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# Compound Interest by Calculation

Compound interest by calculation method involves calculating the interest on the principal outstanding, as well as the interest accumulated at any given point in time during the period under consideration.

Simple interest is calculated only on the principal, and hence grows linearly to the principal. On the other hand, compound interest is calculated not only on the principal, but also on the interest accumulated. Hence compound interest grows exponentially to the principal amount i.e. interest grows rapidly as the time period increases.

There are two ways to calculate compound interest

1. By the repeated process of simple interest calculation (and adding the interest to the principal)
2. Using a compound interest formula

We’ll look at two examples and use both the above methods to calculate the compound interest for both examples. In this section, we’ll look at obtaining compound interest by calculating simple interest repeatedly for each time period (year, say), adding it to the principal, calculating simple interest again, and so on.

## Example 1: Compound Interest by Calculation

Jessica invested \$9000 for 3 years at 13% pa, interest compounded annually. Find the total interest earned by Jessica at the end of 3 years.

Here the Principal P (= \$9000) increases by 13% every year.

Amount after first year

Interest I  =  P x R x N  =  \$9000 x 0.13 x 1 (n = 1 year)

Interest  I =  \$ 1170

Amount A  =  Principal  + Interest  =  \$9000 + \$1170  =  \$10170.

You can also calculate the amount as \$9000 + \$9000 x 0.13

=  \$9000 (1 + 0.13)

=  \$9000 x 1.13  =  \$10170.

Amount after two years

Interest I  =  P x R x N  =  \$10170 x 0.13 x 1 (n = 1 year again)

Interest  I =  \$ 1322.10

Amount A  =  Principal  + Interest  =  \$10170 + \$1322.10  =  \$11492.10

You can also calculate the amount as \$10170 + \$10170 x 0.13

=  \$10170 x 1.13  =  \$11492.10.

Amount after three years

Interest I  =  P x R x N  =  \$11492.10 x 0.13 x 1 (n = 1 year again)

Interest  I =  \$ 1493.97

Amount A  =  Principal  + Interest  =  \$11492.10 + \$1493.97  =  \$12986.07

You can also calculate the amount as \$11492.10 + \$11492.10 x 0.13

=  \$11492.10 x 1.13  =  \$12986.07.

So Interest earned in 3 years  =  Amount accumulated at the end of 3 years  – Principal

=  \$12986.07  –  \$9000  =  \$ 3986.07

Alternatively, you can add the interest earned for each of the three years to get the total interest earned in 3 years. This will be –

= \$1170 + \$ 1322.10 + \$1493.97  =  \$ 3986.07

## Example 2: Compound Interest by Calculation

What is the interest earned for 2 months on \$12000 invested at 9% pa, compounded monthly.

Since N = 2 months, we need to convert the interest rate to monthly as well.

Here P = \$12000,  R = 9% pa  =    = 0.75% pm (per month)  = 0.0075

Amount after one month

Interest I  =  P x R x N  =  \$12000 x 0.0075 x 1 (n = 1 month)

Interest  I =  \$ 90

Amount A  =  Principal  + Interest  =  \$12000 + \$90  =  \$12090.

Alternatively Amount A = \$ 12000 x 1.0075  =  \$12090.

Amount after two months

Interest I  =  P x R x N  =  \$12090 x 0.0075 x 1 (n = 1 month)

Interest  I =  \$90.675  = \$ 90.68

Amount A  =  Principal  + Interest  =  \$12090 + \$90.68  =  \$12180.68.

Alternatively Amount A = \$ 12090 x 1.0075  =  \$12180.68.

So Interest earned in 2 months  =  Amount accumulated at the end of 2 months  – Principal

=  \$12180.68  –  \$12000  =  \$ 180.68

Alternatively, you can add the interest earned for each of the two months to get the total interest earned in 2 months. This will be –

= \$90 + \$ 90.68  =  \$ 188.68

We will now look at the same two examples using the compound interest calculation using a formula.