How Do You Solve for the Number of Years in the Compound Interest Formula?

In another MathFAQ, I looked at how you can find the rate in the compound interest formula. Now let’s look at an example where we solve for the number of years n. This problem is different because what we are looking for appears in a power.

Problem Suppose $5000 is deposited in an account that earns 2% compound interest that is done annually. In how many years will there be $6000 in the account.

Solution This problem requires the use of 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
  • that earns 2% compound interest that is done annually  >  r = 0.02
  • Will there be $6000 in the account  >  A = 6000

Putting these values into the formula above gives us

compound_02_01

Unlike other problems where we solve for P or r, here we need to solve for the power in the right hand side, n. Solving for a value in the power requires the property of logarithms, log(yx) = x logy. It allows us to move the n in the power and change it to a multiplier. But before we can apply this property, we isolate the factor containing the n:

compound_02_02

Now take the logarithm of both sides of the equation:

compound_02_03

This gives us

compound_02_04

or n ≈ 9.21 years.

In WolframAlpha, we could evaluate the logs as follows.

wolfram_log

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 dollars is deposited in an account that earns compound interest that is done annually. If there is 7000 dollars 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 dollars is deposited in an account > P = 5000
  • If there is 7000 dollars in the account after 2 years > A = 7000 and n = 2

Putting these values into the formula above gives us

 

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

 

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

 

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.

 

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

 

The root can be computed on a graphing calculator using the MATH button or put into WolframAlpha:

 

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.

How Do You Find Compound Interest Future Value In Google Sheets?

Spreadsheets have several built in functions for working with compound interest and annuities. To use these functions, we’ll start with a standard sheet.

sheets_financial_01

This worksheet contains the variables used throughout Chapter 8. These variables correspond to these letter used in the text.

  • Number of periods is n
  • Annual interest rate is r
  • Payment is R
  • Present value is P
  • Future value is A
  • Periods per year is m

Values given in a problem will be entered in column B. Values calculated by the spreadsheet will be entered in column C. We will also assume that amounts paid out are negative and amounts received are positive.

Continue reading “How Do You Find Compound Interest Future Value In Google Sheets?”

How Do You Solve For Time in the Compound Interest Formula?

Suppose 5000 dollars is deposited in an account that earns 2% compound interest that is done annually. In how many years will there be 6000 dollars in the account.

This problem requires the use of 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 dollars is deposited in an account  >  P = 5000
  • that earns 2% compound interest that is done annually  >  r = 0.02
  • Will there be 6000 dollars in the account  >  A = 6000

Putting these values into the formula above gives us

compound_02_01

Unlike other problems where we solve for P or r, here we need to solve for the power in the right hand side, n. Solving for a value in the power requires the property of logarithms, log(yx) = x logy. It allows us to move the n in the power and change it to a multiplier. But before we can apply this property, we isolate the factor containing the n:

compound_02_02

Now take the logarithm of both sides of the equation:

compound_02_03

This gives us

compound_02_04

or n ≈ 9.21 years.

In WolframAlpha, we could evaluate the logs as follows.

wolfram_log

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.