How Do You Find the Annual Interest Rate From the Compound Interest Formula?

Suppose $5000 is deposited in an account that earns compound interest that is done annually. If there is $7000 in the account after 2 years, what is the annual interest rate?

The easiest way to approach this problem is to use the compound interest formula,

compound_01

This formula applies when interest is earned on an annual basis and the interest is earned once a year.

Let’s look at the quantities in the problem statement:

$5000 is deposited in an account > P = 5000

If there is $7000 in the account after 2 years > A = 7000 and n = 2

Putting these values into the formula above gives us

compound_02

We need to find the annual interest rate r. Since the r is hidden in the parentheses, we start by isolating the parentheses.

compound_03

To get at the r, we need to remove the square on the parentheses.

compound_04

Using a calculator to do the square root, we get r ≈ 0.183 or 18.3%.

Now what if the interest is earned over six years instead of two years? Instead of a square on the parentheses we now have a sixth power.

compound_05

To solve for r in this equation, we follow similar steps.

compound_06

The root can be computed on a graphing calculator using the MATH button, by raising to the 1/6 power (^(1/6)), or put into WolframAlpha:

wolframalpha_root

Either method gives r ≈ 0.577 or 5.77%. Notice that the annual interest is lower when it is earned over a longer period of time.