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In the first article on interests, we learned about Simple Interest. Well, simple interest is not the only interest mechanism used by lenders and borrowers. There is another method, one which actually runs majority of business operations and goes by the name of Compound Interest.

The basic concept operating behind compound interest is very simple. Let us take a small example to understand its rudiments.

For example, Sham borrows a sum of Rs. 100 from Ghansham for a period of two years. The rate of interest for this loan is 10%. At the end of year one, the amount due is the principal and 10% interest on it, that is a total of Rs. 110. Now, effectively the principal value of the loan for the second year is no longer Rs. 100, it is in fact Rs. 110. That is what Ghansham would say and believe. According to him, for the 2nd year, he has lent Rs. 110 as that was the amount he would have had if he taken back the money at the end of year 1. Now for the 2nd year, the interest becomes Rs. 11 (10% of Rs. 110) and the total amount Ghansham would get would be Rs. 121.

If the same calculation was done using the logic of simple interest, you would see that the interest due for two years would be Rs. 20 (10% of Rs. 100 for two years). Thus, replace a S with a C and there is such a big difference in the calculations carried out.

Effectively, for compound interest, the 2nd term of interest is actually the sum total of the principal and the interest for the first term.

Compound Interest Tooltip 1: The Definitions

Principal (P): The original sum of money loaned/deposited. Also known as capital.
Interest (I): The amount of money that you pay to borrow money or the amount of money that you earn on a deposit.
Time (T): The duration for which the money is borrowed. The duration does not necessarily have to be years. The duration can be semi-annual, quarterly or any which way deemed fit.
Rate of Interest (R):  The percent of interest that you pay for money borrowed, or earn for money deposited

Compound Interest Tooltip 2: The Basic Formula

Amount Due at the end of the time period, A =  P (1+r/100)t

Where:
P: Principal (original amount)
R: Rate of Interest (in %)
T: Time period (yearly, half-yearly etc.)

Compound Interest (CI) =  A- P = P (1+r/100)t -P
= P {(1+r/100)t -1}

Compound Interest Tooltip 3: Basic Problems to explain the concept

Example 1: Maninder took a loan of Rs. 10000 from Prashant . If the rate of interest is 5% per annum compounded annually, find the amount received by Prashant by the end of three years

Solution:
The following is the data given:
Principal, P= 10000
Rate = 5%
Time =3 years

Using the formula for Compound Interest:
A = P(1+R/100)t
So A= 10000(1+5/100)3
A = 10000(1+1/20)3
A = 10000 x 21/20 x 21/20 x21/20 =11576.25
So the total amount paid by Maninder at the end of third year is Rs.11576.25

Example 2: Richa gave Rs. 8100 to Bharat at a rate of 9% for 2 years compounded annually. Find the amount of money which she gained as a compound interest from Bharat at the end of second year.

Solution:
Principal value = 8100
Rate = 9%
Time = 2 years
So the total amount paid by Bharat
= 8100(1+9/100)2
=Rs.  9623.61

The question does not probe the amount, rather, it wants to know the CI paid, that the difference between the total amount and original principal.
The Compound Interest = 9623.61 – 8100 = 1523.61

Compound Interest Tooltip 4: Multiple Compounding in a year

Amount Due at the end of the time period

Where:
A = future value
P = principal amount (initial investment)
r = annual nominal interest rate
n = number of times the interest is compounded per year
t = number of years money borrowed

Amount for Half Yearly Compounding, A = P {1+(R/2)/ 100}2T
(compound interest applied two times an year).

Like Half Yearly Compound Interest, we can calculate the amount for Quarterly Compounding:

A = P {1+(R/4)/ 100}4T

Example 3: Sona deposited Rs. 4000 in a bank for 2 years at 5% rate. Find the amount received at the end of year by her from the bank when compounded half yearly.

Solution:
Principal value = Rs. 4000
Rate = 5%
Time = 2 years
Since the interest is compounded half yearly so 2 years = 4 times in two years
So we have = A = P {1+(R/2)/ 100}2T
A= 4000{1+ (5/2)/100}4
A = 4000 x 41/40 x 41/40 x 41/40 x 41/40
A = Rs. 4415.2
So, Sona received Rs. 4415.2 from the bank after two years

Example 4: Manpreet lent Rs 5000 to Richa at 10% rate for 1 year. But she told her that she will take her money on compound interest. So find the amount of interest received by Manpreet when compounded quarterly?

Solution:
Principal value = Rs. 5000
Rate = 10%
Time = 1 year
Since the interest is compounded quarterly, that is 4 times in 1 year
Using the formula= A = P {1+(R/4)/ 100}4T
A= 5000{1+ (10/4)/100}4
A = 5000 x 41/40 x 41/40 x 41/40 x 41/40
A = Rs. 5519.064
So, Manpreet received Rs. 4415.2 from bank after two years
And the total amount of interest received by her is 5519.064 – 5000 = Rs. 519.06

Compound Interest Tooltip 5: Difference between Simple Interest and Compound Interest
In case the same principle P is invested in two schemes, at the same rate of interest r and for the same time period t, then in that case:

Simple Interest = (P x R x T)/100
Compound Interest = P [(1+R/100)T – 1]
So, the difference between them is
= PRT/100 – P[(1+R/100)T -1]
= P [(1+r/100)T -1-RT/100]

Two shortcuts which we can use:
Difference between CI and SI when time given is 2 years = P(R/100)2
Difference between CI and SI when time given is 3 years = P[(R/100)3 + 3(R/100)2]

Example 5:
The difference between compound interest and simple interest is 2500 for two years at 2% rate, then find the original sum.

Solution:
Given Interest is = 2500
So, Simple Interest = (P X R X T)/100
Compound Interest = P [(1+r/100)t – 1]
So the difference between both of them is
= PRT/100 – P [(1+R/100)T -1]
= P [(1+r/100)T -1-RT/100]
So the sum is 2500 = P [{(1+2/100)2-1}-4/100]
On simplification this equation the sum will be = Rs. 6250000

We can check it by our shortcut method
When time given is 2 years = P(R/100)2
Since we are given by the difference so
2500 = P (2/100)2
=> 2500 = P (1/50)2
=> 2500= P (1/2500)
=> 6250000=P

So the sum is Rs.6250000.

Similarly we can conclude the sum when time given is 3 years.