Money that is invested typically earns compound interest, meaning that interest continues to be earned on the money accumulated so far (not just on the original investment). This is just like the exponential function considered in Exponential functions, f(x) = a(1+r)^x, where a represents the initial investment (called the principal), r represents the interest rate per compounding period, and x represents the number of compounding periods.

The interest rate in a compounding interest problem is typically expressed as an annual interest rate, so to find the interest rate per compounding period simply divide by the number of compounding periods per year. For example, if interest is compounded monthly divide the annual interest rate by 12, while if interest is compounded quarterly divide the annual interest rate by 4.

To find the number of compounding periods, simply multiply the number of years by the number of compounding periods per year. For example, if interest is compounded monthly multiply the number of years by 12, while if interest is compounded quarterly multiply the number of years by 4.

Example 5

• Suppose you invest $2,000 in an account that pays 3% annual interest compounded monthly. How much will be in the account after 4 years?
• Since the initial investment is 2,000, a = 2,000.
• Since 3% annual interest is compounded monthly, the monthly interest rate is r = (3/12)% = 0.25% = 0.0025.
• To find the value of the account in 4 years, set x = 4(12) = 48 months.
• After 4 years, the account value is 2,000(1.0025)⁴⁸ = $2,254.66.

Example 6

• Suppose you’re saving to have $5,000 in 10 years time in an account that pays 6% annual interest compounded quarterly. How much will you need to invest initially?
• Since the initial investment is 2,000, a = 2,000.
• Since 6% annual interest is compounded quarterly, the quarterly interest rate is r = (6/4)% = 1.5% = 0.015.
• To find the value of the account in 10 years, set x = 10(4) = 40 quarters.
• After 10 years, the account value is a(1.015)⁴⁰ = $5,000.
• Solving for a gives: a = 5,000/1.015⁴⁰ = $2,756.31.
• Check: 2,756.31(1.015)⁴⁰ = $5,000.00.