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Compound Interest

Author: Sophia
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what's covered
In this lesson, you will solve problems involving compound interest. When you deposit money into a bank account, interest is added periodically and applied to the amount of money that is in the account at that point in time, which means that not only does your original deposit earn interest but the interest that you earn over time also earns interest! The idea of compound interest means that you are earning interest on interest. Specifically, this lesson will cover:

Table of Contents

1. Using the Compound Interest Formula to Find the Accumulated Amount

To understand the compound interest formula, we first need to understand the idea of compounding.

Suppose that you deposit $1000 dollars into an account that adds 2% in interest at the end of every year. Notice that this is essentially a percent increase problem!

Let A open parentheses t close parentheses equals the accumulated value in the account at the end of t years. Then, A open parentheses t close parentheses equals 1000 open parentheses 1.02 close parentheses to the power of t.

Banks usually compute and add interest to your account more often than yearly. For example, most loans are compounded monthly. In fact, these are the most commonly used interest compounding periods:

How Interest Is Compounded # Times Per Year
Annually 1
Semiannually 2
Quarterly 4
Monthly 12
Daily 360 or 365
(depends on the bank)

Now, here are the quantities that are used in computing compounding interest:

  • P = the principal (the amount invested at the beginning)
  • r = the annual interest rate in decimal form
  • t = the number of years that the money is in the account
  • n = the number of times per year that interest is compounded
To develop the compound interest formula, we use the following facts:

  • The starting value is P, the principal.
  • If the annual interest rate is r, then the interest rate per period is r over n. This is the percent increase from one period to the next.
  • If interest is compounded n times per year, then the total number of times that interest is compounded over t years is nt. This is the exponent.
This information leads to the compound interest formula.

formula to know
Compound Interest
The accumulated value in an account when P dollars is invested is A equals P open parentheses 1 plus r over n close parentheses to the power of n t end exponent comma where:
  • P = the principal (the amount invested at the beginning)
  • r = the annual interest rate in decimal form
  • t = the number of years that the money is in the account
  • n = the number of times per year that interest is compounded

EXAMPLE

Suppose you have an initial investment of $2000 and you decide to deposit it into an account that pays 0.6% annual interest compounded monthly. If the money is left in the account for 10 years, how much money is in the account at that time?

First, identify the quantities:

  • P equals 2000
  • r equals 0.006
  • n equals 12
  • t equals 10
Now, substitute into the formula and simplify (using your calculator).

A equals 2000 open parentheses 1 plus fraction numerator 0.006 over denominator 12 end fraction close parentheses to the power of 12 open parentheses 10 close parentheses end exponent Substitute P equals 2000 comma r equals 0.006 comma n equals 12 comma and t equals 10.
A equals 2000 open parentheses 1.0005 close parentheses to the power of 120 Simplify the base and exponent.
A equals 2123.64 Use a calculator to simplify. Since A is monetary, it should be rounded to the nearest cent.

Thus, after 10 years, there will be $2123.64 in the account.

Another important aspect of investments we can examine here is the interest earned.

formula to know
Interest Earned
Interest Earned = Accumulated Amount - Principal

In the previous example, the interest earned over the 10-year period is $ 2123.64 minus $ 2000 equals $ 123.64.

try it
An investment of $15,000 is deposited into a certificate of deposit that pays an annual interest rate of 2.1% compounded quarterly.
How much money is available in the account at the end of three years?
Use the compound interest formula A equals P open parentheses 1 plus r over n close parentheses to the power of n t end exponent with P equals 15000 comma r equals 0.021 comma n equals 4 comma and t equals 3.

We have A equals 15000 open parentheses 1 plus fraction numerator 0.021 over denominator 4 end fraction close parentheses to the power of 4 open parentheses 3 close parentheses end exponent equals $ 15 comma 972.77. (Use a calculator.)
How much interest is earned over the three-year period?
The interest earned is found by subtracting the principal from the accumulated amount.

Interest equals $ 15 comma 972.77 minus $ 15 comma 000 equals $ 972.77

2. Using the Compound Interest Formula to Find the Principal

The compound interest formula is also useful to determine the principal needed in order to reach an investment goal at the end of a given time period.

EXAMPLE

You want to invest money into an account that pays 1.2% annual interest compounded monthly so that you will have $10,000 available after 20 years. How much do you need to invest?

First, identify the quantities:

  • A equals 10000
  • r equals 0.012
  • n equals 12
  • t equals 20
Next, substitute the quantities into the formula and simplify where possible.

10000 equals P open parentheses 1 plus fraction numerator 0.012 over denominator 12 end fraction close parentheses to the power of 12 open parentheses 20 close parentheses end exponent Substitute A equals 10000 comma r equals 0.012 comma n equals 12 comma and t equals 20.
10000 equals P open parentheses 1 plus 0.001 close parentheses to the power of 240
10000 equals P open parentheses 1.001 close parentheses to the power of 240 		Simplify the base and exponent.
10000 over open parentheses 1.001 close parentheses to the power of 240 equals P Divide both sides by open parentheses 1.001 close parentheses to the power of 240.
P equals 7867.22 Evaluate by using a calculator. Round to the nearest cent.

Thus, $7,867.22 needs to be invested now in order to have $10,000 available in 20 years.

We could also compute the amount of interest earned over the 20-year period. This is $ 10 comma 000 minus $ 7 comma 867.22 equals $ 2 comma 132.78.

try it
You wish to have $20,000 available in an account that pays an annual interest rate of 1.05% compounded monthly.
To the nearest cent, how much needs to be invested now in order to reach this goal in 15 years?
Start with the compound interest formula A equals P open parentheses 1 plus r over n close parentheses to the power of n t end exponent with A equals 20000 comma r equals 0.0105 comma n equals 12 comma and t equals 15. Then, solve for P. 

20000 equals P open parentheses 1 plus fraction numerator 0.0105 over denominator 12 end fraction close parentheses to the power of 12 open parentheses 15 close parentheses end exponent Substitute all given quantities in the compound interest formula.
20000 equals P open parentheses 1 plus fraction numerator 0.0105 over denominator 12 end fraction close parentheses to the power of 180 Simplify the exponent.
20000 equals P open parentheses 1.170500147448 close parentheses Use a calculator to approximate the exponential quantity. For the most accurate results, use as many decimal places as you can when performing subsequent calculations.
P equals 17 comma 086.71 Divide both sides by 1.170500147448, then round P to the nearest penny.

Thus, a principal of $17,086.71 is required to meet the $20,000 goal in 15 years.
How much interest is earned over the 15-year period?
The interest earned is the difference between the accumulated amount and the principal, which is $ 20 comma 000 minus $ 17 comma 086.71 comma or $2,913.29.

3. Using the Compound Interest Formula to Find the Annual Interest Rate

There may be situations in which you know how much you can deposit and your investment goal at the end of some investment period, in which case, you want to know the interest rate required to reach such a goal.

EXAMPLE

You want to deposit $1000 into an account that pays interest compounded annually, and your goal is to have $1200 available after six years. What annual interest rate is required to reach this goal?

We use the compound interest formula as follows:

A equals P open parentheses 1 plus r over n close parentheses to the power of n t end exponent This is the compound interest formula.
1200 equals 1000 open parentheses 1 plus r over 1 close parentheses to the power of open parentheses 1 close parentheses open parentheses 6 close parentheses end exponent Substitute A equals 1200 comma P equals 1000 comma n equals 1 comma and t equals 6.
1200 equals 1000 open parentheses 1 plus r close parentheses to the power of 6 Simplify.
1.2 equals open parentheses 1 plus r close parentheses to the power of 6 Divide both sides by 1000.
root index 6 of 1.2 end root equals 1 plus r Take the 6th root of both sides.
Note the negative solution is not considered since an interest rate must be positive.
r equals root index 6 of 1.2 end root minus 1 Solve for r.
r almost equal to 0.03085332... Approximate r.

This means that an annual interest rate of approximately 3.085% will achieve the investment goal.

try it
A friend borrows $500 from you and pays you $700 three years later.
Find the annual interest rate of this loan, assuming interest is compounded annually. Give your answer in percent form, rounded to three decimal places.
Start with the compound interest formula A equals P open parentheses 1 plus r over n close parentheses to the power of n t end exponent with A equals 700 comma P equals 500 comma n equals 1 comma and t equals 3. Then, solve for r. 

700 equals 500 open parentheses 1 plus r over 1 close parentheses to the power of 1 open parentheses 3 close parentheses end exponent Substitute all given quantities into the compound interest formula.
700 equals 500 open parentheses 1 plus r close parentheses cubed Simplify.
1.4 equals open parentheses 1 plus r close parentheses cubed Divide both sides by 500.
cube root of 1.4 end root equals 1 plus r Apply the cube root to both sides.
r equals cube root of 1.4 end root minus 1 Subtract 1 from both sides.
r almost equal to 0.118688942... Approximate r.

To convert to a percent, multiply by 100 (or simply move the decimal point two spaces to the right) to get r almost equal to 11.8688942... percent sign.

Then, round to 3 decimal places. The interest rate is 11.869%.

EXAMPLE

To save for a child’s college education, $15,000 is deposited into an account that pays annual interest compounded monthly. What annual rate is required to meet an investment goal of $30,000 in 15 years?

We use the compound interest formula as follows:

A equals P open parentheses 1 plus r over n close parentheses to the power of n t end exponent This is the compound interest formula.
30000 equals 15000 open parentheses 1 plus r over 12 close parentheses to the power of open parentheses 12 close parentheses open parentheses 15 close parentheses end exponent Substitute A equals 30000 comma P equals 15000 comma n equals 12 comma and t equals 15.
30000 equals 15000 open parentheses 1 plus r over 12 close parentheses to the power of 180 Simplify.
2 equals open parentheses 1 plus r over 12 close parentheses to the power of 180 Divide both sides by 15000.
root index 180 of 2 equals 1 plus r over 12 Take the 180th root of both sides.
Since the 180 is even, there is a positive and a negative solution to this equation. The negative solution is not considered since an interest rate must be positive.
r over 12 equals root index 180 of 2 minus 1
r equals 12 open parentheses root index 180 of 2 minus 1 close parentheses 		Solve for r.
r almost equal to 0.046298899... Approximate r.

Thus, an annual interest rate of 4.630% is required to meet the investment goal. Given what you know about interest rates offered by banks today, is this a realistic goal?

watch
Check out this video to see how to determine the rate required to turn $6000 to $9500 in eight years when compounded semi-annually.

try it
Three years ago, you opened a certificate of deposit (CD) with a $2000 deposit. Today, you withdrew the funds totaling $2129.70.
If the CD paid annual interest compounded quarterly, what was the annual interest rate? Write your final answer in percent form rounded to the nearest tenth of a percent.
Start with the compound interest formula A equals P open parentheses 1 plus r over n close parentheses to the power of n t end exponent with A equals 2129.70 comma P equals 2000 comma n equals 4 comma and t equals 3. Then, solve for r. 

2129.70 equals 2000 open parentheses 1 plus r over 4 close parentheses to the power of 4 open parentheses 3 close parentheses end exponent Substitute all given quantities into the compound interest formula.
1.0646 equals open parentheses 1 plus r over 4 close parentheses to the power of 12 Divide both sides by 2000; simplify the exponent.
root index 12 of 1.0646 end root equals 1 plus r over 4 Apply the 12th root to both sides. Since we desire a positive interest rate, only the positive solution is considered.
r over 4 equals root index 12 of 1.0646 end root minus 1 Subtract 1 from both sides.
r equals 4 open parentheses root index 12 of 1.0646 end root minus 1 close parentheses Multiply both sides by 4.
r almost equal to 0.020920901... Approximate r.

To convert to a percent, multiply by 100 (or simply move the decimal point two spaces to the right) to get r almost equal to 2.0920901... percent sign.

Rounded to the nearest tenth of a percent, the annual interest rate is 2.1%. 

summary
In this lesson, you began by learning about the concept of compound interest. Banks pay compound interest on deposits, which means that interest is earned periodically, and interest is not only earned on the original investment, but also on any other interest that is earned along the way. The compound interest formula relates five quantities: The accumulated amount, the principal, the annual interest rate, the number of times that the interest is compounded per year, and the time that the money is in the account. You learned how to use the compound interest formula to find the accumulated amount for a given principal, and to find the principal for a given accumulated amount. After both quantities are known, you can determine the interest that was earned over the time that the money was in the account. Lastly, you learned how to use the compound interest formula to find the annual interest rate in order to meet a specific investment goal.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

Formulas to Know
Compound Interest

The accumulated value in an account when P dollars is invested is A equals P open parentheses 1 plus r over n close parentheses to the power of n t end exponent comma where:
• P = the principal (the amount invested at the beginning)
• r = the annual interest rate in decimal form
• t = the number of years that the money is in the account
• n = the number of times per year that interest is compounded

Interest Earned

Interest Earned = Accumulated Amount - Principal

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