{original amount}-\text{new amount}$
Find the percent decrease as a percent of the original amount.

EXAMPLE 11

The average price of a gallon of gas in one city in June $2014$ was $\text{\$3.71}$ . The average price in that city in July was $\text{\$3.64}$ . Find the percent decrease.

Solution

What are you asked to find?	the percent decrease
Choose a variable to represent it.	Let $p=$ percent.
Find the amount of decrease.	$\text{decrease}=\text{original amount}-\text{new amount}$ $\$3.71 - \$3.64 = \$0.07$
Find the percent of decrease.	The decrease is what percent of the original amount?
Translate to an equation.	${\underbrace{\text{0.07}}_{\text{0.07}}} {\underbrace{\text{ is }}_{=}} {\underbrace{\text{what percent}}_{p}} {\underbrace{\text{ of }}_{\cdot}} {\underbrace{3.71?}_{3.71}}$ $0.07=p \cdot 3.71$
Divide both sides by 3.71.	$\dfrac{0.07}{\color{red}3.71}=\dfrac{3.71p}{\color{red}0.07}$
Round to the nearest thousandth.	$0.019=p$
Convert to percent form.	$1.9\%=p$
Write a complete sentence.	The price of gas decreased 1.9%.

TRY IT 11.1

The population of one city was about $672,000$ in $2010$ . The population of the city is projected to be about $630,000$ in $2020$ . Find the percent decrease. (Round to the nearest tenth of a percent.)

Answer

6.3%

TRY IT 11.2

Last year Sheila’s salary was $\text{\$42,000}$ . Because of furlough days, this year her salary was $\text{\$37,800}$ . Find the percent decrease. (Round to the nearest tenth of a percent.)

Answer

10%

Access Additional Online Resources

Percent Increase and Percent Decrease Visualization

Key Concepts

Solve an application.
1. Identify what you are asked to find and choose a variable to represent it.
2. Write a sentence that gives the information to find it.
3. Translate the sentence into an equation.
4. Solve the equation using good algebra techniques.
5. Write a complete sentence that answers the question.
6. Check the answer in the problem and make sure it makes sense.
Find percent increase.
1. Find the amount of increase:
  $\text{increase}=\text{new amount}-\text{original amount}$
2. Find the percent increase as a percent of the original amount.
Find percent decrease.
1. Find the amount of decrease.
  $\text{decrease}=\text{original amount}-\text{new amount}$
2. Find the percent decrease as a percent of the original amount.

Glossary

percent increase: The percent increase is the percent the amount of increase is of the original amount.

percent decrease: The percent decrease is the percent the amount of decrease is of the original amount.

Practice Makes Perfect

Translate and Solve Basic Percent Equations

In the following exercises, translate and solve.

1. What number is $\text{45\%}$ of $120?$	2. What number is $\text{65\%}$ of $100?$
3. What number is $\text{24\%}$ of $112?$	4. What number is $\text{36\%}$ of $124?$
5. $\text{250\%}$ of $65$ is what number?	6. $\text{150\%}$ of $90$ is what number?
7. $\text{800\%}$ of $2,250$ is what number?	8. $\text{600\%}$ of $1,740$ is what number?
9. $28$ is $\text{25\%}$ of what number?	10. $36$ is $\text{25\%}$ of what number?
11. $81$ is $\text{75\%}$ of what number?	12. $93$ is $\text{75\%}$ of what number?
13. $\text{8.2\%}$ of what number is $\text{\$2.87}?$	14. $\text{6.4\%}$ of what number is $\text{\$2.88}?$
15. $\text{11.5\%}$ of what number is $\text{\$108.10}?$	16. $\text{12.3\%}$ of what number is $\text{\$92.25}?$
17. What percent of $260$ is $78?$	18. What percent of $215$ is $86?$
19. What percent of $1,500$ is $540?$	20. What percent of $1,800$ is $846?$
21. $30$ is what percent of $20?$	22. $50$ is what percent of $40?$
23. $840$ is what percent of $480?$	24. $790$ is what percent of $395?$

Solve Applications of Percents

In the following exercises, solve the applications of percents.

25. Geneva treated her parents to dinner at their favorite restaurant. The bill was $\text{\$74.25}$ . She wants to leave $\text{16\%}$ of the total bill as a tip. How much should the tip be?	26. When Hiro and his co-workers had lunch at a restaurant the bill was $\text{\$90.50}$ . They want to leave $\text{18\%}$ of the total bill as a tip. How much should the tip be?
27. Trong has $\text{12\%}$ of each paycheck automatically deposited to his savings account. His last paycheck was $\text{\$2,165}$ . How much money was deposited to Trong’s savings account?	28. Cherise deposits $\text{8\%}$ of each paycheck into her retirement account. Her last paycheck was $\text{\$1,485}$ . How much did Cherise deposit into her retirement account?
29. One serving of oatmeal has $8$ grams of fiber, which is $\text{33\%}$ of the recommended daily amount. What is the total recommended daily amount of fiber?	30. One serving of trail mix has $67$ grams of carbohydrates, which is $\text{22\%}$ of the recommended daily amount. What is the total recommended daily amount of carbohydrates?
31. A bacon cheeseburger at a popular fast food restaurant contains $2,070$ milligrams (mg) of sodium, which is $\text{86\%}$ of the recommended daily amount. What is the total recommended daily amount of sodium?	32. A grilled chicken salad at a popular fast food restaurant contains $650$ milligrams (mg) of sodium, which is $\text{27\%}$ of the recommended daily amount. What is the total recommended daily amount of sodium?
33. The nutrition fact sheet at a fast food restaurant says the fish sandwich has $380$ calories, and $171$ calories are from fat. What percent of the total calories is from fat?	34. The nutrition fact sheet at a fast food restaurant says a small portion of chicken nuggets has $190$ calories, and $114$ calories are from fat. What percent of the total calories is from fat?
35. Emma gets paid $\text{\$3,000}$ per month. She pays $\text{\$750}$ a month for rent. What percent of her monthly pay goes to rent?	36. Dimple gets paid $\text{\$3,200}$ per month. She pays $\text{\$960}$ a month for rent. What percent of her monthly pay goes to rent?

Find Percent Increase and Percent Decrease

In the following exercises, find the percent increase or percent decrease.

37. Tamanika got a raise in her hourly pay, from $\text{\$15.50}$ to $\text{\$17.55}$ . Find the percent increase.	38. Ayodele got a raise in her hourly pay, from $\text{\$24.50}$ to $\text{\$25.48}$ . Find the percent increase.
39. According to Statistics Canada, annual international graduate student fees in Canada rose from about $\text{\$13,000}$ in $2015$ to about $\text{\$15,000}$ in $2019$ . Find the percent increase.	40. The price of a share of one stock rose from $\text{\$12.50}$ to $\text{\$50}$ . Find the percent increase.
41. According to Time magazine $\left(\text{7/19/2011}\right)$ annual global seafood consumption rose from $22$ pounds per person in $1960$ to $38$ pounds per person today. Find the percent increase. (Round to the nearest tenth of a percent.)	42. In one month, the median home price in the Northeast rose from $\text{\$225,400}$ to $\text{\$241,500}$ . Find the percent increase. (Round to the nearest tenth of a percent.)
43. A grocery store reduced the price of a loaf of bread from $\text{\$2.80}$ to $\text{\$2.73}$ . Find the percent decrease.	44. The price of a share of one stock fell from $\text{\$8.75}$ to $\text{\$8.54}$ . Find the percent decrease.
45. Hernando’s salary was $\text{\$49,500}$ last year. This year his salary was cut to $\text{\$44,055}$ . Find the percent decrease.	46. From $2000$ to $2010$ , the population of Detroit fell from about $951,000$ to about $714,000$ . Find the percent decrease. (Round to the nearest tenth of a percent.)
47. In one month, the median home price in the West fell from $\text{\$203,400}$ to $\text{\$192,300}$ . Find the percent decrease. (Round to the nearest tenth of a percent.)	48. Sales of video games and consoles fell from $\text{\$1,150}$ million to $\text{\$1,030}$ million in one year. Find the percent decrease. (Round to the nearest tenth of a percent.)

Everyday Math

49. Tipping At the campus coffee cart, a medium coffee costs $\text{\$1.65}$ . MaryAnne brings $\text{\$2.00}$ with her when she buys a cup of coffee and leaves the change as a tip. What percent tip does she leave?

50. Late Fees Alison was late paying her credit card bill of $\text{\$249}$ . She was charged a $\text{5\%}$ late fee. What was the amount of the late fee?

Writing Exercises

51. Without solving the problem $``44$ is $\text{80\%}$ of what number”, think about what the solution might be. Should it be a number that is greater than $44$ or less than $44?$ Explain your reasoning.	52. Without solving the problem “What is $\text{20\%}$ of $300?''$ think about what the solution might be. Should it be a number that is greater than $300$ or less than $300?$ Explain your reasoning.
53. After returning from vacation, Alex said he should have packed $\text{50\%}$ fewer shorts and $\text{200\%}$ more shirts. Explain what Alex meant.	54. Because of road construction in one city, commuters were advised to plan their Monday morning commute to take $\text{150\%}$ of their usual commuting time. Explain what this means.

Answers

1. 54	3. 26.88	5. 162.5
7. 18,000	9. 112	11. 108
13. $35	15. $940	17. 30%
19. 36%	21. 150%	23. 175%
25. $11.88	27. $259.80	29. 24.2 grams
31. 2,407 grams	33. 45%	35. 25%
37. 13.2%	39. 15%	41. 72.7%
43. 2.5%	45. 11%	47. 5.5%
49. 21.2%	51. The original number should be greater than 44.80% is less than 100%, so when 80% is converted to a decimal and multiplied to the base in the percent equation, the resulting amount of 44 is less. 44 is only the larger number in cases where the percent is greater than 100%.	53. Alex should have packed half as many shorts and twice as many shirts.

Attributions

This chapter has been adapted from “Solve General Applications of Percent” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

License

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Translate and Solve Basic Percent Equations

Solve Applications of Percent

Find Percent Increase and Percent Decrease

Key Concepts

Glossary

Practice Makes Perfect

Translate and Solve Basic Percent Equations

Solve Applications of Percents

Find Percent Increase and Percent Decrease

Everyday Math

Writing Exercises

Answers

Attributions

License

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