Prealgebra 2e — Original English

Solve General Applications of Percent

Translate and Solve Basic Percent Equations

We will solve percent equations by using the methods we used to solve equations with fractions or decimals. In the past, you may have solved percent problems by setting them up as proportions. That was the best method available when you did not have the tools of algebra. Now as a prealgebra student, you can translate word sentences into algebraic equations, and then solve the 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 Aolani and her friends ate dinner at a restaurant, the bill came to $80. They wanted to leave a 20% tip. What amount would the tip be?

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

The figure shows a customer copy of a restaurant receipt with the amount of the bill, $80, and the amount of the tip, $16. There is a group of bills totaling $16.
A 20% tip for an $80 restaurant bill comes out to $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.

What number is 35% of 90?

Solution

Solution

Translate into algebra. Let n=the number. A visual representation of translating the word problem 'What number is 35% of 90?' into the algebraic equation 'n = 0.35 * 90'.
Multiply. A mathematical equation is displayed on a white background, showing the variable 'n' equals '31.5'.
31.5 is 35% of 90

125% of 28 is what number?

Solution

Solution

Translate into algebra. Let a=the number. An image illustrating the conversion of the word problem '125% of 28 is what number?' into the algebraic equation '1.25 * 28 = a', with visual cues mapping words to their mathematical forms.
Multiply. A simple mathematical equation shows '35 = a' with the number 35 equal to the variable 'a'.
125% of 28 is 35.

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

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

Translate and solve: 36 is 75% of what number?

Solution

Solution

Translate. Let b= the number. A diagram illustrates how to translate the word problem '36 is 75% of what number?' into the mathematical equation '36 = 0.75 . b' by breaking down each part of the sentence.
Divide both sides by 0.75. A mathematical equation is displayed where both sides of an equality are divided by 0.75 to solve for the variable 'b', represented as '36 / 0.75 = 0.75b / 0.75'.
Simplify. The image displays two lines of text: '48 = b' and '36 is 75% of 48', illustrating a mathematical relationship between numbers and a percentage.

6.5% of what number is $1.17?

Solution

Solution

Translate. Let b= the number. Translating the percentage word problem '6.5% of what number is $1.17?' into the equation '0.065 * b = 1.17', showing term-by-term correspondence.
Divide both sides by 0.065. A mathematical equation shows '0.065n over 0.065 equals 1.17 over 0.065' on a white background, demonstrating the step of dividing both sides of an equation by 0.065 to solve for 'n'.
Simplify. The image displays two lines of text: 'n = 18' at the top right, and '6.5% of $18 is $1.17.' at the bottom left, illustrating a percentage calculation.

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

What percent of 36 is 9?

Solution

Solution

Translate into algebra. Let p= the percent. A visual guide translating the phrase 'What percent of 36 is 9?' into the algebraic equation 'p . 36 = 9', showing the correspondence between words and mathematical symbols.
Divide by 36. A mathematical equation shows '36p over 36 equals 9 over 36'.
Simplify. The image displays a mathematical equation on a white background, stating 'p = 1/4'. The letter 'p' is shown in italics, followed by an equals sign, and then the fraction one over four.
Convert to decimal form. The image displays the equation 'p = 0.25' in a clear, sans-serif font against a plain white background, indicating a probability or a numerical value.
Convert to percent. The image displays the mathematical statement '25% of 36 is 9.' and also shows 'p = 25%,' illustrating a percentage calculation and its variable representation.

144 is what percent of 96?

Solution

Solution

Translate into algebra. Let p= the percent. Translating the phrase '144 is what percent of 96?' into an algebraic equation '144 = p . 96' to solve for the unknown percentage 'p'.
Divide by 96. A mathematical equation shows '144 divided by 96 equals 96p divided by 96' on a white background.
Simplify. The equation 1.5 = p is displayed in black text on a white background, representing a simple algebraic statement or a given value for the variable 'p'.
Convert to percent. Two lines of text display mathematical concepts: the first line shows '150% = p', and the second line states '144 is 150% of 96.'

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.

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.

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

Solution

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. Algebraic equation representing a calculation: The variable "t" equals zero point one eight times sixty eight dollars and fifty cents. This shows how to convert eighteen percent to a decimal and interpret "of" as multiplication.
Multiply. The image shows a mathematical expression
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.

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

The figures shows the nutrition facts for cereal.
Solution

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. Translating the word problem '85 mg is 2% of a?' into the algebraic equation '85 = 0.02 * a'.
Divide both sides by 0.02. A mathematical equation shows both sides being divided by 0.02: 85/0.02 = 0.02a/0.02.
Simplify. The image displays a mathematical equation: 4,250 equals the variable '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.

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

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. Visual representation of translating 'What percent of 480 is 240?' into the algebraic equation 'p * 480 = 240', showing how words map to mathematical symbols.
Divide both sides by 480. A mathematical equation shows p multiplied by 480 and divided by 480, which is equal to 240 divided by 480.
Simplify. The mathematical expression p = 0.5 is displayed, suggesting a probability or a specific value for the variable p.
Convert to percent form. The image displays the mathematical expression 'p = 50%' in black text against a plain white background, indicating a probability or percentage value.
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.

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.

In 2011, the California governor proposed raising community college fees from $26 per unit to $36 per unit. Find the percent increase. (Round to the nearest tenth of a percent.)

Solution

Solution

What are you asked to find? the percent increase
Choose a variable to represent it. Let p= percent.
Find the amount of increase. A mathematical equation illustrating the calculation of an increase: 36 (new amount) - 26 (original amount) = 10 (increase).
Find the percent increase. The increase is what percent of the original amount?
Translate to an equation. A mathematical problem asks '10 is what percent of 26?' above its algebraic translation: '10 = p . 26'. Each part of the word problem is shown with a bracket above its corresponding mathematical symbol below.
Divide both sides by 26. An algebraic equation showing 10 divided by 26 equals 26p divided by 26, presented in fraction form on a white background.
Round to the nearest thousandth. The equation 0.385 = p is displayed in black text on a white background.
Convert to percent form. A mathematical equation displays '38.5% = p' in a simple, clear font on a white background. This equation assigns the percentage value of 38.5 to the variable 'p', indicating a direct equivalency.
Write a complete sentence. The new fees represent a 38.5% increase over the old fees.

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.

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

Solution

Solution

What are you asked to find? the percent decrease
Choose a variable to represent it. Let p= percent.
Find the amount of decrease. An image displays the subtraction 3.71 - 3.64 = 0.07. The number 3.71 is labeled 'original amount', 3.64 is labeled 'new amount', and 0.07 is labeled 'decrease'.
Find the percent of decrease. The decrease is what percent of the original amount?
Translate to an equation. A visual representation of translating the English phrase '0.07 is what percent of 3.71?' into the algebraic equation '0.07 = p * 3.71' for solving percentages.
Divide both sides by 3.71. A mathematical equation shows both sides being divided by 3.71: 0.07 / 3.71 = 3.71p / 3.71.
Round to the nearest thousandth. A mathematical equation displays '0.019 = p' in black text against a plain white background.
Convert to percent form. The image displays a mathematical equation: 1.9% = p. The equation shows the percentage 1.9% equated to the variable p, suggesting a conversion or a given value for p.
Write a complete sentence. The price of gas decreased 1.9%.

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:
      increase=new amountoriginal amount
    2. Find the percent increase as a percent of the original amount.
  • Find percent decrease.
    1. Find the amount of decrease.
      decrease=original amountnew amount
    2. Find the percent decrease as a percent of the original amount.

Practice Makes Perfect

Translate and Solve Basic Percent Equations

In the following exercises, translate and solve.

What number is 45% of 120?

Solution

54

What number is 65% of 100?

What number is 24% of 112?

Solution

26.88

What number is 36% of 124?

250% of 65 is what number?

Solution

162.5

150% of 90 is what number?

800% of 2,250 is what number?

Solution

18,000

600% of 1,740 is what number?

28 is 25% of what number?

Solution

112

36 is 25% of what number?

81 is 75% of what number?

Solution

108

93 is 75% of what number?

8.2% of what number is $2.87?

Solution

$35

6.4% of what number is $2.88?

11.5% of what number is $108.10?

Solution

$940

12.3% of what number is $92.25?

What percent of 260 is 78?

Solution

30%

What percent of 215 is 86?

What percent of 1,500 is 540?

Solution

36%

What percent of 1,800 is 846?

30 is what percent of 20?

Solution

150%

50 is what percent of 40?

840 is what percent of 480?

Solution

175%

790 is what percent of 395?

Solve Applications of Percents

In the following exercises, solve the applications of percents.

Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. She wants to leave 16% of the total bill as a tip. How much should the tip be?

Solution

$11.88

When Hiro and his co-workers had lunch at a restaurant the bill was $90.50. They want to leave 18% of the total bill as a tip. How much should the tip be?

Trong has 12% of each paycheck automatically deposited to his savings account. His last paycheck was $2,165. How much money was deposited to Trong's savings account?

Solution

$259.80

Cherise deposits 8% of each paycheck into her retirement account. Her last paycheck was $1,485. How much did Cherise deposit into her retirement account?

One serving of oatmeal has 8 grams of fiber, which is 33% of the recommended daily amount. What is the total recommended daily amount of fiber?

Solution

24.2 grams

One serving of trail mix has 67 grams of carbohydrates, which is 22% of the recommended daily amount. What is the total recommended daily amount of carbohydrates?

A bacon cheeseburger at a popular fast food restaurant contains 2,070 milligrams (mg) of sodium, which is 86% of the recommended daily amount. What is the total recommended daily amount of sodium?

Solution

2,407 mg

A grilled chicken salad at a popular fast food restaurant contains 650 milligrams (mg) of sodium, which is 27% of the recommended daily amount. What is the total recommended daily amount of sodium?

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?

Solution

45%

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?

Emma gets paid $3,000 per month. She pays $750 a month for rent. What percent of her monthly pay goes to rent?

Solution

25%

Dimple gets paid $3,200 per month. She pays $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.

Tamanika got a raise in her hourly pay, from $15.50 to $17.55. Find the percent increase.

Solution

13.2%

Ayodele got a raise in her hourly pay, from $24.50 to $25.48. Find the percent increase.

Annual student fees at the University of California rose from about $4,000 in 2000 to about $9,000 in 2014. Find the percent increase.

Solution

125%

The price of a share of one stock rose from $12.50 to $50. Find the percent increase.

According to Time magazine (7/19/2011) 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.)

Solution

72.7%

In one month, the median home price in the Northeast rose from $225,400 to $241,500. Find the percent increase. (Round to the nearest tenth of a percent.)

A grocery store reduced the price of a loaf of bread from $2.80 to $2.73. Find the percent decrease.

Solution

2.5%

The price of a share of one stock fell from $8.75 to $8.54. Find the percent decrease.

Hernando's salary was $49,500 last year. This year his salary was cut to $44,055. Find the percent decrease.

Solution

11%

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.)

In one month, the median home price in the West fell from $203,400 to $192,300. Find the percent decrease. (Round to the nearest tenth of a percent.)

Solution

5.5%

Sales of video games and consoles fell from $1,150 million to $1,030 million in one year. Find the percent decrease. (Round to the nearest tenth of a percent.)

Everyday Math

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

Solution

21.2%

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

Writing Exercises

Without solving the problem 44 is 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.

Solution

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%.

Without solving the problem “What is 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.

After returning from vacation, Alex said he should have packed 50% fewer shorts and 200% more shirts. Explain what Alex meant.

Solution

Alex should have packed half as many shorts and twice as many shirts.

Because of road construction in one city, commuters were advised to plan their Monday morning commute to take 150% of their usual commuting time. Explain what this means.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A self-assessment table titled 'I can...' asks users to rate their ability to 'translate and solve basic percent equations,' 'solve applications of percent,' and 'find percent increase and percent decrease' with options 'Confidently,' 'With some help,' or 'No - I don't get it!'

After reviewing this checklist, what will you do to become confident for all objectives?

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.