Compound Interest: It is the interest earned on the principal (original amount) and the interest already earned. It also keeps multiplying every year. If we investigate our bank statement, some interest amount is credited to our account every year. We can see that interest increases for successive years. The interest calculated by banks is known as Compound Interest (C.I.).

What is Compound Interest?

Compound interest is the interest paid on both principal and interest, compounded at regular intervals. The interest so far accumulated at regular intervals is added with the existing principal amount, and then the interest is calculated for the new principal. The new principal is equal to the sum of the initial principal and the interest accumulated so far.

Compound Interest = Interest on Principal + Compounded Interest at regular intervals

The compound interest is calculated at regular intervals like annually (yearly), semi-annually, quarterly, monthly, etc. Banks and other financial organizations calculate the amount based on compound interest only.

What is the Compound Interest Formula?

After calculating the total amount over a period, the compound interest is calculated based on the rate of interest and the initial principal.

For an initial principal \(P\) rate of interest per annum \(r,\) time period in years, frequency of the number of times the interest is compounded annually \(n.\)

The formula for calculation of amount as follows:

The above formula represents the total amount at the end of the time period and includes the compounded interest and the principal. Further, we can calculate the compound interest by subtracting the principal from this amount.

The formula for calculating the compound interest as follows:

\(C.I. = P{\left( {1 + \frac{{\frac{r}{{100}}}}{n}} \right)^{nt}} – P\)

Terms Related to Compound Interest

There are some terms related to compound interest, which are described in detail below:

1. Principal: The sum of money lent for a certain period at a particular rate of interest
2. Time: It is the duration for which the principle is mostly calculated in years.
3. Interest: It is the profit earned on lending a principal for a certain period.
4. Compound Interest: It is the total annual interest earned on lending a principal for a certain period.
5. Rate: It is the percentage of interest and lending a sum of money.
6. Amount: The amount is the final money left at the end. It is the sum of the original principal and the total compound interest earned.

What are the Notations used in Calculation of Interest(C.I.)?

Compound Interest	\(C.I.\)
Total amount	\(A\)
Principal	\(P\)
Rate of interest	\(r\)
Time period	\(t\)
Compounding frequency per given time period	\(n\)

Compound Interest Simple Calculation

Banks, post offices, and companies lend and accept money deposits in different way.
Apart from the calculation of money, this basic formula is used for some other applications that are given as:

1. Increase or decrease in population.
2. Rise or depreciation in the value of an item.

In the compound interest method, the lender and borrower agree to fix a certain unit of time, for example, quarter year, half-year, one year, two years, to settle the previous account.

In such cases, the amount of interest that occurred during this fixed unit of time is added with the original principal amount to obtain the principal amount for the next unit of time. Then, the interest of this amount is calculated again for the next unit of time and added to the principal amount again. This process continues till all the units of time are over. This method of calculating interest is called compound interest.

Derivation of Compound Interest Formula

We can derive the formula for compound interest from the formula for simple interest, which is given by. \(I = \frac{{PTR}}{{100}}.\)

Before looking into the derivation of the formula for compound interest, let us understand the basic difference between simple interest and compound interest computation.

The principal remains constant over a period for simple interest computation but, for compound interest computation, the interest is added to the principal.

Derivation:

Let the principal is \P\) and the rate of interest be \(r.\)
At the end of the first compounding period, the simple interest on the principal amount is \(\frac{{\Pr }}{{100}}.\)
And hence, the amount is \(P + \frac{{\Pr }}{{100}} = P\left( {1 + \frac{r}{{100}}} \right).\)
The amount is taken as the principal for the second computation period.
At the end of the second compounding period, the simple interest in the principal is \(P\left( {1 + \frac{r}{{100}}} \right) \times \frac{r}{{100}}\;.\)
And the amount is \(P\left( {1 + \frac{r}{{100}}} \right) \times \frac{r}{{100}} + P\left( {1 + \frac{r}{{100}}} \right) = P{\left( {1 + \frac{r}{{100}}} \right)^2}.\)
Continuing in this manner for n compounding periods, the amount at the end of the \({n^{th}}\) compounding period is \(A = P{\left( {1 + \frac{r}{{100}}} \right)^n}\)
From the above formulas and computations, we can observe that compound interest is the same as simple interest for the first interval of time.

Compound Interest Formula for Different Time Periods

What is the Calculation of Interest Compounded Yearly?
For some principal \(P,\) that is borrowed from the person at the rate of \(r\% \) compounded yearly, then the compound interest to be paid is given by
\(C.I. = P\left( {1 + \frac{r}{{100}}} \right) – P\)

What is the Calculation of Interest for Two Years?
If the principal is compounded for two years, the total amount is given by
\(A = P{\left( {1 + \frac{r}{{100}}} \right)^2}\)
And interest is given as:
\(P{\left( {1 + \frac{r}{{100}}} \right)^2} – P\)

What is the Calculation of Interest for \(n\) Years?
If the principal is compounded for \({\rm{n}}\) years, the total amount is given by
\(A = P{\left( {1 + \frac{r}{{100}}} \right)^n}\)
And interest is given as:
\(P{\left( {1 + \frac{r}{{100}}} \right)^n} – P\)

What is the Calculation of Interest for Half-Yearly?
If the principal is compounded for half-yearly, the total amount is given by
\(A = P{\left( {1 + \frac{{\frac{r}{2}}}{{100}}} \right)^{2t}}\)
And interest is given as:
\(P{\left( {1 + \frac{{\frac{r}{2}}}{{100}}} \right)^{2t}} – P\)

What is the Calculation of Interest for Quarterly?
If the principal is compounded quarterly, the total amount is given by
\(A = P{\left( {1 + \frac{{\frac{r}{4}}}{{100}}} \right)^{4t}}\)
And interest is given as:
\(P{\left( {1 + \frac{{\frac{r}{4}}}{{100}}} \right)^{4t}} – P\)

What is The Calculation of Interest Monthly?
If the principal is compounded monthly (\(12\) months in a year), the total amount is given by
\(A = P{\left( {1 + \frac{{\frac{r}{{12}}}}{{100}}} \right)^{12t}}\)
And interest is given as:
\(P{\left( {1 + \frac{{\frac{r}{{12}}}}{{100}}} \right)^{12t}} – P\)

What is the Calculation of Interest Daily?
If the principal is compounded daily (\(365\) days in a year), the total amount is given by
\(A = P{\left( {1 + \frac{{\frac{r}{{365}}}}{{100}}} \right)^{365t}}\)
And interest is given as:
\(P{\left( {1 + \frac{{\frac{r}{{365}}}}{{100}}} \right)^{365t}} – P\)

Compound Interest Table

Now that we have provided the compound interest formulas, let us have a summary of the formulas in the table below:

Time	Compound Interest	Amount
\(1\) year [Compounded annually]	\(P{\left( {1 + \frac{r}{{100}}} \right)^t} – P\)	\(P{\left( {1 + \frac{r}{{100}}} \right)^t}\)
\(6\) months [Compounded half yearly]	\(P{\left( {1 + \frac{{\frac{r}{2}}}{{100}}} \right)^{2t}} – P\)	\(P{\left( {1 + \frac{{\frac{r}{2}}}{{100}}} \right)^{2t}}\)
\(3\) months [Compounded quarterly]	\(P{\left( {1 + \frac{{\frac{r}{4}}}{{100}}} \right)^{4t}} – P\)	\(P{\left( {1 + \frac{{\frac{r}{4}}}{{100}}} \right)^{4t}}\)
\(1\) month [Monthly compound interest formula]	\(P{\left( {1 + \frac{{\frac{r}{{12}}}}{{100}}} \right)^{12t}} – P\)	\(P{\left( {1 + \frac{{\frac{r}{{12}}}}{{100}}} \right)^{12t}}\)
\(365\) days [Daily compound interest formula]	\(P{\left( {1 + \frac{{\frac{r}{{365}}}}{{100}}} \right)^{365t}} – P\)	\(P{\left( {1 + \frac{{\frac{r}{{365}}}}{{100}}} \right)^{365t}}\)

Compound Interest vs Simple Interest

Now, let us understand the difference between the amount earned through compound interest and simple interest on a certain amount of money, say \(₹ 100\) in \(3\) years, and the rate of interest is \(10%\) per annum.

The following table shows the process of calculating interest and the total amount in two different cases.

Solved Examples – Compound Interest

Q.1. Find the compound interest on \(₹ 5000\) for \(1\) year at \(8%\) per annum, compounded half-yearly.

Ans : Principal amount \(= ₹ 5000\)
Rate of interest \(= 8%\)
Time \(=1\) year \(=2\) half years
We know that compound interest for half-yearly is \(P{\left( {1 + \frac{{\frac{r}{2}}}{{100}}} \right)^{2t}}\)
\( = 5000{\left( {1 + \frac{8}{{200}}} \right)^2}\)
\(= ₹ 5408\)

Q.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 \(600000.\)

Ans : Given, initial population \(\left( P \right) = 600000.\)
Rate \(\left( R \right) = 2\% \)
Since the population of bacteria increases at the rate of \(2%\) per hour, we use the formula
\(A = P{\left( {1{\rm{\;}} + \frac{R}{{100}}} \right)^n}\)
Thus, the population at the end of \(2\) \( = {\rm{\;}}600000{\left( {1 + \frac{2}{{100}}} \right)^2}{\rm{\;}}\)
\( = 60000{\left( {1 + 0.02} \right)^2}\)
\( = 624240\)

Q.3. A sum of \( ₹5000\) is borrowed at the rate is What is the monthly compound interest for \(2\) years?

Ans : Given, principal \(\left( P \right) = 5000\)
Rate of interest \(\left( R \right) = 8{\rm{\% }}\)
Time period \(\left( T \right) = 2\) years
Compound interest after two years is = Principal \({\left( {1 + \frac{{\frac{r}{{12}}}}{{100}}} \right)^{12 \times 2}} – \) Principal
\( = 5000{\left( {1 + \frac{8}{{1200}}} \right)^{12 \times 2}} – 5000\)
\( = 5000{\left( {1 + \frac{8}{{1200}}} \right)^{24}} – 5000\)
\( = 5000 \times 1.173 – 5000\)
\( = 5865 – 5000 = {\rm{Rs}}.865\)
The monthly compound interest for \(2\) years is \( ₹865.\)

Q.4. A town had \(10,000\) residents in \(2000.\) Its population declines at a rate of \(10%\) per annum. What will be its total population in \(2005\)?

Ans : The population of the town decreases by \(10%\) every year. Thus, it has a new population every year.
The population for the next year is calculated on the current year population.
For the decrease, we have the formula
\(A = P{\left( {1{\rm{\;}}–\frac{r}{{100}}} \right)^n}{\rm{\;}}\)
Therefore, the population at the end of \(5\) years \( = 10000{\left( {1{\rm{\;}}–\frac{{10}}{{100}}} \right)^5}\)
\( = 10000{\left( {1\;–\;0.1} \right)^5}\)
\( = 10000{\rm{\;x\;}}{0.95^5}{\rm{\;}}\)
\( = {\rm{\;}}5904{\rm{\;}}\left( {{\rm{Approx}}.} \right)\)

Q.5. Keerthi borrowed a certain sum at the rate of \(15%\) per annum. If she paid at the end of two years \( ₹1290\) as interest compounded annually, find the sum she borrowed?

Ans : Given:
Rate of interest \(= 15%\)
Time period \(= 2\) years
Compound interest \(\left( {C.I} \right)\; = ₹ 1290\)
\(C.I = P{\left( {1 + \frac{R}{{100}}} \right)^T} – P\)
\( \Rightarrow 1290 = P{\left( {1 + \frac{{15}}{{100}}} \right)^2} – P\)
\( \Rightarrow 1290 = P\left( {0.3225} \right)\)
\( \Rightarrow P = \frac{{1290}}{{0.3225}}\)
\( \Rightarrow P = ₹ 4000\)
Therefore, the sum is \(₹ 4000.\)

Q.6. The difference between SI and CI of a certain sum of money \( ₹ 48\) at \(20%\) per annum for two years, find the principal.

Ans : Simple interest \(S.I = \frac{{PTR}}{{100}} = \frac{{P \times 2 \times 20}}{{100}} = \frac{{2P}}{5}\)
Compound interest \(C.I = P\left[ {{{\left( {1 + \frac{r}{{100}}} \right)}^{\rm{t}}} – 1} \right] = P\left[ {{{\left( {1 + \frac{{20}}{{100}}} \right)}^2} – 1} \right] = P\left[ {{{\left( {\frac{6}{5}} \right)}^2} – 1} \right] = \frac{{11P}}{{25}}\)
Given, \(C.I – S.I = ₹ 48\)
\(\frac{{11P}}{{25}} – \frac{{2P}}{5} = 48\)
\( \Rightarrow \frac{{11{\rm{P}} – 10{\rm{P}}}}{{25}} = 48\)
\( \Rightarrow {\rm{P}} = 25 \times 48\)
\( \Rightarrow {\rm{P}} = {\rm{Rs}}.1200\)
Therefore, the principal amount is \({\rm{Rs}}.1200.\)

Summary

In this article, we have provided the formula to calculate compound interest. Also, we have provided the compound interest examples so that you become confident on this topic. The compound interest is calculated at regular intervals like annually (yearly), semi-annually, quarterly, monthly, etc. Banks and other financial organizations calculate the amount based on compound interest only.

Frequently Asked Questions (FAQ) – Compound Interest

Q.1. How do you calculate interest per month?

Ans: If the principal is compounded monthly (\(12\)months in a year), the interest is given by
\({\rm{Compound\;interest}}\; = \;{\rm{Principle}}{\left( {1 + \frac{{\frac{r}{{12}}}}{{100}}} \right)^{12 \times {\rm{time}}}} – \;{\rm{Principle}}\)

Q.2. What is the compound interest?

Ans: Compound interest is the interest paid on both principal and interest, compounded at regular intervals. 

Q.3. What is the compounded daily formula?

Ans : The compound interest formula when the interest is compounded daily is given by:
\(P{\left( {1 + \frac{{\frac{r}{{365}}}}{{100}}} \right)^{365t}} – P\)

Q.4. Namita borrowed \( ₹ 50, 000\) for \(3.5%\) per annum. Find the interest accumulated at the end of \(3\) years.

Ans : \(P \; = \; ₹ 50,000\)

\(R\; = \;3.5\% \) and \(T\; = \;3\;{\rm{years}}\)

\(\;SI\; = \frac{{\left( {P\; \times \;R\; \times T} \right)}}{{100}}\; = \frac{{\;\left( {50,000 \times \;3.5\; \times 3} \right)}}{{100}}\; = \; ₹ 5250\;\)

Therefore, simple interest is \(₹ 5250.\) 

Q.5. Who benefits from compound interest?

Ans: The investors benefit from the compound interest since the interest is paid here on the principal plus on the interest which they already earned. 

Q.6. What is compound interest with example?

Ans : For some principal \(\left( P \right),\) that is borrowed from the person at the rate of \(r%\) compounded yearly, then the compound interest to be paid is given b­y
\(C.I = P\left( {1 + \frac{r}{{100}}} \right) – P\)

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