1. Learning objectives

Learning objectives

Translate and solve basic percent equations

Solve percent applications

Find percent increase and percent decrease

Solve simple interest applications

Solve applications with discount or mark-up

2. Translate and Solve Basic Percent Equations

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.

3. Example 3.12

Example 3.12

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

Solution

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 Example 3.12. We will again use direct translation to write the equation.

4. Example 3.13

Example 3.13

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

Solution

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

5. Example 3.14

Example 3.14

Translate and solve: 144 is what percent of 96?

Solution

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

6. Solve Applications of Percent

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.

Use a Problem-Solving Strategy to Solve an Application.

Step 1. Read the problem. Make sure all the words and ideas are understood.
Step 2. Identify what we are looking for.
Step 3. Name what we are looking for. Choose a variable to represent that quantity.
Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
Step 5. Solve the equation using good algebra techniques.
Step 6. Check the answer in the problem and make sure it makes sense.
Step 7. Answer the question with a complete sentence.

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.

7. Example 3.15

Example 3.15

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

Notice that we used t to represent the unknown tip.

8. Example 3.16

Example 3.16

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?

Solution

9. Example 3.17

Example 3.17

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

10. Find Percent Increase and Percent Decrease

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.

Find the Percent Increase.

Step 1. Find the amount of increase. $\mathrm{new amount}-\mathrm{original amount}=\mathrm{increase}$
Step 2. Find the percent increase. The increase is what percent of the original amount?

11. Example 3.18

Example 3.18

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

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.

Find the Percent Decrease.

Step 1. Find the amount of decrease. $\mathrm{original amount}-\mathrm{new amount}=\mathrm{decrease}$
Step 2. Find the percent decrease. Decrease is what percent of the original amount?

12. Example 3.19

Example 3.19

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

13. Solve Simple Interest Applications

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, $I=Prt.$

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}{lcclcclcccccccc} I=Prt & & & \mathrm{where} & & & \begin{array}{rll} I & = & \mathrm{interest} \\ P & = & \mathrm{principal} \\ r & = & \mathrm{rate} \\ t & = & \mathrm{time} \\ \end{array} \\ \end{array}

Interest earned according to this formula is called simple interest.

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.

14. Example 3.20

Example 3.20

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}{rll} I & = & ? \\ P & = & \textdollar 12,500 \\ r & = & 4\% \\ t & = & 5\,\mathrm{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.	$\begin{array}{c} \\ I=Prt \\ I=(12,500)(.04)(5) \\ \end{array}$
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.

Table

3.2

There may be times when we know the amount of interest earned on a given principal over a certain length of time, but we don’t know the rate. To find the rate, we use the simple interest formula, substitute in the given values for the principal and time, and then solve for the rate.

15. Example 3.21

Example 3.21

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}{rll} I & = & \textdollar 660 \\ P & = & \$3,000 \\ r & = & ? \\ t & = & 4\,\mathrm{years} \\ \end{array}

Solution


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.	$\begin{array}{c} \\ I=Prt \\ 660=(3,000)r(4) \\ \end{array}$
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? $\begin{array}{c} \\ I=Prt \\ 660\overset{?}{=}(3,000)(0.055)(4) \\ 660=660\mathrm{✓} \\ \end{array}$
Step 7. Answer the question with a complete sentence.	The rate of interest was 5.5%.

Table

3.3

Notice that in this example, Loren’s brother paid Loren interest, just like a bank would have paid interest if Loren invested his money there.

16. Example 3.22

Example 3.22

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?

Solution


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.	$\begin{array}{rll} I & = & Prt \\ 6,596.25 & = & P(0.075)(5) \\ \end{array}$
Step 5. Solve the equation. Divide.	$\begin{array}{rll} 6,596.25 & = & 0.375P \\ 17,590 & = & P \\ \end{array}$
Step 6. Check: Does this make sense? $\begin{array}{rll} \,I & = & Prt \\ \,6,596.25 & \overset{?}{=} & (17,590)(0.075)(5) \\ \,6,596.25 & = & 6,596.25\mathrm{✓} \\ \end{array}$
Step 7. Answer the question with a complete sentence.	The principal was $17,590.

Table

3.4

17. Solve Applications with Discount or Mark-up

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.

\begin{array}{l} \mathrm{amount of discount}=\mathrm{discount rate}\,\times\,\mathrm{original price} \\ \mathrm{sale price}=\mathrm{original price}-\mathrm{amount of discount} \\ \end{array}

Keep in mind that the sale price should always be less than the original price.

18. Example 3.23

Example 3.23

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


ⓐ $\begin{array}{rll} \mathrm{Original price} & = & \textdollar 140 \\ \mathrm{Discount rate} & = & 35\% \\ \mathrm{Discount} & = & ? \\ \end{array}$
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.

Table

3.5

ⓑ Read the problem again.

There may be times when we know the original price and the sale price, and we want to know the discount rate. To find the discount rate, first we will find the amount of discount and then use it to compute the rate as a percent of the original price. Example 3.24 will show this case.

19. Example 3.24

Example 3.24

Jeannette bought a swimsuit at a sale price of $13.95. The original price of the swimsuit was$ 31. Find the ⓐ amount of discount and ⓑ discount rate.

Solution


ⓐ $\begin{array}{rll} \mathrm{Original price} & = & \textdollar 31 \\ \mathrm{Discount} & = & ? \\ \mathrm{Sale Price} & = & \textdollar 13.95 \\ \end{array}$
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 the difference between the original price and the sale price. $d=31-13.95$
Step 5. Solve the equation.	$d=17.05$
Step 6. Check: Does this make sense? Is 17.05 less than 31? Yes.
Step 7. Answer the question with a complete sentence.	The amount of discount was $17.05.

Table

3.6

ⓑ Read the problem again.

Applications of mark-up are very common in retail settings. The price a retailer pays for an item is called the original cost. The retailer then adds a mark-up to the original cost to get the list price, the price he sells the item for. The mark-up is usually calculated as a percent of the original cost. To determine the amount of mark-up, multiply the mark-up rate by the original cost.

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

\begin{array}{l} \mathrm{amount of mark-up}=\mathrm{mark-up rate}\,\times\,\mathrm{original cost} \\ \mathrm{list price}=\mathrm{original cost}+\mathrm{amount of mark up} \\ \end{array}

Keep in mind that the list price should always be more than the original cost.

20. Example 3.25

Example 3.25

Adam’s art gallery bought a photograph at original cost $250. Adam marked the price up 40%. Find the ⓐ amount of mark-up and ⓑ the list price of the photograph.

Solution

ⓐ

ⓑ

21. Section 3.2 Exercises

Section 3.2 Exercises

Translate and Solve Basic Percent Equations

In the following exercises, translate and solve.

Solve Percent Applications

In the following exercises, solve.

Find Percent Increase and Percent Decrease

In the following exercises, solve.

Solve Simple Interest Applications

In the following exercises, solve.

Solve Applications with Discount or Mark-up

In the following exercises, find the sale price.

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

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

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

Everyday Math

Writing Exercises

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

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

22. Practice 1

Translate and solve:

7.5% of what number is $1.95?

Translate and solve:

8.5% of what number is $3.06?

Translate and solve:

110 is what percent of 88?

Translate and solve:

126 is what percent of 72?

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?

23. Practice 2

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?

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?

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?

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?

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?

24. Practice 4

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?

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?

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?

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?

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

25. Practice 5

Find ⓐ the amount of discount and ⓑ the sale price:

Sergio bought a belt that was discounted 40% from an original price of $29.

Find ⓐ the amount of discount and ⓑ the sale price:

Oscar bought a barbecue that was discounted 65% from an original price of $395.

Find ⓐ the amount of discount and ⓑ the discount rate.

Lena bought a kitchen table at the sale price of $375.20. The original price of the table was$ 560.

Find ⓐ the amount of discount and ⓑ the discount rate.

Nick bought a multi-room air conditioner at a sale price of $340. The original price of the air conditioner was$ 400.

Find ⓐ the amount of mark-up and ⓑ the list price.

Jim’s music store bought a guitar at original cost $1,200. Jim marked the price up 50%.

26. Practice 6

Find ⓐ the amount of mark-up and ⓑ the list price.

The Auto Resale Store bought Pablo’s Toyota for $8,500. They marked the price up 35%.

800% of 2250 is what number?

8.2% of what number is $2.87?

11.5% of what number is $108.10?

What percent of 260 is 78?

27. Practice 7

What percent of 1500 is 540?

30 is what percent of 20?

840 is what percent of 480?

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?

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?

28. Practice 8

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?

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?

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

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?

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

29. Practice 9

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?

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

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

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.

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.)

30. Practice 10

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

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.)

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

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

31. Practice 11

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?

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?

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

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?

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

32. Practice 12

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

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

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

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

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

33. Practice 3

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.

Find the percent increase.

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

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.

Find the percent decrease.

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

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

34. Practice 13

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

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?

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

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

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

35. Practice 14

Leaving a Tip 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?

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.

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