6.1 Simple Interest

Robert Foth

6.1 Simple Interest

6.1: Simple Interest

Learning Objectives

Upon completion of this section, you should be able to

Solve application problems using simple interest.
Calculate interest amounts, total amounts, and determine unknown variables in simple interest scenarios.

Simple Interest

What is Interest?

Interest is the cost of using someone else’s money. When you borrow money, you pay interest as a fee for using that money. Conversely, when you save or invest money, you can earn interest as a reward for allowing others (like banks or businesses) to use your money. Interest is typically expressed as a percentage of the original amount (called the principal) over a specific period, usually a year.

Principal and Interest

Principal, often denoted with $P$ is the original amount of money involved in a financial transaction. This could be:

The amount of money borrowed in a loan
The initial amount deposited into a savings account
The face value of a bond or other financial instrument

Interest, often denoted as $I$ , is the cost of using the money over time. It can be viewed from two perspectives:

For borrowers: Interest is the additional amount paid on top of the principal when repaying a loan.
For lenders or investors: Interest is the amount earned on top of the principal for allowing others to use their money.

Interest is usually expressed as a percentage of the principal, known as the interest rate, over a specific time period (often annually).

There are different ways that interest can be calculated when borrowing or when investing. In the context of simple interest, the interest amount is calculated solely based on the original principal, regardless of the duration of the agreement or how often interest is paid out or charged.

Simple interest is found in various aspects of personal and business finance:

Short-term Loans: Many personal loans, auto loans, and some mortgages use simple interest calculations, especially when payments are made frequently (e.g., biweekly or monthly).
Credit Cards: Some credit card companies use simple interest to calculate charges on outstanding balances, particularly for cash advances.
Microloans and Peer-to-Peer Lending: Many online platforms offering small, short-term loans use simple interest to keep calculations transparent and easy to understand for borrowers.
Treasury Bills: These short-term government securities often use simple interest in their yield calculations.
Legal Judgments: Courts often award simple interest on monetary judgments.
Business Agreements: Some business contracts, particularly those involving short-term financing or accounts receivable, may specify simple interest terms.

Discussing interest starts with the principal, or amount your account starts with. This could be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: $$ 100 (0.05) = $ 5$ . The total amount you would repay would be $105, the original principal plus the interest.

Simple Interest over Time

$\begin{array}{l} I = P r t \\ F V = P (1 + r t) \end{array}$

$I$ is the interest
$F V$ is the future value: principal plus interest
$P$ is the principal (starting amount also called Present Value)
$r$ is the (annual) interest rate in decimal form
$t$ is time (in years)

The units of measurement (years, months, etc.) for the time should match the time period for the interest
rate.

In the above definition we are assuming the interest rate is given as an annual rate as well as the time period. There are situations where you may be given an interest rate for a shorter period of time (say monthly). In those cases you would need to adjust the units of time to math that of the interest rate for the formula to yield the correct results.

Example 1

A friend asks to borrow $500 and agrees to repay it in one year with an annual 3% interest rate. How much interest will you earn?

Solution

$\begin{array}{l} P = $ 500 & the principal \\ r = 0.03 & 3% rate \\ t = 1 & 1 year \\ I = $ 500 \cdot (0.03) \cdot 1 = $ 15 . & You will earn $ 15 interest . \end{array}$

Example 2

If you invest $10,000 at a simple interest rate of 4% per year, how much will you have after 15 months?

Solution

In this situation we are given an interest rate in years, but the time is in months. One of the first things we will need to do is convert that time into the same units as the given interest rate (years).

$\begin{array}{l} t = 15 months \\ t = 15 months \cdot (\frac{1 year}{12 months}) \\ t = 1.25 years \end{array}$

Now lets write out the given information:

$\begin{array}{l} P = $ 10, 000 \\ t = 1.25 years \\ r = 0.04 \end{array}$

Plug the given information into the simple interest formula to find the future value in 1.25 years.

$\begin{array}{l} F V = 10, 000 (1 + 0.04 \cdot 1.25) \\ F V = 10, 000 (1.05) \\ F V = $ 10, 500 \end{array}$

After 1.25 years (15 months) you would have $10,500. $500 of this amount is the interest that was earned from the investment.

Payday loan companies often obscure their true interest rates by presenting fees instead of conventional interest percentages. This clever tactic can make their loans seem more affordable at first glance. To illustrate how dramatically high the actual interest rates can be when converted to an annual percentage, let’s consider the next example.

Example 3

A pay day loan company offers a $250 short term loan that charges a fee of $15. The loan is due in two weeks. What is the equivalent annual simple interest rate for that loan? Treat the fee as the interest charged for two weeks.

Solution

To find the annual simple interest rate we will start with treating the fee of $15 as the interest earned (charged) for the two week time period. Before we use the interest formula we do have to convert the time into years.

$\begin{array}{l} t = 2 weeks \\ t = 2 weeks \cdot (\frac{1 year}{52 weeks}) \\ t = \frac{1}{26} years \\ t \approx 0.0384 years \end{array}$

Here is the known information:

$\begin{array}{l} P = $ 250 \\ t = 0.0384 years \\ I = $ 15 \end{array}$

Substitute the known information into the interest formula and solve for $r$ :

$\begin{array}{l} I = P \cdot r \cdot t \\ 15 = 250 \cdot r \cdot (0.0384) \\ 15 = 9.6 \cdot r \\ r = \frac{15}{9.6} \\ r = 1.5625 \end{array}$

Recall $r$ (the annual interest rate) is in decimal form, so we convert it to a percent to find that the annual simple interest rate that is being charged is $r = 156.25 %$

APR – Annual Percentage Rate

Interest rates are usually given as an annual percentage rate (APR) – the total
interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will
be divided up.

For example, a 6% APR paid monthly would be divided into twelve 0.5% payments.
A 4% annual rate paid quarterly would be divided into four 1% payments.

In the Try It Now problem below we will need to use a generalization of 30 days per month. It is common to use 30 days in a month for financial computations as it was a standard adopted to make it easier to do calculations by hand (Wikipedia: Day count convention). We will continue to use a convention of 365 days per year when needed.

Try it Now 1

A credit card charges 1.5% simple interest per month on cash advances. If you take a cash advance of $1,000 and pay it back after 45 days, how much interest will you owe?

What is the equivalent annual simple interest rate?

Hint 1 (click to Show/Hide)

Take a close look at how the interest rate was described. It states that it charges 1.5% simple interest per month (a monthly rate), so we need to ensure any formulas we use that we convert the time into months (instead of years as we have done before). In addition we will need to use the information from above about using 30 days for a month.

Answer (click to Show/Hide)

Start with converting our time into months as the given simple interest rate is 1.5% per month and we need the interest rate and the time units to match.

$\begin{array}{l} t = 45 days \\ t = 45 days \cdot (\frac{1 month}{30 days}) \\ t = \frac{3}{2} months \\ t = 1.5 months \end{array}$

Write out the given information that is known:

$\begin{array}{l} P = $ 1000 \\ t = 1.5 months \\ r = 0.015 \end{array}$

Plug the known values into the interest formula:

$\begin{array}{l} I = P \cdot r \cdot t \\ I = 1000 \cdot (0.015) \cdot (1.5) \\ I = $ 22.50 \end{array}$

After 45 days the interest charged would be $22.50.

It’s important to note that credit card companies might have additional fees for cash advances that we are not considering in our work here.

Now for the second part of the question we revisit the interest formula with the interest being known as $22.50, but we need to convert the time into years as we are looking for an annual interest rate that would have yielded that interest charge for a cash advance of 45 days.

$\begin{array}{l} t = 45 days \\ t = 45 days \cdot (\frac{1 year}{365 days}) \\ t \approx 0.1233 years \end{array}$

Write out the given information that is known:

$\begin{array}{l} t = 0.1233 years \\ P = $ 1000 \\ I = $ 22.50 \\ r = ? \end{array}$

Plug the give information into the interest formula:

$\begin{array}{l} I = P \cdot r \cdot t \\ 22.50 = 1000 \cdot r \cdot (0.1233) \\ 22.50 = 123.2 \cdot r \\ r = \frac{22.50}{123.2} \\ r \approx 0.1826 \end{array}$

This problem illustrates why cash advances can be expensive – in just 45 days, you’re paying $22.50 on a $1,000 advance, which is equivalent to an APR of about 18.26%!

More Application with Simple Interest

Bonds are essentially a loan made to the bond issuer (a company or government) by you, the bond holder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which time the issuer pays back the original bond value.

A bond is a financial instrument that represents a loan made by an investor (the bond holder) to a borrower (the bond issuer), typically a company or government entity. The key components of a bond that relate to simple interest are:

Face Value (Principal): This is the amount the investor pays upfront to purchase the bond, and it’s also the amount the bond issuer promises to repay the bond holder at the end of the bond’s term. It serves as the base amount for interest calculations. For example, a $1,000 bond means the investor pays $1,000 initially and will receive $1,000 back at maturity (ignoring the interest payments received).
Interest Rate (Coupon Rate): Bonds pay interest at a fixed rate, often annually. This rate is applied to the face value using simple interest calculations.
Term: Bonds have a predetermined lifespan, known as the term or maturity, after which the issuer repays the face value to the bond holder.
Periodic Interest Payments: Many bonds make regular interest payments (often annually or semi-annually) based on simple interest calculations.

Example 4

Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn?

Solution

Each year, you would earn 5% interest: $1000(0.05) = $50 in interest. So over the course of five years, you would earn a total of $250 in interest. When the bond matures, you would receive back the $1,000 you originally paid, leaving you with a total of $1,250.

Video Solution (2 mins 23 secs – CC)

Example 5

Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?

Solution

Start with writing out the given information:

$\begin{array}{l} P = $ 1000 \\ r = 0.04 \\ t = 4 years \\ I = ? \end{array}$

Plug the values into the interest formula:

$\begin{array}{l} I = P \cdot r \cdot t \\ I = 1000 \cdot (0.04) \cdot 4 \\ I = $ 160 \end{array}$

Alternatively we could have solved this by looking at the time period being in half-years (how often interest is paid out). When doing this we need to convert the interest rate to half-years (semi-annually) as well, so the 4% interest will be divided into two 2% payments. Translate the information of four years to 8 half-years and the interest rate to semi-annually to solve for the interest:

$\begin{array}{l} P = $ 1000 & the principal \\ r = 0.02 & 2 % rate per half - year \\ t = 8 & 4 years = 8 half - years \\ I = $ 1000 (0.02) (8) = $ 160. \end{array}$

Video Solution (2 mins 2 secs – CC)

Try it Now 2

On November 7, 2023, the TUSD community voted to support a $480 million bond for Tucson Unified School District. The 10-year bond program will renovate aging schools, upgrade security and safety systems and provide every student with state-of-the-art technology (TUSD Bond Home). If we assume the bond pays an annual 4.5% simple interest, with interest payments made semi-annually, then answer the following questions:

What is the semi-annual interest payment on this bond?
How much total interest will be paid over the life of the bond?

Hint 1 (click to Show/Hide)

When finding the interest payment on the bond we can find the annual payment of interest and then divide that into two equal parts to find the semi-annual interest payments.

Answer (click to Show/Hide)

To find the semi-annual interest payment we can first find the annual interest charges and then divide that by two for two equal interest payments:
$\begin{array}{l} I = P \cdot r \cdot t \\ I = 480, 000, 000 \cdot (0.045) \cdot (1) \\ I = $ 21, 600, 000 \end{array}$

The annual interest was found to be $21,600,000, so this leaves us with $\frac{$ 21, 600, 000}{2} = $ 10, 800, 000$ paid semi-annually.
To find the total interest paid over the life of the bond we can use the interest formula:
$\begin{array}{l} I = P \cdot r \cdot t \\ I = 480, 000, 000 \cdot (0.045) \cdot (10) \\ I = $ 216, 000, 000 \end{array}$

The interest on the bond after 10 years (if not called in early) would be $216,000,000.

Simple Interest Add On Loans

Another approach for smaller loans on items like appliances, furniture, or electronics is a loan where the interest is calculated for the initial principal for a set period of time. This interest is then added on to that principal as the amount to be paid over fixed payments during for the term of the loan (or in one lump sum at the end). This type of loan is sometimes seen for borrowers with very poor credit scores on cars and homes as well.

Add-On or Precomputed Simple Interest Loan

With an Add-On loan the future value $FV$ (or total amount owed) is first calculated at the beginning by finding the simple interest for a given annual rate $r$ , time $t$ , and
principal $P$ . After the interest is found it is added to the principal and the payments pymts is found
by dividing by the number of months in the term of the loan.

Find the Interest
$I = P \cdot r \cdot t$
Add the interest to the principal to find the future value
$F V = P + I$
or you can use this formula to skip the first step

$F V = P (1 + r \cdot t)$
Determine the payments by dividing the future value by the number of months $n$ in the term for the loan
$pymts = \frac{F V}{n}$

You can find $n$ by multiplying the number of years $t$ of the loan by 12 if needed.

Example 6

Samantha purchases new furniture for her first apartment. The furniture costs $1350 (including the tax and delivery). If the store is offering an annual rate of 8.95% to finance the purchase with a simple interest add-on loan with a term of 2 years, then what would her payments be?

Solution

We first calculate the Interest owed where the principal is $1350, rate of 8.95%, and time of 2 years

$\begin{array}{l} I = P \cdot r \cdot t \\ I = ($ 1350) \cdot (0.0895) \cdot (2) \\ I = $ 241.65 \end{array}$

Add that interest to the principal to find the future value or total amount owed

$\begin{array}{l} F V = P + I \\ F V = $ 1350 + $ 241.65 \\ F V = $ 1591.6 \end{array}$

Next divide that total amount owed by 24 since there are 24 months in 2 years.

$\begin{array}{l} pymts = \frac{F V}{n} \\ pymts = \frac{$ 1591.65}{24} \\ pymts = $ 66.32 \end{array}$

The monthly payments will be $66.32 for the purchase. Now the very last payments may not be exactly that total as the payment amount was rounded.

With an add-on loan the amount of interest paid and the amount paid on the principal will be the same every month while making the payments. This means on the very last payment you paid the exact same amount of interest as you did on the first payment. If the agreement allows you to pay off the loan early you would only need to determine the total number of principal payments left, but many of these agreements either will not let you pay the loan off early or will apply a fee for early payoff of the loan.

Try it Now 3

Sam bought a new TV for $690 and stand for $350 at a home electronics store. If the sales tax in the city is 5.7% and the store offers to finance the purchases with an add-on loan at an annual rate of 11.9% for 10 months, then how much is the total cost of the purchase going to be for Sam and what would the payments be?

Hint 1 (click to Show/Hide)

Start by finding the total amount of the purchase including sales tax. That would be the starting principal for the add-on loan.

Answer (click to Show/Hide)

The cost of the purchase with sales tax is the principal $P$ for the loan. Add the two charges and
the sales tax to find $P$ .

$\begin{array}{l} P = ($ 690 + $ 350) + ($ 690 + $ 350) \cdot (0.057) \\ P = $ 1099.28 \end{array}$

The interest on the purchase is found below with the time given in years

$\begin{array}{l} I = ($ 1099.28) \cdot (0.119) \cdot (\frac{10}{12}) \\ I = $ 109.01 \end{array}$

The future value or total cost is

$\begin{array}{l} F V = $ 1099.28 + $ 109.01 \\ F V = $ 1208.29 \end{array}$

Divide by 10 for the amount of the payment each month

$\begin{array}{l} pymts = \frac{$ 1208.29}{10} \\ pymts = $ 120.83 \end{array}$

The total cost of the purchase was found to be $1208.29 with monthly payments of $120.83 for 10 months.

Exercises

If you invest $10,000 at a simple interest rate of 4% per year, how much will you have after 3 years?
If you invest $5,000 at a simple interest rate of 3.5% per year, how much will you have after 18 months?
A loan of $2,500 accumulates $375 in simple interest over a period of 15 months. Determine the simple annual interest rate.
A friend lends you $200. After two weeks it is repaid with 5% interest added on to the amount borrowed.
1. How much will you have to repay?
2. What is the equivalent annual simple interest rate?
Suppose you obtain a $3,000 T-note with a 3% annual rate, paid quarterly, with maturity in 5 years. How much interest will you earn?
A T-bill is a type of bond that is sold at a discount over the face value. For example, suppose you buy a 13-week T-bill with a face value of $10,000 for $9,800. This means that in 13 weeks, the government will give you the face value, earning you $200. What annual simple interest rate have you earned?
You borrow $1000 from a friend with a promise to pay back $1300 in two years. What annual simple interest rate will you be paying?
A pay day loan company offers a $500 short term loan that charges a fee of $25. The loan is due in 2 weeks. What is the equivalent annual simple interest rate for that loan? Treat the fee as the interest charged for two weeks.
In California a payday lender can loan up to $300 and charge a maximum of $45 in fees (Source). If that payday loan is for two weeks what is the equivalent annual simple interest rate for the loan? Teach the fee as the interest charged for two weeks.
A bank is offering a CD that pays an annual simple interest rate of 4.5%. How much money would you need to invest in this CD now in order to have $2500 two years from now for a down payment on a used car?
A new furnace and AC unit was purchased for a home. The cost for the unit (including sales tax and installation) was $7,981. If the company is offering financing with a simple interest add-on loan with an annual rate of 6.9% and a term of two years, then what are the monthly payments? What was the total cost of the purchase if that financing is used?
A TV is purchased and financed through an electronics store. The cost of the TV is $899.99 and there is a local sales tax of 7.2%. The entire purchase is financed through the store with a simple interest add-on loan. If the annual rate is 10.9% and the term is 15 months, then what are the monthly payments?

Attributions

This page contains modified content from David Lippman, “Math In Society, 2nd Edition.” Licensed under CC BY-SA 4.0.
This page contains content by Robert Foth, Math Faculty, Pima Community College, 2021. Licensed under CC BY 4.0.

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Simple Interest

Solution

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More Application with Simple Interest

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Simple Interest Add On Loans

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Exercises

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