Solve Percent Applications

Translate and Solve Basic Percent Equations

We will solve percent equations using the methods we used to solve equations with fractions or decimals. Without the tools of algebra, the best method available to solve percent problems was by setting them up as proportions. Now as an algebra student, you can just translate English sentences into algebraic equations and then solve the equations.

We can use any letter you like as a variable, but it is a good idea to choose a letter that will remind us of what you are looking for. We must be sure to change the given percent to a decimal when we put it in the equation.

Example

Try it.

Translate and solve: What number is 35% of 90?

Solution

Translate into algebra. Let \(n\)= the number.
Remember "of" means multiply, "is" means equals.
Multiply.
	\(31.5\) is \(35\%\) of \(90\)

We must be very careful when we translate the words in the next example. The unknown quantity will not be isolated at first, like it was in . We will again use direct translation to write the equation.

Example

Try it.

Translate and solve: 6.5% of what number is $1.17?

Solution

Translate. Let \(n=\) the number.
Multiply.
Divide both sides by 0.065 and simplify.
	\(6.5\%\) of\(\$18\) is \(\$1.17\)

In the next example, we are looking for the percent.

Example

Try it.

Translate and solve: 144 is what percent of 96?

Solution

Translate into algebra. Let \(p=\) the percent.
Multiply.
Divide by 96 and simplify.
Convert to percent.
	\(144\) is \(150\%\) of \(96\)

Note that we are asked to find percent, so we must have our final result in percent form.

Solve Applications of Percent

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

We will restate the problem solving strategy we used earlier for easy reference.

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 will involve everyday situations, you can rely on your own experience.

Example

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

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.		the amount of tip should Dezohn leave
Step 3. Name what we are looking for.
Choose a variable to represent it.		Let t = amount of tip.
Step 4. Translate into an equation.
Write a sentence that gives the information to find it.
Translate the sentence into an equation.
Step 5. Solve the equation. Multiply.
Step 6. Check. Does this make sense?
Yes, 20% of $70 is $14.
Step 7. Answer the question with a complete sentence.		Dezohn should leave a tip of $12.33.

Notice that we used t to represent the unknown tip.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

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

Example

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

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.		the percent increase
Step 3. Name what we are looking for.
Choose a variable to represent it.		Let \(p=\) the percent.
Step 4. Translate. Write a sentence that gives the information to find it.
First find the amount of increase.		new amount − original amount = increase
		\(36-26=10\)
Find the percent.		Increase is what percent of the original amount?
Translate into an equation.
Step 5. Solve the equation.
Divide by 26.
Change to percent form; round to the nearest tenth.
Step 6. Check. Does this make sense?
Yes, 38.4% is close to \(\frac{1}{3}\), and 10 is close to \(\frac{1}{3}\) of 26.
Step 7. Answer the question with a complete sentence.		The new fees represent a 38.5% increase over the old fees.

Notice that we rounded the division to the nearest thousandth in order to round the percent to the nearest tenth.

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

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Solve Simple Interest Applications

Do you know that banks pay you to keep your money? The money a customer puts in the bank is called the principal, P, and the money the bank pays the customer is called the interest. The interest is computed as a certain percent of the principal; called the rate of interest, r. We usually express rate of interest as a percent per year, and we calculate it by using the decimal equivalent of the percent. The variable t, (for time) represents the number of years the money is in the account.

To find the interest we use the simple interest formula, \(I=Prt.\)

Interest may also be calculated another way, called compound interest. This type of interest will be covered in later math classes.

The formula we use to calculate simple interest is \(I=Prt.\) To use the formula, we substitute in the values the problem gives us for the variables, and then solve for the unknown variable. It may be helpful to organize the information in a chart.

Example

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Nathaly deposited $12,500 in her bank account where it will earn 4% interest. How much interest will Nathaly earn in 5 years?

\[\begin{array}{lll}I & = & ? \\ P & = & \$12,500 \\ r & = & 4\text{\%} \\ t & = & 5\ \text{years}\end{array}\]

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.	the amount of interest earned
Step 3. Name what we are looking for. Choose a variable to represent that quantity.	Let \(I=\) the amount of interest.
Step 4. Translate into an equation. Write the formula. Substitute in the given information.	\(\begin{array}{l} \\ I=Prt \\ I=(12,500)(.04)(5)\end{array}\)
Step 5. Solve the equation.	\(I=2,500\)
Step 6. Check: Does this make sense? Is $2,500 a reasonable interest on $12,500? Yes.
Step 7. Answer the question with a complete sentence.	The interest is $2,500.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Solve Applications with Discount or Mark-up

Applications of discount are very common in retail settings. When you buy an item on sale, the original price has been discounted by some dollar amount. The discount rate, usually given as a percent, is used to determine the amount of the discount. To determine the amount of discount, we multiply the discount rate by the original price.

We summarize the discount model in the box below.

Example

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Elise bought a dress that was discounted 35% off of the original price of $140. What was ⓐ the amount of discount and ⓑ the sale price of the dress?

Solution

ⓐ \(\begin{array}{lll}\text{Original price} & = & \text{\$}140 \\ \text{Discount rate} & = & 35\text{\%} \\ \text{Discount} & = & ?\end{array}\)
Step 1. Read the problem.
Step 2. Identify what we are looking for.	the amount of discount
Step 3. Name what we are looking for. Choose a variable to represent that quantity.	Let \(d=\) the amount of discount.
Step 4. Translate into an equation. Write a sentence that gives the information to find it. Translate into an equation.	The discount is 35% of $140. \(d=0.35(140)\)
Step 5. Solve the equation.	\(d=49\)
Step 6. Check: Does this make sense? Is a $49 discount reasonable for a $140 dress? Yes.
Step 7. Write a complete sentence to answer the question.	The amount of discount was $49.

ⓑ
Read the problem again.

Step 1. Identify what we are looking for.		the sale price of the dress
Step 2. Name what we are looking for.
Choose a variable to represent that quantity.		Let \(s=\) the sale price.
Step 3. Translate into an equation.
Write a sentence that gives the information to find it.
Translate into an equation.
Step 4. Solve the equation.
Step 5. Check. Does this make sense?
Is the sale price less than the original price?
Yes, $91 is less than $140.
Step 6. Answer the question with a complete sentence.		The sale price of the dress was $91.

We summarize the mark-up model in the box below.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

• Percent Increase To find the percent increase:
  1. Find the amount of increase. \(\text{increase}=\text{new amount}-\text{original}\ \text{amount}\)
  2. Find the percent increase. Increase is what percent of the original amount?
• Percent Decrease To find the percent decrease:
  1. Find the amount of decrease. \(\text{decrease}=\text{original amount}-\text{new}\ \text{amount}\)
  2. Find the percent decrease. Decrease is what percent of the original amount?
• Simple Interest If an amount of money, P, called the principal, is invested for a period of t years at an annual interest rate r, the amount of interest, I, earned is
  \[\begin{array}{lll}I & = & Prt \\ \text{where}\ I & = & \text{interest} \\ P & = & \text{principal} \\ r & = & \text{rate} \\ t & = & \text{time}\end{array}\]
• Discount
  • amount of discount is discount rate \(\cdot\) original price
  • sale price is original price – discount
• Mark-up
  • amount of mark-up is mark-up rate \(\cdot\) original cost
  • list price is original cost + mark up

Solve Percent Applications

Translate and Solve Basic Percent Equations

In the following exercises, translate and solve.

Try it.

What number is 45% of 120?

Solution

54

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What number is 65% of 100?

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What number is 24% of 112?

Solution

26.88

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What number is 36% of 124?

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250% of 65 is what number?

Solution

162.5

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150% of 90 is what number?

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800% of 2250 is what number?

Solution

18,000

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600% of 1740 is what number?

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28 is 25% of what number?

Solution

112

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36 is 25% of what number?

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81 is 75% of what number?

Solution

108

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93 is 75% of what number?

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8.2% of what number is $2.87?

Solution

$35

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6.4% of what number is $2.88?

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11.5% of what number is $108.10?

Solution

$940

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12.3% of what number is $92.25?

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What percent of 260 is 78?

Solution

30%

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What percent of 215 is 86?

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What percent of 1500 is 540?

Solution

36%

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What percent of 1800 is 846?

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30 is what percent of 20?

Solution

150%

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50 is what percent of 40?

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840 is what percent of 480?

Solution

175%

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790 is what percent of 395?

Solve Percent Applications

In the following exercises, solve.

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

Solution

$11.88

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

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Trong has 12% of each paycheck automatically deposited to his savings account. His last paycheck was $2165. How much money was deposited to Trong’s savings account?

Solution

$259.80

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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?

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

Solution

24.2 g

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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?

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A bacon cheeseburger at a popular fast food restaurant contains 2070 milligrams (mg) of sodium, which is 86% of the recommended daily amount. What is the total recommended daily amount of sodium?

Solution

2407 mg

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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?

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After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa’s original weight?

Solution

175 lb.

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Tricia got a 6% raise on her weekly salary. The raise was $30 per week. What was her original salary?

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Yuki bought a dress on sale for $72. The sale price was 60% of the original price. What was the original price of the dress?

Solution

$120

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Kim bought a pair of shoes on sale for $40.50. The sale price was 45% of the original price. What was the original price of the shoes?

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Tim left a $9 tip for a $50 restaurant bill. What percent tip did he leave?

Solution

18%

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Rashid left a $15 tip for a $75 restaurant bill. What percent tip did he leave?

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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%

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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?

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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%

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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, solve.

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Tamanika got a raise in her hourly pay, from $15.50 to $17.36. Find the percent increase.

Solution

12%

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

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Annual student fees at the University of California rose from about $4,000 in 2000 to about $12,000 in 2010. Find the percent increase.

Solution

200%

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

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According to Time magazine annual global seafood consumption rose from 22 pounds per person in the 1960s to 38 pounds per person in 2011. Find the percent increase. (Round to the nearest tenth of a percent.)

Solution

72.7%

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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.)

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

Solution

2.5%

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

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

Solution

11%

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In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.

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In 1 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%

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

Solve Simple Interest Applications

In the following exercises, solve.

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Casey deposited $1,450 in a bank account with interest rate 4%. How much interest was earned in two years?

Solution

$116

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Terrence deposited $5,720 in a bank account with interest rate 6%. How much interest was earned in 4 years?

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Robin deposited $31,000 in a bank account with interest rate 5.2%. How much interest was earned in 3 years?

Solution

$4,836

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Carleen deposited $16,400 in a bank account with interest rate 3.9%. How much interest was earned in 8 years?

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Hilaria borrowed $8,000 from her grandfather to pay for college. Five years later, she paid him back the $8,000, plus $1,200 interest. What was the rate of interest?

Solution

3%

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Kenneth loaned his niece $1,200 to buy a computer. Two years later, she paid him back the $1,200, plus $96 interest. What was the rate of interest?

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Lebron loaned his daughter $20,000 to help her buy a condominium. When she sold the condominium four years later, she paid him the $20,000, plus $3,000 interest. What was the rate of interest?

Solution

3.75%

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Pablo borrowed $50,000 to start a business. Three years later, he repaid the $50,000, plus $9,375 interest. What was the rate of interest?

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In 10 years, a bank account that paid 5.25% earned $18,375 interest. What was the principal of the account?

Solution

$35,000

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In 25 years, a bond that paid 4.75% earned $2,375 interest. What was the principal of the bond?

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Joshua’s computer loan statement said he would pay $1,244.34 in interest for a 3-year loan at 12.4%. How much did Joshua borrow to buy the computer?

Solution

$3,345

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Margaret’s car loan statement said she would pay $7,683.20 in interest for a 5-year loan at 9.8%. How much did Margaret borrow to buy the car?

Solve Applications with Discount or Mark-up

In the following exercises, find the sale price.

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Perla bought a cell phone that was on sale for $50 off. The original price of the cell phone was $189.

Solution

$139

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Sophie saw a dress she liked on sale for $15 off. The original price of the dress was $96.

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Rick wants to buy a tool set with original price $165. Next week the tool set will be on sale for $40 off.

Solution

$125

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Angelo’s store is having a sale on televisions. One television, with original price $859, is selling for $125 off.

In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.

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Janelle bought a beach chair on sale at 60% off. The original price was $44.95.

Solution

ⓐ $26.97 ⓑ $17.98

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Errol bought a skateboard helmet on sale at 40% off. The original price was $49.95.

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Kathy wants to buy a camera that lists for $389. The camera is on sale with a 33% discount.

Solution

ⓐ $128.37 ⓑ $260.63

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Colleen bought a suit that was discounted 25% from an original price of $245.

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Erys bought a treadmill on sale at 35% off. The original price was $949.95 (round to the nearest cent.)

Solution

ⓐ $332.48 ⓑ $617.47

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Jay bought a guitar on sale at 45% off. The original price was $514.75 (round to the nearest cent.)

In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent if needed.)

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Larry and Donna bought a sofa at the sale price of $1,344. The original price of the sofa was $1,920.

Solution

ⓐ $576 ⓑ 30%

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Hiroshi bought a lawnmower at the sale price of $240. The original price of the lawnmower is $300.

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Patty bought a baby stroller on sale for $301.75. The original price of the stroller was $355.

Solution

ⓐ $53.25 ⓑ 15%

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Bill found a book he wanted on sale for $20.80. The original price of the book was $32.

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Nikki bought a patio set on sale for $480. The original price was $850. To the nearest tenth of a percent, what was the rate of discount?

Solution

ⓐ $370 ⓑ 43.5%

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Stella bought a dinette set on sale for $725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount?

In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.

Try it.

Daria bought a bracelet at original cost $16 to sell in her handicraft store. She marked the price up 45%.

Solution

ⓐ $7.20 ⓑ $23.20

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Regina bought a handmade quilt at original cost $120 to sell in her quilt store. She marked the price up 55%.

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Tom paid $0.60 a pound for tomatoes to sell at his produce store. He added a 33% mark-up.

Solution

ⓐ $0.20 ⓑ $0.80

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Flora paid her supplier $0.74 a stem for roses to sell at her flower shop. She added an 85% mark-up.

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Alan bought a used bicycle for $115. After re-conditioning it, he added 225% mark-up and then advertised it for sale.

Solution

ⓐ $258.75 ⓑ $373.75

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Michael bought a classic car for $8,500. He restored it, then added 150% mark-up before advertising it for sale.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Convert 4.5% to a decimal.
   If you missed this problem, review .

   Bonisa impendulo

   \(0.045\)

2. Convert 0.6 to a percent.
   If you missed this problem, review .

   Bonisa impendulo

   \(60\%\)

3. Round 0.875 to the nearest hundredth.
   If you missed this problem, review .

   Bonisa impendulo

   \(0.88\)

4. Multiply (4.5)(2.38).
   If you missed this problem, review .

   Bonisa impendulo

   \(10.71\)

5. Solve \(3.5=0.7n.\)
   If you missed this problem, review .

   Bonisa impendulo

   \(n=5\)

6. Subtract \(50-37.45.\)
   If you missed this problem, review .

   Bonisa impendulo

   \(12.55\)

7. Translate and solve: What number is 35% of 90?

   Bonisa impendulo

   Translate into algebra. Let \(n\)= the number.
   Remember "of" means multiply, "is" means equals.
   Multiply.
   	\(31.5\) is \(35\%\) of \(90\)

8. Translate and solve:

   What number is 45% of 80?

   Bonisa impendulo

   36

9. Translate and solve:

   What number is 55% of 60?

   Bonisa impendulo

   33

10. Translate and solve: 6.5% of what number is $1.17?

    Bonisa impendulo

    Translate. Let \(n=\) the number.
    Multiply.
    Divide both sides by 0.065 and simplify.
    	\(6.5\%\) of\(\$18\) is \(\$1.17\)

11. Translate and solve:

    7.5% of what number is $1.95?

    Bonisa impendulo

    $26

12. Translate and solve:

    8.5% of what number is $3.06?

    Bonisa impendulo

    $36

13. Translate and solve: 144 is what percent of 96?

    Bonisa impendulo

    Translate into algebra. Let \(p=\) the percent.
    Multiply.
    Divide by 96 and simplify.
    Convert to percent.
    	\(144\) is \(150\%\) of \(96\)

    Note that we are asked to find percent, so we must have our final result in percent form.

14. Translate and solve:

    110 is what percent of 88?

    Bonisa impendulo

    125%

15. Translate and solve:

    126 is what percent of 72?

    Bonisa impendulo

    175%

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

    Bonisa impendulo

    Step 1. Read the problem.
    Step 2. Identify what we are looking for.		the amount of tip should Dezohn leave
    Step 3. Name what we are looking for.
    Choose a variable to represent it.		Let t = amount of tip.
    Step 4. Translate into an equation.
    Write a sentence that gives the information to find it.
    Translate the sentence into an equation.
    Step 5. Solve the equation. Multiply.
    Step 6. Check. Does this make sense?
    Yes, 20% of $70 is $14.
    Step 7. Answer the question with a complete sentence.		Dezohn should leave a tip of $12.33.

    Notice that we used t to represent the unknown tip.

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

    Bonisa impendulo

    $14.67

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

    Bonisa impendulo

    $2.16

19. 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?

    Bonisa impendulo

    Step 1. Read the problem.
    Step 2. Identify what we are looking for.		the total amount of potassium that is recommended
    Step 3. Name what we are looking for.
    Choose a variable to represent it.		Let \(a=\) total amount of potassium.
    Step 4. Translate. Write a sentence that gives the information to find it.
    Translate into an equation.
    Step 5. Solve the equation.
    Step 6. Check. Does this make sense?
    Yes, 2% is a small percent and 85 is a small part of 4,250.
    Step 7. Answer the question with a complete sentence.		The amount of potassium that is recommended is 4,250 mg.

20. One serving of wheat square cereal has seven grams of fiber, which is 28% of the recommended daily amount. What is the total recommended daily amount of fiber?

    Bonisa impendulo

    25 grams

21. One serving of rice cereal has 190 mg of sodium, which is 8% of the recommended daily amount. What is the total recommended daily amount of sodium?

    Bonisa impendulo

    2,375 mg

22. 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?

    Bonisa impendulo

    Step 1. Read the problem.
    Step 2. Identify what we are looking for.		the percent of the total calories from fat
    Step 3. Name what we are looking for.
    Choose a variable to represent it.		Let \(p=\) percent of fat.
    Step 4. Translate. Write a sentence that gives the information to find it.
    Translate into an equation.
    Step 5. Solve the equation.
    Divide by 480.
    Put in a percent form.
    Step 6. Check. Does this make sense?
    Yes, 240 is half of 480, so 50% makes sense.
    Step 7. Answer the question with a complete sentence.		Of the total calories in each brownie, 50% is fat.

23. Solve. Round to the nearest whole percent.

    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?

    Bonisa impendulo

    26%

24. Solve. Round to the nearest whole percent.

    The mix Ricardo plans to use to make brownies says that each brownie will be 190 calories, and 76 calories are from fat. What percent of the total calories are from fat?

    Bonisa impendulo

    40%

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

    Bonisa impendulo

    Step 1. Read the problem.
    Step 2. Identify what we are looking for.		the percent increase
    Step 3. Name what we are looking for.
    Choose a variable to represent it.		Let \(p=\) the percent.
    Step 4. Translate. Write a sentence that gives the information to find it.
    First find the amount of increase.		new amount − original amount = increase
    		\(36-26=10\)
    Find the percent.		Increase is what percent of the original amount?
    Translate into an equation.
    Step 5. Solve the equation.
    Divide by 26.
    Change to percent form; round to the nearest tenth.
    Step 6. Check. Does this make sense?
    Yes, 38.4% is close to \(\frac{1}{3}\), and 10 is close to \(\frac{1}{3}\) of 26.
    Step 7. Answer the question with a complete sentence.		The new fees represent a 38.5% increase over the old fees.

    Notice that we rounded the division to the nearest thousandth in order to round the percent to the nearest tenth.

26. Find the percent increase. (Round to the nearest tenth of a percent.)

    In 2011, the IRS increased the deductible mileage cost to 55.5 cents from 51 cents.

    Bonisa impendulo

    8.8%

27. Find the percent increase.

    In 1995, the standard bus fare in Chicago was $1.50. In 2008, the standard bus fare was $2.25.

    Bonisa impendulo

    50%

28. 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.

    Bonisa impendulo

    Step 1. Read the problem.
    Step 2. Identify what we are looking for.		the percent decrease
    Step 3. Name what we are looking for.
    Choose a variable to represent that quantity.		Let \(p=\) the percent decrease.
    Step 4. Translate. Write a sentence that gives the information to find it.
    First find the amount of decrease.		\(3.71-3.64=0.07\)
    Find the percent.		Decrease is what percent of the original amaount?
    Translate into an equation.
    Step 5. Solve the equation.
    Divide by 3.71.
    Change to percent form; round to the nearest tenth.
    Step 6. Check. Does this make sense?
    Yes, if the original price was $4, a 2% decrease would be 8 cents.
    Step 7. Answer the question with a complete sentence.		The price of gas decreased 1.9%.

29. Find the percent decrease. (Round to the nearest tenth of a percent.)

    The population of North Dakota was about 672,000 in 2010. The population is projected to be about 630,000 in 2020.

    Bonisa impendulo

    6.3%

30. Find the percent decrease.

    Last year, Sheila’s salary was $42,000. Because of furlough days, this year, her salary was $37,800.

    Bonisa impendulo

    10%

31. Nathaly deposited $12,500 in her bank account where it will earn 4% interest. How much interest will Nathaly earn in 5 years?

    \[\begin{array}{lll}I & = & ? \\ P & = & \$12,500 \\ r & = & 4\text{\%} \\ t & = & 5\ \text{years}\end{array}\]
    Bonisa impendulo

    Step 1. Read the problem.
    Step 2. Identify what we are looking for.	the amount of interest earned
    Step 3. Name what we are looking for. Choose a variable to represent that quantity.	Let \(I=\) the amount of interest.
    Step 4. Translate into an equation. Write the formula. Substitute in the given information.	\(\begin{array}{l} \\ I=Prt \\ I=(12,500)(.04)(5)\end{array}\)
    Step 5. Solve the equation.	\(I=2,500\)
    Step 6. Check: Does this make sense? Is $2,500 a reasonable interest on $12,500? Yes.
    Step 7. Answer the question with a complete sentence.	The interest is $2,500.

32. Areli invested a principal of $950 in her bank account with interest rate 3%. How much interest did she earn in 5 years?

    Bonisa impendulo

    $142.50

33. Susana invested a principal of $36,000 in her bank account with interest rate 6.5%. How much interest did she earn in 3 years?

    Bonisa impendulo

    $7,020

34. Loren loaned his brother $3,000 to help him buy a car. In 4 years his brother paid him back the $3,000 plus $660 in interest. What was the rate of interest?

    \[\begin{array}{lll}I & = & \$660 \\ P & = & \$3,000 \\ r & = & ? \\ t & = & 4\ \text{years}\end{array}\]
    Bonisa impendulo

    Step 1. Read the problem.
    Step 2. Identify what we are looking for.	the rate of interest
    Step 3. Name what we are looking for. Choose a variable to represent that quantity.	Let \(r=\) the rate of interest.
    Step 4. Translate into an equation. Write the formula. Substitute in the given information.	\(\begin{array}{l} \\ I=Prt \\ 660=(3,000)r(4)\end{array}\)
    Step 5. Solve the equation. Divide. Change to percent form.	\(\begin{array}{l}660=(12,000)r \\ 0.055=r \\ 5.5\%=r\end{array}\)
    Step 6. Check: Does this make sense? \(\begin{array}{l} \\ I=Prt \\ 660\overset{?}{=}(3,000)(0.055)(4) \\ 660=660✓\end{array}\)
    Step 7. Answer the question with a complete sentence.	The rate of interest was 5.5%.

35. Jim loaned his sister $5,000 to help her buy a house. In 3 years, she paid him the $5,000, plus $900 interest. What was the rate of interest?

    Bonisa impendulo

    6%

36. Hang borrowed $7,500 from her parents to pay her tuition. In 5 years, she paid them $1,500 interest in addition to the $7,500 she borrowed. What was the rate of interest?

    Bonisa impendulo

    4%

37. Eduardo noticed that his new car loan papers stated that with a 7.5% interest rate, he would pay $6,596.25 in interest over 5 years. How much did he borrow to pay for his car?

    Bonisa impendulo

    Step 1. Read the problem.
    Step 2. Identify what we are looking for.	the amount borrowed (the principal)
    Step 3. Name what we are looking for. Choose a variable to represent that quantity.	Let \(P=\) principal borrowed.
    Step 4. Translate into an equation. Write the formula. Substitute in the given information.	\(\begin{array}{lll} \\ I & = & Prt \\ 6,596.25 & = & P(0.075)(5)\end{array}\)
    Step 5. Solve the equation. Divide.	\(\begin{array}{lll} \\ \\ 6,596.25 & = & 0.375P \\ 17,590 & = & P\end{array}\)
    Step 6. Check: Does this make sense? \(\begin{array}{lll}\ I & = & Prt \\ 6,596.25 & \overset{?}{=} & (17,590)(0.075)(5) \\ 6,596.25 & = & 6,596.25✓\end{array}\)
    Step 7. Answer the question with a complete sentence.	The principal was $17,590.

38. Sean’s new car loan statement said he would pay $4,866.25 in interest from an interest rate of 8.5% over 5 years. How much did he borrow to buy his new car?

    Bonisa impendulo

    $11,450

39. In 5 years, Gloria’s bank account earned $2,400 interest at 5%. How much had she deposited in the account?

    Bonisa impendulo

    $9,600

40. Elise bought a dress that was discounted 35% off of the original price of $140. What was ⓐ the amount of discount and ⓑ the sale price of the dress?

    Bonisa impendulo

    ⓐ \(\begin{array}{lll}\text{Original price} & = & \text{\$}140 \\ \text{Discount rate} & = & 35\text{\%} \\ \text{Discount} & = & ?\end{array}\)
    Step 1. Read the problem.
    Step 2. Identify what we are looking for.	the amount of discount
    Step 3. Name what we are looking for. Choose a variable to represent that quantity.	Let \(d=\) the amount of discount.
    Step 4. Translate into an equation. Write a sentence that gives the information to find it. Translate into an equation.	The discount is 35% of $140. \(d=0.35(140)\)
    Step 5. Solve the equation.	\(d=49\)
    Step 6. Check: Does this make sense? Is a $49 discount reasonable for a $140 dress? Yes.
    Step 7. Write a complete sentence to answer the question.	The amount of discount was $49.

    ⓑ
    Read the problem again.
    
    

    Step 1. Identify what we are looking for.		the sale price of the dress
    Step 2. Name what we are looking for.
    Choose a variable to represent that quantity.		Let \(s=\) the sale price.
    Step 3. Translate into an equation.
    Write a sentence that gives the information to find it.
    Translate into an equation.
    Step 4. Solve the equation.
    Step 5. Check. Does this make sense?
    Is the sale price less than the original price?
    Yes, $91 is less than $140.
    Step 6. Answer the question with a complete sentence.		The sale price of the dress was $91.

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

i

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Solve Percent Applications

1. Translate and solve basic percent equations
2. Solve percent applications
3. Find percent increase and percent decrease
4. Solve simple interest applications
5. Solve applications with discount or mark-up
6. Find the amount of increase.
7. Find the percent increase.
8. Find the amount of decrease.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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