We come across compound interest in our day-to-day lives. To know about the calculation of compound interest and how does compound interest works is much essential. To help you with this we have given this article covering everything on What is meant by Compound Interest, Formula for Compound Interest, How to Calculate Compound Interest for various periods of intervals such as monthly, quarterly, half-yearly, yearly, etc. Go through the solved examples on calculating compound interest in our real-life examples and learn how to solve similar kinds of problems.
List of Compound Interest Topics
- Introduction to Compound Interest
- Compound Interest as Repeated Simple Interest
- Formulae for Compound Interest
- Comparison between Simple Interest and Compound Interest
- Worksheet on Compound Interest as Repeated Simple Interest
- Worksheet on Use of Formula for Compound Interest
What is Compound Interest?
Compound Interest is the interest calculated on both principal and interest for regular intervals. For regular intervals, interest obtained is summed with the principal to get the new principal. Usually, we calculate the compound interest for regular intervals like monthly, quarterly, half-yearly, yearly, etc. The majority of the Banks and Financial Organizations calculate their interest on basis of compound interest only.
Compound Interest Formula
The Formula for finding the Compound Interest is given by Compound Interest = Amount- Principal
The amount is given by formula A = (1+r/n)nt
Where A = Amount
P = Principal
r = rate of interest
n = number of times interest is compounded per year
t = time (in years)
Alternatively, we can write the formula as given below:
CI = A – P
and
\(CI=P\left ( 1+\frac{r}{n} \right )^{nt}-P\)
Also, See:
- What is Simple Interest
- To find Rate When Principal Interest and Time are given
- To find Time Principal Interest and Rate are given
Compound Interest Formula for Different Time Periods
Compound Interest can be found for different intervals using different formulas. The formulas are listed below for your reference
CI Formula – Half Yearly
In the case of calculating compound interest half-yearly we find interest for every 6 months and the amount is compounded twice a year.
Here the rate of interest r is divided by 2 and the time period is doubled.
The formula to calculate the amount when the principal is compounded half-yearly is given by
| CI = P(1+\(\frac{r/2}{100}\))^2t – P A= P(1+\(\frac{r/2}{100}\))^2t |
CI Formula – Quarterly
In the case of time period calculation on a quarterly basis, we find interest for every 3 months, and the amount is compounded 4 times a year. Compound Interest Formula when Principal is compounded quarterly is
| CI = P(1+\(\frac{r/4}{100}\))^4t – P A= P(1+\(\frac{r/4}{100}\))^4t |
Monthly Compound Interest Formula
The monthly Compound Interest Formula is the interest calculated for every month i.e n = 12. Monthly Compound Interest can be found using the formula CI = P (1 + r/12)12t – P
Daily Compound Interest Formula
When the amount compounds daily, it means that the amount compounds 365 times in a year. i.e., n = 365. The daily compound interest formula is expressed as:
CI = P (1 + r/365)365t – P
Compound Interest Derivation
In order to derive the formula for the compound interest, we use the simplest interest formula. We know the simple interest calculated for 1 year is the same as compound interest calculated for 1 year.
Let, Principal amount = \(P\), Time = \(n\) years, Rate = \(R\)
Simple Interest (SI) for the first year:
\(SI_1\) = \(\frac{P~×~R~×~T}{100}\)
Amount after first year = \(P~+~SI_1\)
= \(P ~+~ \frac{P~×~R~×~T}{100}\)
= \(P \left(1+ \frac{R}{100}\right)\) = \(P_2\)
Simple Interest (SI) for second year:
\(SI_2\) = \(\frac{P_2~×~R~×~T}{100}\)
Amount after second year = \(P_2~+~SI_2\)
= \(P_2 ~+~ \frac{P_2~×~R~×~T}{100}\)
= \(P_2\left(1~+~\frac{R}{100}\right)\)
= \(P\left(1~+~\frac{R}{100}\right) \left(1~+~\frac{R}{100}\right)\)
= \(P \left(1~+~\frac{R}{100}\right)^2\)
Similarly, if we proceed further to \(n\) years, we can deduce:
\(A\) = \(P\left(1~+~\frac{R}{100}\right)^n\)
\(CI\) = \(A~–~P\) = \(P \left[\left(1~+~ \frac{R}{100}\right)^n~ –~ 1\right]\)
Compound Interest Examples
Example 1.
Nazma lends $3000 to Hemanth at an interest rate of 5% per annum, compounded half-yearly for a period of 1 year. Can you help her find out how much amount she gets after a period of 1 year from Hemanth?
Solution:
Given Data Principal P = $3000
Rate of Interest r = 5%
Time Period = 1 Year
We know the formula to find Compound Interest Half-Yearly CI = P(1+\(\frac{r/2}{100}\))^2t – P
Substituting the known values we have
CI =3000(1+\(\frac{5/2}{100}\))^2.1 – 3000
CI = 3151.88-3000
CI = $151.88
Therefore, Compound Interest earned for the given data is $151.88
Example 2.
The price of a radio is Rs. 1500 and it depreciates by 4% per month. Find its value after 6 months?
Solution:
Given Data Price of Radio = Rs.1500
Rate of Depreciation = 4%
Time Interval = 6 months
We know the formula of Depreciation A = P(1 – R/100)n.
Thus, the price of the radio after 6 months = 1500(1 – 4/100)6
= 1500(1 – 0.04)3 = 1500(0.96)3 = Rs. 1327 (Approx.)
Example 3.
A town had 20,000 residents in 2005. Its population declines at a rate of 10% per annum. What will be its total population in 2010?
Solution:
Population in a Town decreases by 10% every year
Depreciation Formula is given by A = P(1 – R/100)n
Substituting the known data and calculating the population by the end of 5 years we have 20000(1 – 10/100)5
= 20000(1-0.1)5
= 20000(0.9)5
= 11809 Approximately
Therefore, the total population in 2010 is around 11809 residents.
FAQs on Compound Interest
1. What is the formula for calculating Compound Interest?
Compound Interest can be calculate using the formula CI = A – P Where A = P(1 + r/n)^{nt}P
2. What is meant by Compound Interest?
Compound Interest is the interest calculated on both principal and interest for regular intervals.
3. What is Monthly Compound Interest Formula?
Monthly Compound Interest is given by the Formula CI = P (1 + r/12)12t – P