# Compound Interest

## What is Compound Interest?

When we observe our bank statements, we generally notice that some interest is credited to our account every year. This interest varies with each year for the same principal amount. We can see that interest increases for successive years. Hence, we can conclude that the interest charged by the bank is not simple interest, this interest is known as compound interest.

Compound interest is the interest calculated on the principal and the interest accumulated over the previous period. It is unlike simple interest where interest is not added to the principal while calculating the interest during the next period. Compound interest finds its usage in most of the transactions in the banking and finance sectors and also in other areas as well. Some of its applications are:

1. Increase or decrease in population.
2. The growth of bacteria.
3. Rise or depreciation in the value of an item.

## How to calculate compound interest?

The compound interest formula is given below:

Compound Interest = Amount – Principal

Where the amount is given by:

Where, A= amount; P= principal; R= rate of interest; n= number of years

It is to be noted that the above formula is the general formula for the number of times the principal is compounded in an year. If the amount is compounded annually, the amount is given as-

$A = P \left (1 + \frac{R}{100} \right )^t$

Let us get to know the values of Amount and Interest in case of Compound Interest for different years-

 Time (in years) Amount Interest 1 $P\left (1+\frac{R}{100} \right )$ $\frac{PR}{100}$ 2 $P\left (1+\frac{R}{100} \right )^{2}$ $P\left (1+ \frac{R}{100} \right )$ 3 $P\left (1+\frac{R}{100} \right )^{3}$ $P\left (1+ \frac{R}{100} \right )^{2}$ 4 $P\left (1+\frac{R}{100} \right )^{4}$ $P\left (1+ \frac{R}{100} \right )^{3}$ n $P\left (1+\frac{R}{100} \right )^{n}$ $P\left (1+ \frac{R}{100} \right )^{n}$

#### NOTE-

From the data it is clear that the interest rate for the first year in compound interest is same as that in case of simple interest, ie. $\frac{PR}{100}$.

Other than the first year, the interest compound annually is always greater than that in case of simple interest.

Let us solve various examples to understand the concepts in a better manner.

## Increase or decrease in population

Examples 1: A town has 10,000 residents in 2000. Its population declines at the rate of 10% per annum. What will be its total population in 2005?

Solution:

The population of the town decreases by 10% every year. Thus, it has a new population every year. So the population for the next year is calculated on the current year population. For the decrease, we have the formula A = P(1 – R/100)n

Therefore, the population at the end of 5 years = 10000(1 – 10/100)5

= 10000(1 – 0.1)5 = 10000 x 0.95 = 5904 (Approx.)

## The growth of bacteria

Examples 2: The count of a certain breed of bacteria was found to increase at the rate of 2% per hour. Find the bacteria at the end of 2 hours if the count was initially 600000.

Solution:

Since the population of bacteria increases at the rate of 2% per hour, we use the formula

A = P(1 + R/100)n

Thus, the population at the end of 2 hours = 600000(1 + 2/100)2

= 600000(1 + 0.02)2 = 600000(1.02)2 = 624240

## Rise or depreciation in the value of an item

Examples 3: The price of a radio is Rs 1400 and it depreciates by 8% per month. Find its value after 3 months.

Solution:

For the depreciation, we have the formula A = P(1 – R/100)n.

Thus, price of the radio after 3 months = 1400(1 – 8/100)3

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

## Derivation of formula for Compound Interest

Let Principal amount = $P$, Time = $n$ years, Rate = $R$

Simple Interest (S.I.) 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 (S.I.) 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:

$A$ = $P\left(1~+~\frac{R}{100}\right)^n$

$CI$ = $A~–~P$ = $P \left[\left(1~+~ \frac{R}{100}\right)^n~ –~ 1\right]$

Here we are providing examples on how to calculate compound interest.

Illustration 1: A sum of $Rs.10000$ is borrowed by Akshit for $2$ years at an interest of $10%$ compounded annually. Calculate the compound interest and amount he has to pay at the end of $2$ years.

Solution: Given,

Principal/ Sum = Rs. 10000,  Rate = 10%, andTime = 2 years

From the table shown above it is easy to calculate the amount and interest for the second year, which is given by-

Amount($A_{2}$) = $P\left (1+\frac{R}{100} \right )^{2}$

$A_{2}$= $= 10000 \left ( 1 + \frac{10}{100} \right )^{2} = 10000 \left ( \frac{11}{10} \right )\left ( \frac{11}{10} \right )= Rs.12100$

Compound Interest (for 2nd year) = $A_{2} – P$ = 12100 – 10000 = Rs. 2100

Illustration 2: Calculate the compound interest ($CI$) on $Rs.5000$ for $2$ years at $10%$ per annum compounded annually.

Solution: Principal (P) = Rs.5000 , Time (T)= 2 year, Rate (R) = 10 %

We have, Amount, $A = P \left ( 1 + \frac{R}{100} \right )^{T}$

$A = 5000 \left ( 1 + \frac{10}{100} \right )^{2} = 5000 \left ( \frac{11}{10} \right )\left ( \frac{11}{10} \right ) 50 \times 121 = Rs. 6050$

Interest (Second Year) = A – P = 6050 – 5000 = Rs.1050

OR

Directly we can use the formular for calculating the interest for second year, which will give us the same result.

Interest (I1) = $P\times \frac{R}{100} = 5000 \times \frac{10}{100} =500$

Interest (I2) = $P\times \frac{R}{100}\left (1 + \frac{R}{100} \right ) = 5000 \times \frac{10}{100}\left ( 1 + \frac{10}{100} \right ) = 550$

Total Interest = I1+ I2 = 500 + 550 = Rs. 1050

## Calculation of compound interest when the rate is compounded half yearly:

Let us calculate the compound interest on a principal, $P$ kept for $1$ year at interest rate $R$ % compounded half yearly.

Since interest is compounded half yearly, the principal amount will change at the end of first 6 months. The interest for the next six months will be calculated on the amount remaining after the first six months. Simple interest at the end of first six months,

$SI_1$ = $\frac{P~×~R~×~1}{100~×~2}$

Amount at the end of first six months,

$A_1$ = $P~ + ~SI_1$ = $P ~+~ \frac{P~×~R~×~1}{2~×~100}$ = $P \left(1~+~\frac{R}{2~×~100}\right)$ = $P_2$

Simple interest for next six months, now the principal amount has changed to $P_2$

$SI_2$ = $\frac{P_2~×~R~×~1}{100~×~2}$

Amount at the end of 1 year,

$A_2$ = $P_2~ +~ SI_2$ = $P_2 ~+~ \frac{P_2~×~R~×~1}{2~×~100}$ = $P_2\left(1~+~\frac{R}{2~×~100}\right)$ = $P \left(1~+~ {R}{2~×~100}\right)\left(1~+~\frac{R}{2~×~100}\right)$ = $P \left(1~+~\frac{R}{2~×~100}\right)^2$

Now we have the final amount at the end of 1 year:

$A$ = $P\left(1~+~\frac{R}{2~×~100}\right)^2$

Rearranging the above equation,

$A$ = $P\left(1~+~\frac{\frac{R}{2}}{100}\right)^{2~×~1}$

Let $\frac{R}{2}$ = $R'$; $2T$ = $T’$, the above equation can be written as, [for the above case $T$ = $1$ year]

$A$ = $P\left(1~+~\frac{R’}{100}\right)^{T’}$

Hence, for the cases, when the rate is compounded half yearly, we divide the rate by $2$ and multiply the time by $2$ before using the general formula for amount in case of compound interest.

Illustration 3: Calculate the compound interest to be paid on a loan of $Rs.2000$ for $\frac{3}{2}$ years at $10%$ per annum compounded half yearly?

Solution: Principal, $P$ = $Rs.2000$, Time, $T’$ = $2~×~\frac{3}{2}$ years = 3 years, Rate, $R’$ = $\frac{10%}{2}$ = $5%$, amount, $A$ can be given as:

$A = P ~\left(1~+~\frac{R}{100}\right)^n$

$A = 2000~×~\left(1~+~\frac{5}{100}\right)^3$

= $2000~×~\left(\frac{21}{20}\right)^3 = Rs.2315.25$

$CI = A – P = Rs.2315.25~ –~ Rs.2000$ = $Rs.315.25$<

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