# 3.2: Solve Percent Applications

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##### Learning Objectives

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

• 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
##### Be prepared

Before you get started, take this readiness quiz.

1. Convert 4.5% to a decimal.
If you missed this problem, review [link].
2. Convert 0.6 to a percent.
If you missed this problem, review [link].
3. Round 0.875 to the nearest hundredth.
If you missed this problem, review [link].
4. Multiply (4.5)(2.38).
If you missed this problem, review [link].
5. Solve 3.5=0.7n.
If you missed this problem, review [link].
6. Subtract 50−37.45.
If you missed this problem, review [link].

## 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.

##### Exercise $$\PageIndex{1}$$

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

 Translate into algebra. Let n= the number. Remember "of" means multiply, "is" means equals. Multiply. 31.5 is 35% of 90
##### Exercise $$\PageIndex{2}$$

Translate and solve:

What number is 45% of 80?

36

##### Exercise $$\PageIndex{3}$$

Translate and solve:

What number is 55% of 60?

33

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

##### Exercise $$\PageIndex{23}$$

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.

6.3%

##### Exercise $$\PageIndex{24}$$

Find the percent decrease.

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

10%

## 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.

##### 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

$\begin{array}{lllll} {} &{} &{I} &{=} &{\text { interest }}\\ {I = Prt} &{\text{where}} &{P} &{=} &{\text { principle }}\\ {} &{} &{r} &{=} &{\text { rate }}\\ {} &{} &{t} &{=} &{\text { time }} \end{array}$

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.

$7020 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. ##### Exercise $$\PageIndex{28}$$ 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?

$\begin{array}{lll} {I} &{=} &{\ 660} \\ {P} &{=} &{\ 3000} \\ {r} &{=} &{?} \\ {t} &{=} &{4 \text { years } }\end{array}$

$$\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the rate of interest}} \\\\ {\textbf{Step 3. Name} \text{ what we are looking for. Choose}} &{\text{Let r = the rate of interest.}} \\ {\text{ a variable to represent that quantity}} &{\text{}} \\\\ {\textbf{Step 4.} \text{ Translate into an equation.}} &{} \\ {\qquad\text{Write the formula.}} &{I = Prt} \\ {\qquad\text{Substitute in the given information.}} &{660 = (3000)r(4)} \\ \\ {\textbf{Step 5. Solve} \text{ the equation.}} &{} \\ {} &{660 = (12000)r} \\ {\text{Divide.}} &{0.055 = r} \\ {\text{Change to percent form.}} &{5.5\% = r} \\\\ {\textbf{Step 6. Check} \text{: Does this make sense?}} &{} \\\\ {I = Prt} &{} \\ {660 \stackrel{?}{=} (3000)(0.055)(4)} &{} \\ {660 = 660\checkmark} &{} \\ {\textbf{Step 7. Answer} \text{ the question with a}} &{\text{The rate of interest was }5.5%} \\ {\text{complete sentence.}} &{} \end{array}$$

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.

4%

$9,600 ## 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. ##### DISCOUNT $\begin{array}{l}{\text { amount of discount }=\text { discount rate } \times \text { original price }} \\ {\text { sale price }=\text { original price - amount of discount }}\end{array}$ Keep in mind that the sale price should always be less than the original price. ##### Exercise $$\PageIndex{34}$$ 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?

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

2.

 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. ##### Exercise $$\PageIndex{35}$$ Find ⓐ the amount of discount and ⓑ the sale price: Sergio bought a belt that was discounted 40% from an original price of$29.

ⓐ $11.60 ⓑ$17.40

##### Exercise $$\PageIndex{36}$$

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

2. 33%
##### Exercise $$\PageIndex{39}$$

Find

1. the amount of discount and
2. 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.

1. $60 2. 15% 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. ##### MARK-UP $\begin{array}{l}{\text { amount of mark-up }=\text { mark-up rate } \times \text { original cost }} \\ {\text { list price }=\text { original cost }+\text { amount of mark up }}\end{array}$ Keep in mind that the list price should always be more than the original cost. ##### Exercise $$\PageIndex{40}$$ Adam’s art gallery bought a photograph at original cost$250. Adam marked the price up 40%. Find the

1. amount of mark-up and
2. the list price of the photograph.

1.

 Step 1. Read the problem. Step 2. Identify what we are looking for. the amount of mark-up Step 3. Name what we are looking for. Choose a variable to represent it. Let m= the amount of markup. Step 4. Translate into an equation. 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, 40% is less than one-half and 100 is less than half of 250. Step 7. Answer the question with a complete sentence. The mark-up on the photograph was $100. 2.  Step 1. Read the problem again. Step 2. Identify what we are looking for. the list price Step 3. Name what we are looking for. Choose a variable to represent it. Let p= the list price. Step 4. Translate into an equation. 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? Is the list price more than the net price? Is$350 more than $250? Yes. Step 7. Answer the question with a complete sentence. The list price of the photograph was$350.
##### Exercise $$\PageIndex{41}$$

Find

1. the amount of mark-up and
2. the list price.

Jim’s music store bought a guitar at original cost $1,200. Jim marked the price up 50%. Answer 1.$600
2. $1,800 ##### Exercise $$\PageIndex{42}$$ Find 1. the amount of mark-up and 2. the list price. The Auto Resale Store bought Pablo’s Toyota for$8,500. They marked the price up 35%.

1. $2,975 2.$11,475

## Key Concepts

• Percent Increase To find the percent increase:
1. Find the amount of increase. increase=new amount−originalamountincrease=new amount−originalamount
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. decrease=original amount−newamountdecrease=original amount−newamount
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{aligned} I &=P r t \\ \text { where } I &=\text { interest } \\ P &=\text { principal } \\ r &=\text { rate } \\ t &=\text { time } \end{aligned}

• Discount
• amount of discount is discount rate ·· original price
• sale price is original price – discount
• Mark-up
• amount of mark-up is mark-up rate ·· original cost
• list price is original cost + mark up

## Glossary

amount of discount
The amount of discount is the amount resulting when a discount rate is multiplied by the original price of an item.
discount rate
The discount rate is the percent used to determine the amount of a discount, common in retail settings.
interest
Interest is the money that a bank pays its customers for keeping their money in the bank.
list price
The list price is the price a retailer sells an item for.
mark-up
A mark-up is a percentage of the original cost used to increase the price of an item.
original cost
The original cost in a retail setting, is the price that a retailer pays for an item.
principal
The principal is the original amount of money invested or borrowed for a period of time at a specific interest rate.
rate of interest
The rate of interest is a percent of the principal, usually expressed as a percent per year.
simple interest
Simple interest is the interest earned according to the formula I=Prt.

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