Section PF.7 – Simple Interest
Discussing interest starts with the principal, or amount your account starts with initially. 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.
| APR – Annual Percentage Rate |
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Interest rates are usually given as an annual percentage rate (APR) – the total interest that will be paid in the year. |
For most long term, simple interest loans, it is common for interest to be paid on an annual basis. In that case, interest would be earned each year on the principal.
For example, suppose you borrowed $10,000.00 from a friend and agree to repay it with 3% annual interest, in 5 years. You would not only repay your friend the $10,000.00 you borrowed. You would also pay simple interest for each year you had borrowed the money.
|
Year |
Starting balance |
Interest earned |
Ending Balance |
|
1 |
10,000.00 |
300.00 |
10,300.00 |
|
2 |
10,300.00 |
300.00 |
10,600.00 |
|
3 |
10,600.00 |
300.00 |
10,900.00 |
|
4 |
10,900.00 |
300.00 |
11,200.00 |
|
5 |
11,200.00 |
300.00 |
11,500.00 |
The total amount you would repay your friend would be $11,500.00, which is the original principal plus the interest over 5 years.
This process can be generalized with the following formulas:
| Simple Interest over Time |
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I = P0rt A = P0 + I = P0 + P0rt = P0(1 + rt) I is the dollar amount of 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, along with any outstanding interest due.
| Example 1 |
| Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $5,000.00 bond that pays 4.5% simple interest annually, and matures in 5 years. Find the future value of the bond after 5 years. |
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A = 5,000(1 + 0.045(5)) A = 5,000(1 + 0.225) A = 5,000(1.225) A = $6,125 When the bond matures after 5 years, you would have received the $5,000.00 you originally paid, plus $1,125.00 in interest, for a total of $6,125.00. |
| You Try PF.7.A |
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Maria invests $16,000.00 at 3% simple interest for 19 years.
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| Example 2 |
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Adam invests $4,000.00 at 4% simple interest for 9 months. How much is in the account at the end of 9 months? A = 4,000(1 + 0.04([latex]\frac{9}{12}[/latex])) Note: The time must be entered as a fraction of a year A = 4,000(1 + 0.03) A = 4,000(1.03) A = $4,120 At the end of 10 months Adam will have $4,120.00 in his account. To use the simple interest formula, the value of t must be time in years. So we had to write 9 months as a fraction of a year. |
| Example 3 | ||||||||
| Martha decides to invest $4500.00 into a savings account earning 2.5% simple interest. How long will it take her investment to double in value? | ||||||||
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Since the account is earning 2.5% interest, r = 0.025 In this problem, we are not specifically given the future balance in the account, A, but we are told that we want to find the amount of time it will take for the initial investment to double in value. Since the initial investment amount was $4,500, we are looking for the value of t where A = 2($4,500) = $9,000
It will take 40 years for Martha’s initial investment of $4,500.00 to double in value. |
| Example 4 | ||||||||
| Rocio decides to invest $1,000.00 into a savings account that earns simple interest. What interest rate does she need for her investment to double in 10 years? | ||||||||
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Since we don’t know the interest rate we will leave it as r. In this problem, we are not specifically given the future balance in the account, A, but we are told that we want to find the amount of time it will take for the initial investment to double in value. Since the initial investment amount was $1,000, we are looking for the value of t where A = 2($1,000) = $2,000
Rocio needs a 10% simple interest rate for her initial investment of $1,000.00 to double in value in 10 years. |
A fee based loan is one type of loan that you may encounter. Many payday/title loan businesses and pawn shops charge you a set fee for borrowing money from them. This fee can make it difficult for you to know what interest rate you are really paying. A loan with simple interest works just like an investment except you owe the future value.
| Example 5 | ||||||||
| Lucille takes her car to a title loan business to borrow some money. She is given $1,500.00. She must pay back the $1,500.00 in addition to a $300.00 fee in 6 months. What simple interest rate is she being charged? | ||||||||
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Since we don’t know the interest rate we will leave it as r. We do know that t = [latex]\frac{1}{2}[/latex] since 6 months is half a year. The initial amount invested is $1,500, so P0 = 1,500 In this problem, we are not specifically given the future balance in the account, A, but we are told that Lucille must pay back $1,500 + $300. So A = $1,800.
Lucille is paying a 40% simple interest rate for her loan. |
Section PF.7 – Answers to You Try Problems
PF.7.A
At the end of the 19 year period, she will have earned $9,120.00 in interest. The balance in her account will be $25,120.00.
