Compound Interest Formula: How to Calculate, Example with Solution

Compound Interest Formula: As students progress to higher grades in school, the curriculum starts introducing various concepts of practical usage to students such as profit and loss, probability, Interest. There are two types of interest, compound interest and simple interest. Compound interest is a financial concept that allows an initial sum of money to grow over time. The difference between simple interest and compound interest (CI) is that CI is the sum of the interest earned or paid on both the principle amount and the accumulated interest. Whether it's a savings account, investment, or loan, compound interest can significantly impact financial outcomes. Studying and understanding compound interest is important for students and adults both. This compounding effect can lead to exponential growth, making it a powerful tool for saving and investing.

Compound Interest Definition

Compound interest is the interest calculated based upon the principle amount as well as the accumulated interest over the previous period. 

What is compound Interest?

Compound interest is also commonly known as ‘interest over interest’  because it is levied not only on the principle amount but also upon the interest existing on it.

Hence, in layman’s terms, compound interest = interest on principle + interest on existing interest

Compound Interest Formula

To calculate compound interest, one has to calculate the compounded sum total over n number of years, at r rate on the principle amount P.

Formula of Compound Interest

Compound Interest = A – P

Where A is the total amount over a period of time and P is the last/ latest principle.

The formula given below is to calculate the complete amount or the new principle after n number of years, at r rate on the principle amount P.

Total Amount (A) = P (1 + r/n)nt 

Therefore, CI = P [1 + (r/100)]n - P

How Compound Interest Works

As mentioned above, Compound interest is paid both on principle and the interest accumulated. 

The interest earned at the end of each time period is added to the existing principle and the new amount is obtained.

This new amount becomes the principle for the next time period. Hence, the principle amount keeps increasing with each time period.

Compound Interest Formula Derivation

Taking,

Principle as “P”

Rate of interest as “R”. 

The simple interest on principle at the end of 1st time period = P*r/100. 

Total amount after 1st time period = P+P*r/100 = P(1+r/100). 

Total amount becomes the new principle.

After the end of second compounding period, the simple interest on new principle = P(1+r/100) x (r/100)

Total amount after 2nd time period = P(1+r/100)x(r/100) + P(1+r/100) + P(1+r/100)x(r/100) = P(1+r/100)2.

Thus, the total amount at the end of the nth compounding period = A = P(1+r/100)ⁿ

How to Calculate Compound Interest for Different Time Periods?

Compound interest is calculated annually(yearly), semi-annually (twice a year), quarterly (every four months), monthly, etc.

All you have to do is divide the rate percent by the time period and multiple the time period with the time period. Check below:

Half-yearly Compound Interest formula

Here, interest is calculated after every 6 months:

Therefore, Half-yearly Compound Interest formula = P [ 1+ (r/2)/ 100)2t - P

Quarterly Compound Interest formula

Here, interest is calculated after every 4 months:

Therefore, Quarterly Compound Interest formula = P [ 1+ (r/4)/ 100)4t - P

Monthly Compound Interest Formula

Here, interest is calculated each month:

Therefore, Monthly Compound Interest formula = P [ 1+ (r/12)/ 100)12t - P

Difference between Compound Interest and Simple Interest: CI vs SI

Read more about Compound Interest vs Simple Interest

Solved Examples 

Q1 Calculate the compound interest on Rs 8000 at 2% per annum in 2 years?

Solution: 

principle P = 8000

rate r = 2% 

time = 2 years 

Using formula A = P (1 + R/100)n

    = 8000 (1 + 2/100)2 = 8000 (102/100)2

    = 8323

Compound interest = A – P 

                               = 8323 – 8000

                               = Rs 323

Q2 The price of an anti-tarnish ring is Rs. 1400 and it is reducing by 8% per month. Find its price after 3 months.

Solution:

Using the formula A = P(1 – R/100)n.

Here, since the rate is reducing by 8 percent every month, we are subtracting in the formula rather than adding.

The price of the the ring after 3 months = 1400(1 – 8/100)3

= 1400(1 – 0.08)3 = 1400(0.92)3 = Rs. 1090

Q3 Calculate the total amount on Rs. 2000 at the rate of half-yearly compounded interest rate 4% per annum for 1.5 years. 

Solution:

Given,

p = 2000

r = 4%

t = 1.5 or 3 half years

A = P (1 + R/200)23

   =  2000 (1 + 4/200)3

   = 2000 (204/200)3

   = 2122

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