SECTION 3.1: Simple Interest
SECTION 3.1: 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
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 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 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 formulas below.
Simple Interest over Time
Simple Interest over Time
I = Prt
A = P + I = P + Prt = P(1 + rt)
I is the dollar amount of interest
A is the balance in the account after t years
P is the balance in the account at the beginning (starting amount, or principal).
r is the annual interest rate (APR) in decimal form (Example: 5% = 0.05)
t is the number of years we plan to leave the money in the account
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 bond that pays 4.5% simple interest annually, and matures in 5 years. Find the future value of the bond after 5 years.
Solution:
A = 5,000(1 0.045(5))
A = 5,000(1 0.225)
A = 5,000(1.225)
A = $6,125
Answer: When the bond matures after 5 years, you would have received the $5,000 you originally paid, plus $1,125 in interest, for a total of $6,125.
Example 2
Martha decides to invest $4500 into a savings account earning 2.5% simple interest. How long will it take her investment to double in value?
Solution: Since the account is earning 2.5% interest, r = 0.025. The initial amount invested is $4,500, so P = 4,500
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, which corresponds to the equation below.
9,000 = 4,500(1 + 0.025t)
2 = 1 + 0.025t
1 = 0.025t
t = 40
Answer: It will take 40 years for Martha’s initial investment of $4,500 to double in value.
Section 3.1 You Try Problems:
Maria invests $16,000 at 3% simple interest for 19 years.
A) How much interest will she earn?
B) How much is in the account at the end of the 19 year period?
3.1 – Answers to You Try Problems
a) At the end of the 19 year period, she will have earned $9,120 in interest.
b) The balance in her account will be $25,120.