How Do You Solve For The Rate In The Compound Interest Formula?

Problems that ask you to solve for the rate r in the compound interest formula require the use of roots or creative use of exponents. Let’s look at an example.

Problem 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?

Solution 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%.

Although most calculators have a square root key, when removing powers it is often useful to raise both sides to a power. For instance, we could remove the square by raising both sides to the ½ power:

When you raise a power to another power, you multiply the exponents 2 ∙ ½ = 1. The right side simply becomes 1 + r. Now we can solve for r:

Using the power key on your calculator gives the same answer as before. Make sure the 1/2 power is entirely in the power. You can make sure this happens using parentheses: (7000/5000)^(1/2)-1.

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 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.

If we use a 1/6 power to solve for r, we would carry out the steps below:

Using a 1/6 power on your calculator gives the same answer as above.