Contemporary Mathematics

# 6.3Simple Interest

Contemporary Mathematics6.3 Simple Interest

Figure 6.4 Interest is how savings earns money. (credit: “Interest Rates” by Mike Mozart/Flickr, CC BY 2.0)

## Learning Objectives

After completing this section, you should be able to:

1. Compute simple interest.
2. Understand and compute future value.
3. Compute simple interest loans with partial payments.
4. Understand and compute present value.

There is truth in the phrase “You need to have money to make money.” In essence, if you have money to lend, you can lend it at a cost to a borrower and make money on that transaction.

When money is borrowed, the person borrowing the money (borrower) typically has to pay the person or entity that lent the money (the lender) more than the amount of money that was borrowed. This extra money is the interest that is to be paid. Interest is sometimes referred to as the cost to borrow, the cost of the loan, or the finance cost.

This idea also applies when someone deposits money in a bank account or some other form of investment. That person is essentially lending the money to the bank or company. The money earned by the depositor is also called interest. The interest is typically based on the amount borrowed, or the principal.

The pairing of borrower and lender can take various forms. The borrower may be a consumer using a credit card or taking out a loan from a bank, the lender. Companies also borrow from lending banks. Someone who invests in a company’s stock is the lender in this case; the company is essentially the borrower.

In this section, we examine the basic building block of interest paid on loans and borrowed credit and also the returns on investments like bank accounts, simple interest.

## Compute Simple Interest

Let’s get some terminology understood. Interest to be paid by a borrower is often expressed as an annual percentage rate, which is the percent of the principal that is paid as interest for each year the money is borrowed. This means that the more that is borrowed, the more that must be paid back. Sometimes, the interest to be paid back is simple interest, which means that the interest is calculated on the amount borrowed only.

The length of time until the loan must be paid off is the term of the loan. The date when the loan must be paid off is when the loan is due. The day that the loan is issued is the origination date. We’ll put this terminology to use in the following examples. Note that in this section we will use letters, called variables, to represent the different parts of the formulas we’ll be using. This will help keep our formulas and calculations manageable.

### Simple Interest Loans with Integer Year Terms

Calculating simple interest is similar to the percent calculations we made in Understanding Percent and Discounts, Markups, and Sales Tax, but must be multiplied by the term of the loan (in years, if dealing with an annual percentage rate).

## FORMULA

The simple interest, $II$, to be paid on a loan with annual interest rate $rr$ for a number of years (term of the loan) $tt$, with principal $PP$, is found using $I=P×r×tI=P×r×t$, where the decimal form of the interest rate, $rr$, is used. The total repaid, then is $T=P+IT=P+I$ or, more directly, $T=P+P×r×tT=P+P×r×t$. This total is often referred to as the loan payoff amount, or more simply just the payoff.

When the annual interest rate, the principal, and the number of years that the money is borrowed is known, the interest to be paid can be found and from there the total to be repaid can be calculated.

## Checkpoint

Be aware, interest paid to a lender is almost uniformly rounded up to the next cent.

## Example 6.27

### Simple Interest on Loans with Integer Year Terms

Calculate the simple interest to be paid on a loan with the given principal, annual percentage rate, and number of years. Then, calculate the loan payoff amount.

1. Principal $PP$ = $4,000, annual interest rate $rr$ = 5.5%, and number of years $tt$ = 4 2. Principal $PP$ =$14,800, annual interest rate $rr$ = 7.9%, and number of years $tt$ = 7

Calculate the simple interest to be paid on a loan with the given principal, annual percentage rate, and number of years. Then calculate the loan payoff, or total to be repaid.
1.
Principal $P$ = $6,700, annual interest rate $r$ = 11.99%, and number of years $t$ = 3 2. Principal $P$ =$25,800, annual interest rate $r$ = 6.9%, and number of years $t$ = 5

Riley runs an auto repair shop, and needs to purchase a new brake lathe, which costs $11,995. She takes out a two-year, simple interest loan at an annual interest rate of 14.9%. How much interest will she pay and how much total will she repay on the loan? ## Your Turn 6.28 1. Beth is the owner of a small retail store in downtown St. Louis. The windows in the storefront need replacing, so she needs to take out a$9,500 loan to get the repairs done. The rate she secures is 9.25% and the term of the loan is one year. How much interest will she pay and how much total will she repay on the loan?

### Simple Interest Loans with Other Lengths of Terms

In the previous example and Your Turn exercise, the loans were paid back in one payment after an integer number of years. However, there are also loans lasting a length of time not equal to an integer number of years (like 1, 2, or 3 years or more), but in a number of months (like 4 months, 18 months, and so on). What model would apply to these situations?

When the loan is paid back after a term that is not an integer number of years but is instead a number of months, the term of the loan, or time, $tt$, is expressed as a fraction of the year. So for a 2-month loan, the time, in years, is 2/12 = 1/6. For a 5-month loan, the time in years is 5/12. For an 18-month term, the term in years is 18/12 = 1.5.

Abeje needs a loan to purchase equipment for the gym she is going to open. She visits the bank and secures a 4-month loan of $20,000. Her annual percentage rate is 6.75%. How much interest will Abeje pay and what is her loan payoff amount? ## Your Turn 6.29 1. Samuel needs to borrow$8,400 to pay for repairs to his small manufacturing facility. He manages to get a simple interest loan at 17.33%, to be paid after 6 months. How much interest will Samuel pay and what is Samuel’s loan payoff amount?

Those examples dealt in months. However, some loans are for days only (45 days, 60 days, 120 days). In such cases, we find the daily interest rate. The fraction we will use for the daily interest rate is the interest rate (as a decimal) divided by 365. This may be referred to as Actual/365. In order to find the term of the loan, divide the number of days in the term of the loan by 365.

## FORMULA

To determine the interest, $II$, on a loan with term $tt$ expressed in days, with principal of $PP$, and interest rate in decimal form of $rr$, calculate $I=P×r365×tI=P×r365×t$. Here, $r365r365$ represents the daily interest rate.

Alternately, the above formula is equivalent to $I=P×r×t365I=P×r×t365$, where the interest rate remains an annual rate, but the time is expressed as a fraction of the year.

## Who Knew?

It seems reasonable to use 365 as the number of days in the year, since there are 365 days in most years. However, sometimes, banks have used (and continue to use) 360 as the number of days in a year. They may also treat all months as if they have 30 days. These differences lead to (sometimes small) differences in how much interest is paid. Since the number of days is in the denominator, a smaller denominator (360) will result in larger numbers (interest) that is 365 is used for the denominator. See this page from ACRE for a comparison.

## Understand and Compute Future Value

Money can be invested for a specific amount of time and earn simple interest while invested. The terminology and calculations are the same as we’ve already seen. However, instead of the total to be paid back, the investor is interested in the total value of the investment after the interest is added. This is called the future value of the investment.

## FORMULA

The future value, $FVFV$, of an investment that yields simple interest is $FV=P+I=P+P×r×tFV=P+I=P+P×r×t$, where $PP$ is the principal (amount invested at the start), $rr$ is the annual interest rate in decimal form, and $tt$ is the length of time the money is invested. The time $tt$ will be an integer if the term of the deposit is an integer number of years, will be number of months/12 if the term is in months, will be actual/365 if the deposit is for a number of days.

## Example 6.31

### Simple Interest on a Deposit

In the following, determine how much interest was earned on the investment and the future value of the investment, if the investment yields simple interest.

1. Principal is $1,000, annual interest rate is 2.01%, and time is 5 years 2. Principal is$5,000, annual interest rate is 1.85%, and time is 30 years
3. Principal is $10,000, annual interest rate is 1.25%, and time is 18 months 4. Principal is$7,000, annual interest rate is 3.26%, and time is 100 days

In the following, determine how much interest was earned on the investment and the future value of the investment if the investment yields simple interest.
1.
Principal is $4,500, annual interest rate is 1.88%, and time is 3 years 2. Principal is$2,000, annual interest rate is 2.03%, and time is 10 years
3.
Principal is $120,000, annual interest rate is 3.1%, and time is 100 days 4. Principal is$4,680, annual interest rate is 1.55%, and time is 42 months

You may have noticed that for these problems, the future value was rounded down. When the future value is paid, the amount is typically rounded down.

A certificate of deposit (CD) is a savings account that holds a single deposit (the principal) for a fixed term at a fixed interest rate. Once the term of the CD is over, the CD may be redeemed (cashed in or withdrawn) and the owner of the CD receives the original principal plus the interest earned. The deposit often cannot be withdrawn until the term is up; if it can be withdrawn early, there is often a penalty imposed to do so.

## WORK IT OUT

The reason CD (certificate of deposit) rates look so small is because they are extremely safe investments. Though overall interest rates for CDs change over time and individual returns vary with the terms of the CD, investors are offered predictable interest income for their investments.

To investigate this yourself, search online to determine the strengths and weaknesses of CDs (investopedia.com offers good, basic information on investing). Then, online, identify five national banks and two local banks who offer CDs.

• Track the interest rates for the CDs at various terms (1 year, 3 years, 5 years) for each of the banks you found that offer CDs.
2.

1.
Russell buys a new car, by taking a loan out from the dealership. After all of the discussions are over, he finances (gets a loan for) $23,660. The annual interest rate that Russell will pay is 4.76%. He plans to pay this off over 5 years. How much will Russell’s monthly payment be? ## Tech Check The calculation of payments is long, and involves many steps. However, most spreadsheet programs, including Google Sheets, have a payment function. In Google Sheets, that function is PMT. To find the payment for an installment loan (like for a car), you need to enter the interest rate per period, the number of payments, and the loan amount. From Your Turn 6.36, the rate was 0.0476/12, the number of payments was 60, and the loan amount was$23,660. In Google Sheets, select any cell and enter the following:

=PMT(0.0476/12,60,23660)

And click the enter key. Immediately, in the cell you selected, the payment of $443.90 appears, though with a negative sign. The negative sign indicates it is a payment out of an account. Since we want to know the payment amount, we ignore the negative sign. The result, with the formula in the formula bar, is shown in Figure 6.5. Figure 6.5 Google Sheets payment function In general, to use the PMT function in Google Sheets, enter =PMT($rr$/12,$tt$*12,$PP$) where $rr$ is the annual interest rate, $tt$ is the number of years, and $PP$ is the principal of the loan. ## Understand and Compute Present Value for Simple Interest Investments When finding the future value of an investment, we know how much is deposited, but we have no idea how much that money will be worth in the future. If we set a goal for the future, it would be useful to know how much to deposit now so an account reaches the goal. The amount that needs to be deposited now to hit a goal in the future is called the present value. ## FORMULA The present value, $PVPV$, of money deposited at an annual, simple interest rate of $rr$ (in decimal form) for time $tt$ (in years) with a specified future value of $FVFV$, is calculated with the formula $PV=FV(1+rt)PV=FV(1+rt)$. Note: Present value, in this calculation, is always rounded up. Otherwise, future value may fall short of the target future value. Understanding what this tells you is important. When you find the present value, that is how much you need to invest now to reach the goal $FVFV$, under the conditions (time and rate) at which the money will be invested. ## Example 6.37 Compute the present value of the investment described. Interpret the result. 1. $FVFV$ =$10,000, $tt$ = 15 years, annual simple interest rate of 5.5%
2. $FVFV$ = $150,000, $tt$ = 20 years, annual simple interest rate of 6.25% 3. $FVFV$ =$250,000, $tt$ = 486 months, annual simple interest rate of 4.75%

Compute the present value of the investment described. Interpret the result.
1.
$FV$ = $25,000, $t$ = 10 years, annual simple interest rate of 7.5% 2. $FV$ =$320,000, $t$ = 35 years, annual simple interest rate of 6.5%
3.

1.
Kentaro will invest some money in a CD that yields 2.75% simple interest when invested for 5 years. How much must Kentaro invest so that after those 5 years his CD is worth $12,000? ## Check Your Understanding 13. What is interest? 14. What is the principal in a loan? 15. Calculate the simple interest to be paid for a 6-year loan with principal$1,500.00 and annual interest rate of 12.99%
16.
A simple interest loan for $24,200 is taken out at 10.55% annual percentage rate. A partial payment of$13,000 is made 25 days into the loan period. After this payment, what will the remaining balance of the loan be?
17.
Find the monthly payment for a $9,800.00 loan at a 13.8% interest for 4 years. 18. Find the present value of an investment with future value$30,000 with a simple interest rate of 3.75% invested for 10 years.

## Section 6.3 Exercises

1 .
If $1,500.00 is invested in an account bearing 3.5% interest, what is the principal? 2 . If$1,500 is invested in an account bearing 3.5% interest, what is the interest rate?
3 .
What is simple interest?
4 .
What is the present value of an investment?
5 .
What is the future value of an investment?
6 .
What is a partial payment on a loan?
In the following exercises, calculate the simple interest and payoff for the loan with the given principal, simple interest rate, and time.
7 .
Principal $P$ = $5,000, annual interest rate $r$ = 6.5%, and number of years $t$ = 6 8 . Principal $P$ =$3,500, annual interest rate $r$ = 12%, and number of years $t$ = 7
9 .
Principal $P$ = $7,800, annual interest rate $r$ = 11.5%, and number of years $t$ = 10 10 . Principal $P$ =$62,500, annual interest rate $r$ = 4.88%, and number of years $t$ = 4
11 .
Principal $P$ = $4,600, annual interest rate $r$ = 9.9%, for 18 months 12 . Principal $P$ =$19,000, annual interest rate $r$ = 16.9%, for 14 months
13 .
Principal $P$ = $8,500, annual interest rate $r$ = 10.66%, for 6 months 14 . Principal $P$ =$17,600, annual interest rate $r$ = 17.9%, for 20 months
15 .
Principal $P$ = $4,000, annual interest rate $r$ = 8.5%, for 130 days 16 . Principal $P$ =$9,900, annual interest rate $r$ = 15.9%, for 90 days
17 .
Principal $P$ = $600, annual interest rate $r$ = 16.8%, for 25 days 18 . Principal $P$ =$890, annual interest rate $r$ = 9.75%, for 200 days
In the following exercises, find the future value of the investment with the given principal, simple interest rate, and time.
19 .
Principal is $5,300, annual interest rate is 2.07%, and time is 18 years. 20 . Principal is$14,700, annual interest rate is 3.11%, and time is 10 years.
21 .
Principal is $5,600, annual interest rate is 2.55%, for 30 months. 22 . Principal is$10,000, annual interest rate is 1.99%, for 15 months.
23 .
Principal is $2,000, annual interest rate is 3.22%, for 100 days. 24 . Principal is$900, annual interest rate is 3.75%, for 175 days.
In the following exercises, determine the amount applied to principal for the indicated partial payment on the loan with the given principal, interest rate, and time when the partial payment was made.
25 .
A simple interest loan for $2,700 is taken out at 11.6% annual percentage rate. A partial payment of$1,500 is made 28 days into the loan period.
26 .
A simple interest loan for $900 is taken out at 18.9% annual percentage rate. A partial payment of$400 is made 30 days into the loan period.
27 .
A simple interest loan for $13,500 is taken out at 14.8% annual percentage rate. A partial payment of$8,000 is made 75 days into the loan period.
28 .
A simple interest loan for $9,900 is taken out at 9.875% annual percentage rate. A partial payment of$4,000 is made 65 days into the loan period.
In the following exercises, determine the remaining principal for the indicated partial payment on the loan with the given principal, interest rate, and time when the partial payment was made.
29 .
A simple interest loan for $2,700 is taken out at 11.6% annual percentage rate. A partial payment of$1,500 is made 28 days into the loan period.
30 .
A simple interest loan for $900 is taken out at 18.9% annual percentage rate. A partial payment of$400 is made 30 days into the loan period
31 .
A simple interest loan for $13,500 is taken out at 14.8% annual percentage rate. A partial payment of$8,000 is made 75 days into the loan period.
32 .
A simple interest loan for $9,900 is taken out at 9.875% annual percentage rate. A partial payment of$4,000 is made 65 days into the loan period.
In the following exercises, find the payoff value of the loan with the given principal, annual simple interest rate, term, partial payment, and time at which the partial payment was made.
33 .
Principal = $1,500, rate = 6.99%, term is 5 years, partial payment of$900 made 2 years into the loan.
34 .
Principal = $21,500, rate = 7.44%, term is 10 years, partial payment of 15,000 made after 6 years. 35 . Principal =$6,800, rate = 11.9%, term is 200 days, partial payment of $4,000 made after 100 days. 36 . Principal =$800, rate = 13.99%, term is 150 days, partial payment of $525 made after 50 days. In the following exercises, find the monthly payment for a loan with the given principal, annual simple interest rate and number of years. 37 . Principal =$4,500, rate = 8.75%, years = 3
38 .
Principal = $2,700, rate = 15.9%, years = 5 39 . Principal =$13,980, rate = 10.5%, years = 4
40 .
Principal = $8,750, rate = 9.9%, years = 10 In the following exercises, find the present value for the given future value, $FV$, annual simple interest rate $r$, and number of years $t$. 41 . $FV$ =$25,000, $t$ = 15 years, annual simple interest rate of 6.5%
42 .
$FV$ = $12,000, $t$ = 10 years, annual simple interest rate of 4.5% 43 . $FV$ =$15,000, $t$ = 16 years, annual simple interest rate of 3.5%
44 .
$FV$ = $100,000, $t$ = 30 years, annual simple interest rate of 5.5% 45 . Rita takes out a simple interest loan for$4,000 for 5 years. Her interest rate is 7.88%. How much will Rita pay when the loan is due?
46 .
Humberto runs a private computer networking company, and needs a loan of $31,500 for new equipment. He shops around for the lowest interest rate he can find. He finds a rate of 8.9% interest for a 10-year term. How much will Humberto’s payoff be at the end of the 10 years? 47 . Jaye needs a short-term loan of$3,500. They find a 75-day loan that charges 14.9% interest. What is Jaye’s payoff?
48 .
Theethat’s car needs new struts, which cost $1,189.50 installed, but he doesn’t have the money to do so. He asks the repair shop if they offer any sort of financing. It offers him a short-term loan at 18.9% interest for 60 days. What is Theethat’s payoff for the struts? 49 . Michelle opens a gaming shop in her small town. She takes out an$8,500 loan to get started. The loan is at 9.5% interest and has a term of 5 years. Michelle decides to make a partial payment of $4,700 after 3 years. What will Michelle pay when the loan is due? 50 . A small retailer borrows$3,750 for a repair. The loan has a term of 100 days at 13.55% interest. If the retailer pays a partial payment of $2,000 after 30 days, what will the loan payoff be when the loan is due? 51 . Sharon invests$2,500 in a CD for her granddaughter. The CD has a term of 5 years and has a simple interest rate of 3.11%. After that 5-year period, how much will the CD be worth?
52 .
Jen and Fred have a baby, and deposit $1,500 in a savings account bearing 1.76% simple interest. How much will the account be worth in 18 years? 53 . Yasmin decides to buy a used car. Her credit union offers 7.9% interest for 5-year loans on used cars. The cost of the car, including taxes and fees, is$11,209.50. How much will Yasmin’s monthly payment be?
54 .
Cleo runs her own silk-screening company. She needs new silk-screening printing machines, and finds two that will cost her, in total, $5,489.00. She takes out a 3-year loan at 8.9% interest. What will her monthly payments be for the loan? 55 . Kylie wants to invest some money in an account that yields 4.66% simple interest. Her goal is to have$20,000 in 15 years. How much should Kylie invest to reach that goal?
56 .
Ishraq wants to deposit money in an account that yields 3.5% simple interest for 10 years, to help with a down payment for a home. Her goal is to have $25,000 for the down payment. How much does Ishraq need to deposit to reach that goal? In the following exercises, use the Cost of Financing. The difference between the total paid for a loan, along with all other charges paid to obtain the loan, and the original principal of the loan is the cost of financing. It measures how much more you paid for an item than the original price. In order to find the cost of financing, find the total paid over the life of the loan. Add to that any fees paid for the loan. Then subtract the principal. ${\text{Cost of Financing }} = {\text{ Total of payments }} + {\text{ fees }} - {\text{ principal}}.$ 57 . Yasmin decides to buy a used car. Her credit union offers 7.9% interest for 5-year loans on used cars. The cost of the car, including taxes and fees, is$11,209.50. How much did she pay the credit union over the 5 years? What was the cost of financing for Yasmin?
58 .
Cleo runs her own silk-screening company. She needs new silk-screening printing machines, and finds two that will cost her, in total, \$5,489.00. She takes out a 3-year loan at 8.9% interest. What was the cost of financing for Cleo?
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