6.5: Solve Simple Interest Applications

Learning Objectives

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

• Use the simple interest formula
• Solve simple interest applications

Be Prepared 6.9

Before you get started, take this readiness quiz.

Solve $0.6y=45.$
If you missed this problem, review Example 5.43.

Be Prepared 6.10

Solve $\frac{n}{1.45}=4.6.$
If you missed this problem, review Example 5.44.

Use the Simple Interest Formula

Do you know that banks pay you to let them keep your money? The money you put in the bank is called the principal, $P,$ and the bank pays you interest, $I.$ The interest is computed as a certain percent of the principal; called the rate of interest, $r.$ The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable for time, $t,$ represents the number of years the money is left in the account.

Simple Interest

If an amount of money, $P,$ the principal, is invested for a period of $t$ years at an annual interest rate $r,$ the amount of interest, $I,$ earned is

$$I=$$

where

$$\begin{array}{cc}\hfill I& =\\ \text{interest}\hfill \\ \hfill P\\ =\\ \text{principal}\hfill \\ \hfill r\\ =\\ \text{rate}\hfill \\ \hfill t\\ =\end{array}$$

Interest earned according to this formula is called simple interest.

The formula we use to calculate simple interest is $I=Prt.$ To use the simple interest formula we substitute in the values for variables that are given, and then solve for the unknown variable. It may be helpful to organize the information by listing all four variables and filling in the given information.

Example 6.33

Find the simple interest earned after $3$ years on $\text{\$500}$ at an interest rate of $\text{6\%.}$

Answer

Organize the given information in a list.

$\begin{array}{ccc}\hfill I& =& ?\hfill \\ \hfill P& =& \text{\$500}\hfill \\ \hfill r& =& \text{6\%}\hfill \\ \hfill t& =& \text{3 years}\hfill \end{array}$

We will use the simple interest formula to find the interest.

Write the formula.	$I=$
Substitute the given information. Remember to write the percent in decimal form.	$I=(500)(0.06)(3)$
Simplify.	$I=$
Check your answer. Is $90 a reasonable interest earned on $500 in 3 years?
In 3 years the money earned 18%. If we rounded to 20%, the interest would have been 500(0.20) or $100. Yes, $90 is reasonable.
Write a complete sentence that answers the question.	The simple interest is $90.

Try It 6.65

Find the simple interest earned after $4$ years on $\text{\$800}$ at an interest rate of $\text{5\%.}$

Try It 6.66

Find the simple interest earned after $2$ years on $\text{\$700}$ at an interest rate of $\text{4\%.}$

In the next example, we will use the simple interest formula to find the principal.

Example 6.34

Find the principal invested if $\text{\$178}$ interest was earned in $2$ years at an interest rate of $\text{4\%.}$

Answer

Organize the given information in a list.

$\begin{array}{ccc}\hfill I& =& \text{\$178}\hfill \\ \hfill P& =& ?\hfill \\ \hfill r& =& \text{4\%}\hfill \\ \hfill t& =& \text{2 years}\hfill \end{array}$

We will use the simple interest formula to find the principal.

Write the formula.	$I=$
Substitute the given information.	$178=P(0.04)(2)$
Divide.	$\frac{178}{0.08}=\frac{0.08P}{0.08}$
Simplify.	$\mathrm{2,225}=P$
Check your answer. Is it reasonable that $2,225 would earn $178 in 2 years?
$I=$
$178\stackrel{?}{=}\mathrm{2,225}(0.04)(2)$
$178=178✓$
Write a complete sentence that answers the question.	The principal is $2,225.

Try It 6.67

Find the principal invested if $\text{\$495}$ interest was earned in $3$ years at an interest rate of $\text{6\%.}$

Try It 6.68

Find the principal invested if $\text{\$1,246}$ interest was earned in $5$ years at an interest rate of $\text{7\%}.$

Now we will solve for the rate of interest.

Example 6.35

Find the rate if a principal of $\text{\$8,200}$ earned $\text{\$3,772}$ interest in $4$ years.

Answer

Organize the given information.

$\begin{array}{ccc}\hfill I& =& \text{\$3,772}\hfill \\ \hfill P& =& \text{\$8,200}\hfill \\ \hfill r& =& ?\hfill \\ \hfill t& =& \text{4 years}\hfill \end{array}$

We will use the simple interest formula to find the rate.

Write the formula.	$I=$
Substitute the given information.	$\mathrm{3,772}=\mathrm{8,200}r(4)$
Multiply.	$\mathrm{3,772}=\mathrm{32,800}r$
Divide.	$\frac{\mathrm{3,772}}{\mathrm{32,800}}=\frac{\mathrm{32,800}r}{\mathrm{32,800}}$
Simplify.	$0.115=r$
Write as a percent.	$\text{11.5\%}=r$
Check your answer. Is 11.5% a reasonable rate if $3,772 was earned in 4 years?
$I=$
$\mathrm{3,772}\stackrel{?}{=}\mathrm{8,200}(0.115)(4)$
$\mathrm{3,772}=\mathrm{3,772}✓$
Write a complete sentence that answers the question.	The rate was 11.5%.

Try It 6.69

Find the rate if a principal of $\text{\$5,000}$ earned $\text{\$1,350}$ interest in $6$ years.

Try It 6.70

Find the rate if a principal of $\text{\$9,000}$ earned $\text{\$1,755}$ interest in $3$ years.

Solve Simple Interest Applications

Applications with simple interest usually involve either investing money or borrowing money. To solve these applications, we continue to use the same strategy for applications that we have used earlier in this chapter. The only difference is that in place of translating to get an equation, we can use the simple interest formula.

We will start by solving a simple interest application to find the interest.

Example 6.36

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

Answer

We are asked to find the Interest, $I.$

Organize the given information in a list.

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

Write the formula.	$I=$
Substitute the given information.	$I=(\mathrm{12,500})(0.04)(5)$
Simplify.	$I=\mathrm{2,500}$
Check your answer. Is $2,500 a reasonable interest on $12,500 over 5 years?
At 4% interest per year, in 5 years the interest would be 20% of the principal. Is 20% of $12,500 equal to $2,500? Yes.
Write a complete sentence that answers the question.	The interest is $2,500.

Try It 6.71

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

Try It 6.72

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

There may be times when you know the amount of interest earned on a given principal over a certain length of time, but you don't know the rate. For instance, this might happen when family members lend or borrow money among themselves instead of dealing with a bank. In the next example, we'll show how to solve for the rate.

Example 6.37

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

Answer

We are asked to find the rate of interest, $r.$

Organize the given information.

$\begin{array}{ccc}\hfill I& =& 660\hfill \\ \hfill P& =& \text{\$3,000}\hfill \\ \hfill r& =& ?\hfill \\ \hfill t& =& \text{4 years}\hfill \end{array}$

Write the formula.	$I=$
Substitute the given information.	$660=(\mathrm{3,000})r(4)$
Multiply.	$660=(\mathrm{12,000})r$
Divide.	$\frac{660}{\mathrm{12,000}}=\frac{(\mathrm{12,000})r}{\mathrm{12,000}}$
Simplify.	$0.055=r$
Change to percent form.	$\text{5.5\%}=r$
Check your answer. Is 5.5% a reasonable interest rate to pay your brother?
$I=$
$660\stackrel{?}{=}(\mathrm{3,000})(0.055)(4)$
$660=660✓$
Write a complete sentence that answers the question.	The rate of interest was 5.5%.

Try It 6.73

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

Try It 6.74

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

There may be times when you take a loan for a large purchase and the amount of the principal is not clear. This might happen, for instance, in making a car purchase when the dealer adds the cost of a warranty to the price of the car. In the next example, we will solve a simple interest application for the principal.

Example 6.38

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

Answer

We are asked to find the principal, $P.$

Organize the given information.

$\begin{array}{ccc}\hfill I& =& \mathrm{6,596.25}\hfill \\ \hfill P& =& ?\hfill \\ \hfill r& =& \text{7.5\%}\hfill \\ \hfill t& =& \text{5 years}\hfill \end{array}$

Write the formula.	$I=$
Substitute the given information.	$\mathrm{6,596.25}=P(0.075)(5)$
Multiply.	$\mathrm{6,596.25}=0.375P$
Divide.	$\frac{\mathrm{6,596.25}}{0.375}=\frac{0.375P}{0.375}$
Simplify.	$\mathrm{17,590}=P$
Check your answer. Is $17,590 a reasonable amount to borrow to buy a car?
$I=$
$\mathrm{6,596.25}\stackrel{?}{=}(\mathrm{17,590})(0.075)(5)$
$\mathrm{6,596.25}=\mathrm{6,596.25}✓$
Write a complete sentence that answers the question.	The amount borrowed was $17,590.

Try It 6.75

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

Try It 6.76

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

In the simple interest formula, the rate of interest is given as an annual rate, the rate for one year. So the units of time must be in years. If the time is given in months, we convert it to years.

Example 6.39

Caroline got $\text{\$900}$ as graduation gifts and invested it in a $\text{10-month}$ certificate of deposit that earned $\text{2.1\%}$ interest. How much interest did this investment earn?

Answer

We are asked to find the interest, $I.$

Organize the given information.

$\begin{array}{ccc}\hfill I& =& ?\hfill \\ \hfill P& =& \text{\$900}\hfill \\ \hfill r& =& \text{2.1\%}\hfill \\ \hfill t& =& \text{10 months}\hfill \end{array}$

Write the formula.	$I=$
Substitute the given information, converting 10 months to $\frac{10}{12}$ of a year.	$I=\mathrm{\$900}(0.021)\left(\frac{10}{12}\right)$
Multiply.	$I=15.75$
Check your answer. Is $15.75 a reasonable amount of interest?
If Caroline had invested the $900 for a full year at 2% interest, the amount of interest would have been $18. Yes, $15.75 is reasonable.
Write a complete sentence that answers the question.	The interest earned was $15.75.

Try It 6.77

Adriana invested $\text{\$4,500}$ for $8$ months in an account that paid $\text{1.9\%}$ interest. How much interest did she earn?

Try It 6.78

Milton invested $\text{\$2,460}$ for $20$ months in an account that paid $\text{3.5\%}$ interest How much interest did he earn?

Section 6.4 Exercises

Practice Makes Perfect

Use the Simple Interest Formula

In the following exercises, use the simple interest formula to fill in the missing information.

201.

Interest	Principal	Rate	Time (years)
	$\text{\$1200}$	$\text{3\%}$	$5$

Table 6.5

202.

Interest	Principal	Rate	Time (years)
	$\text{\$1500}$	$\text{2\%}$	$\text{4}$

Table 6.6

203.

Interest	Principal	Rate	Time (years)
$\mathrm{\$4410}$		$\text{4.5\%}$	$\text{7}$

Table 6.7

204.

Interest	Principal	Rate	Time (years)
$\mathrm{\$2112}$		$\text{3.2\%}$	$\text{6}$

Table 6.8

205.

Interest	Principal	Rate	Time (years)
$\mathrm{\$577.08}$	$\text{\$4580}$		$\text{2}$

Table 6.9

206.

Interest	Principal	Rate	Time (years)
$\mathrm{\$528.12}$	$\text{\$3260}$		$\text{3}$

Table 6.10

In the following exercises, solve the problem using the simple interest formula.

207.

Find the simple interest earned after $5$ years on $\text{\$600}$ at an interest rate of $\text{3\%.}$

208.

Find the simple interest earned after $4$ years on $\text{\$900}$ at an interest rate of $\text{6\%.}$

209.

Find the simple interest earned after $2$ years on $\text{\$8,950}$ at an interest rate of $\text{3.24\%}.$

210.

Find the simple interest earned after $3$ years on $\text{\$6,510}$ at an interest rate of $\text{2.85\%}.$

211.

Find the simple interest earned after $8$ years on $\text{\$15,500}$ at an interest rate of $\text{11.425\%}.$

212.

Find the simple interest earned after $6$ years on $\text{\$23,900}$ at an interest rate of $\text{12.175\%}.$

213.

Find the principal invested if $\text{\$656}$ interest was earned in $5$ years at an interest rate of $\text{4\%}.$

214.

Find the principal invested if $\text{\$177}$ interest was earned in $2$ years at an interest rate of $\text{3\%}.$

215.

Find the principal invested if $\text{\$70.95}$ interest was earned in $3$ years at an interest rate of $\text{2.75\%.}$

216.

Find the principal invested if $\text{\$636.84}$ interest was earned in $6$ years at an interest rate of $\text{4.35\%.}$

217.

Find the principal invested if $\text{\$15,222.57}$ interest was earned in $6$ years at an interest rate of $\text{10.28\%}.$

218.

Find the principal invested if $\text{\$10,953.70}$ interest was earned in $5$ years at an interest rate of $\text{11.04\%.}$

219.

Find the rate if a principal of $\text{\$5,400}$ earned $\text{\$432}$ interest in $2$ years.

220.

Find the rate if a principal of $\text{\$2,600}$ earned $\text{\$468}$ interest in $6$ years.

221.

Find the rate if a principal of $\text{\$11,000}$ earned $\text{\$1,815}$ interest in $3$ years.

222.

Find the rate if a principal of $\text{\$8,500}$ earned $\text{\$3,230}$ interest in $4$ years.

Solve Simple Interest Applications

In the following exercises, solve the problem using the simple interest formula.

223.

Casey deposited $\text{\$1,450}$ in a bank account with interest rate $\text{4\%.}$ How much interest was earned in $2$ years?

224.

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

225.

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

226.

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

227.

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

228.

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

229.

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

230.

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

231.

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

232.

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

233.

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

234.

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

235.

Caitlin invested $\text{\$8,200}$ in an $\text{18-month}$ certificate of deposit paying $\text{2.7\%}$ interest. How much interest did she earn form this investment?

236.

Diego invested $\text{\$6,100}$ in a $\text{9-month}$ certificate of deposit paying $\text{1.8\%}$ interest. How much interest did he earn form this investment?

237.

Airin borrowed $\text{\$3,900}$ from her parents for the down payment on a car and promised to pay them back in $15$ months at a $\text{4\%}$ rate of interest. How much interest did she owe her parents?

238.

Yuta borrowed $\text{\$840}$ from his brother to pay for his textbooks and promised to pay him back in $5$ months at a $\text{6\%}$ rate of interest. How much interest did Yuta owe his brother?

Everyday Math

239.

Interest on savings Find the interest rate your local bank pays on savings accounts.

1. ⓐ What is the interest rate?
2. ⓑ Calculate the amount of interest you would earn on a principal of $\text{\$8,000}$ for $5$ years.

240.

Interest on a loan Find the interest rate your local bank charges for a car loan.

1. ⓐ What is the interest rate?
2. ⓑ Calculate the amount of interest you would pay on a loan of $\text{\$8,000}$ for $5$ years.

Writing Exercises

241.

Why do banks pay interest on money deposited in savings accounts?

242.

Why do banks charge interest for lending money?

Self Check

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

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?