Interest is the amount borrowed that a lender charges a borrower. The annual percentage rate (APR) describes the interest rate on loan (APR). An interest rate can also be applied to money earned via a savings account or a certificate of deposit at a bank or credit union (CD). The income generated on these deposit accounts is the annual percentage yield (APY). 

Interest Rates can be of two types; Simple Interest and Compound Interest. Lets us understand more about calculating interest, different types with examples in this article.

What is Interest Rate?

Definition: The amount of money lent or borrowed is called the Principal amount. The amount of money charged extra for the amount lent or borrowed is called the interest.

Types of Interest

There are two types of interest:

1. Simple Interest

\[{\rm{(S}}{\rm{.I) = }}\frac{{{\rm{PTR}}}}{{{\rm{100}}}}\]

2. Compound interest

\[{\rm{(C}}{\rm{.I)=P(1+R}}{{\rm{)}}^{\rm{T}}}{\rm{-P}}\]

Simple Interest

If the amount of interest is calculated uniformly based upon the original principal amount throughout the loan period, then it is called the simple interest.

Formula of Simple Interest

Simple interest is calculated by multiplying the interest rate by the principal amount and the time period which is generally in years. The S.I. formula is given as:

After the calculation of S.I., the principal has to be added to it to get the total amount that the borrower has to give or the lender will collect. This is called the total amount and its formula is given as:

Amount\(\left( {\bf{A}} \right)=\)Principal\(\left({\bf{P}}\right)+\)Simple interest\(\left({{\bf{S}}.{\bf{I}}.}\right)\)

What are the notations used in Calculation of Interest (S.I.)?

Simple Interest	\(\left({{\bf{S}}.{\bf{I}}.}\right)\)
Total amount	\(A\)
Principal	\(P\)
Rate of interest	\({\rm{R}}\)
Time period in years	\({\rm{T}}\)

Compound Interest

Banks, post offices and companies lend and accept money deposits in different ways. 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 certain money for some time, for example, quarter year, half-year, one year, two years.

In such kind of cases, the amount of interest occurred during \({{\rm{1}}^{{\rm{st}}}}\) year is added with the original principal amount to obtain the principal amount for the next year. Then, the interest of this amount is calculated again for the next year and added to the principal amount again. This process continues till the total time is over. This method of calculating interest is called compound interest.

Formula of Compound Interest

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

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

Interest Calculation Methods

For the principal amount at the rate of interest \(10\% \) for a year, the simple interest and compound interest for the first three years are calculated as follows:

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

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

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 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:

\(A=P{\left({1+\frac{{\frac{r}{2}}}{{100}}}\right)^{2t}}\)

What is the Calculation of Interest for Quarterly?

If the principal is compounded for 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:

\(P{\left({1+\frac{{\frac{r}{{365}}}}{{100}}}\right)^{365t}}-P\)

Solved Examples

Question-1: Find the interest on ₹\({\rm{\;}}500\) for a period of \(4\) years at the rate of \(8\% \) per annum. Also, find the amount to be paid at the end of the period?

Answer: Given, principal amount \((P) = ₹ 500\)

Time period \((T) = 4\) years.

Rate of interest \((R{\rm{\%}})=8{\rm{\%}}\)

We know that, simple interest \({\rm{\;S}}{\rm{.\;}}I.=\frac{{PTR}}{{100}}\).

\(= \frac{{500\times4\times 8}}{{100}}\)

\(=Rs.160\)

Amount\(=\)Principal\(+\)interest

\(= ₹ 500+Rs.160\)

\(=Rs.660\)

Question-2 Find the compound interest on ₹ \(25000\) rupees for \(3\) years at \(10\%\) per annum, compounded annually.

Answer:

Principal amount for the first year \(= ₹ 25000\)

Interest for the first year \(=\frac{{25000\times10\times1}}{{100}}= ₹ 2500\)

Amount at the end of first year \(= ₹ 25000+ ₹ 2500= ₹ 27500\)

The principal amount for the second year equals the amount at the end of the first year.

So, the Principal amount for the second-year \(= ₹ 27500\)

Interest for the second year \(=\frac{{27500\times10\times1}}{{100}}=₹2750\)

Amount at the end of the second year \(=₹27500+₹2750= ₹30250\)

Principal amount for the third year \(=₹30250\)

Interest for the third year \(=\frac{{30250\times10\times1}}{{100}}= ₹ 3025\)

Amount at the end of the third year \(=₹(30250+3025)=₹33275\)

Therefore, compound interest \(=₹(33275-25000)=₹8275\)

Question-3: Find the compound interest on \( ₹5000\) for \(1\) year at \(8\% \) per annum, compounded half-yearly.

Answer:

Principal amount \(=₹5000\)

Rate of interest \(=8\%\)

Time \(=1\)year \(=2\)half-years

We know that, compound interest for half-yearly is \(P{\left({1+\frac{{\frac{r}{2}}}{{100}}}\right)^{2t}}\)

\(=5000{\left({1+\frac{8}{{200}}} \right)^2}\)

\(= ₹ 5408\)

Question-4: 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 \(600000\).

Answer:

Given, initial population \((P)=600000\)

Rate \((R)=2{\rm{\% }}\).

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

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

Thus, the population at the end of \(2\) hours\(=600000{\left({1+\frac{2}{{100}}}\right)^2}\)

\(=60000{(1+0.02)^2}\)

\(= 624240\)

Question-5: 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\(2005\)?

Answer:

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

\(A=P{\left({1-\frac{r}{{100}}} \right)^n}\)

Therefore, the population at the end of \(5\) years \(=10000{\left({1-\frac{{10}}{{100}}} \right)^5}\)

\(=10000(1-0.1)^5\)

\(=10000\times 0.9^5\)

\(=5904\) (Approx.)

Question-6: 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?

Answer:

Given:

Rate of interest \(=15\% \)

Time period \(= 2\) years

Compound interest \(({\rm{\;C}}{\rm{.I}}{\rm{.\;}}) = ₹ 1290\)

\(C.I.=P{\left({1+\frac{R}{{100}}}\right)^T}-P\)

\(\Rightarrow1290=P{\left({1+\frac{{15}}{{100}}}\right)^2}-P\)

\(\Rightarrow1290=P(0.3225)\)

\(\Rightarrow P=\frac{{1290}}{{0.3225}}\)

\(\Rightarrow P=Rs.4000\)

Therefore, the sum is ₹\(4000\).

Question-7: The difference between \(S.I.\) and \(C.I.\) of a certain sum of money is \(Rs.48\) at \(20\%\) per annum for two year, find the principal

Answer:

Simple interest \(S.I.=\frac{{PTR}}{{100}}=\frac{{P\times20\times2}}{{100}}=\frac{{2P}}{5}\)

Compound interest \({\rm{\;C}}{\rm{.I}}{\rm{.\;}}=P\left[ {{{\left({1+\frac{r}{{100}}} \right)}^t}-1}\right]=P\left[{{{\left({1+\frac{{20}}{{100}}}\right)}^2}-1}\right]=P\left[{{{\left({\frac{6}{5}}\right)}^2}-1} \right]=\frac{{11P}}{{25}}\)

Given, \({\rm{\;C}}{\rm{. I}}{\rm{.\;}}-S.I.= ₹ 48\)

\(\frac{{11P}}{{25}}-\frac{{2P}}{5}=48\)

\(\Rightarrow\frac{{11P-10P}}{{25}}=48\)

\(\Rightarrow P=25\times48\)

\(\Rightarrow P=Rs.1200\)

Therefore, principal amount is \(Rs.1200\)

Question-8: 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?

Answer:

Principal, \(P= ₹ 2000\)

Time, \(T=2\times\frac{3}{2}\), years \(= \,3\) years,

Rate, \(R=\frac{{10}}{2}=5\),

Amount, \(A\) can be given as:

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

\(\Rightarrow \)\(A=2000\times{\left({1+\frac{5}{{100}}}\right)^3}=2000\times {\left({\frac{{21}}{{20}}}\right)^3}= ₹ 2315.25\)

So, \({\rm{\;C}}{\rm{. I}}{\rm{.\;}}=A-P=Rs.2315.25-{\rm{ ₹ 2000\;}}= ₹ 315\)\((approx)\)

Summary

This article has discussed the types of interests, simple interest and compound interest, and their formulas to calculate the interest. In Simple Interest, the Principal remains constant while the Principal will change because of compounding in the case of compound interest.
The growth rate of simple interest is lower than the compound interest. The comprehensive knowledge about the topics covered will help the students understand the concept of interest better.

Study About Simple Interest Calculator

FAQs

Q1: What is the difference between simple interest and compound interest?
Answer: Simple interest is the interest paid only on the principal, whereas compound interest is the interest paid on both principal and interest compounded at regular intervals.

Q2: Is interest compounded daily better than monthly?
Answer: The interest compounded daily has \(365\) compounding cycles a year. It will generate more money compared to interest compounded monthly, which has only \(12\) compounding cycles per year.

Q3: How does compound interest depend on the time period?
Answer: The compound interest depends on the time interval of the calculation of interest. The time interval for the calculation of interest can be a day, a week, month, quarterly, half-yearly. For the shorter time of calculation, the net accumulated compound interest is higher.

Q4: What is the formula to calculate interest?
Answer: There is a direct formula for the calculation of compound interest.

\({\rm{\;C}}{\rm{.I}}{\rm{.\;}}=P{\left({1+\frac{R}{{400}}}\right)^n}-P\)

The formulae to calculate simple interest is

\({\rm{\;S}}{\rm{. I}}{\rm{.\;}}=\frac{{PTR}}{{{\rm{\;100\;}}}}\)

Q5: Is interest compounded daily better than monthly?
Answer: The interest compounded daily has \(365\) compounding cycles a year. It will generate more money compared to interest compounded monthly, which has only \(12\) compounding cycles per year.

Q6: What is the information required to calculate compound interest?
Answer: The calculation of compound interest requires us to know the principal, rate of interest, and time. Also, we need to know the time interval for which the interest is to be calculated.

Q7: What is \(PNR\) in simple interest?
Answer: We know the formula for interest is \(I=PNR\)

Q8: How do I calculate \(S.I.\)?
Answer: To calculate the \(S.I.\) for a certain amount of money \((P)\), rate of interest \((R)\) and time \((T)\), the formula is:

\(S\,.I.\, = \,\frac{{PTR}}{{100}}\)

Here,

\(S.{\rm{ }}I.=\) Simple interest

\(P=\) Principal (sum of money borrowed)

\(R=\) Rate of interest per annum

\(T=\) Time (in years)

Q9: Who benefits from compound interest?
Answer: The investors benefit from the compound interest since the interest paid here on the principal plus on the interest which they already earned.

Q10: How do you calculate interest per month?
Answer: If the principal is compounded monthly ((12) months in a year), the interest is given by
Compound interest(= )Principle({\left({1+\frac{{\frac{r}{{12}}}}{{100}}} \right)^{12\times time}}) Principle
Here (‘I’) is the interest, (‘P’) is the principal amount, (‘N’) is the time period, and (‘R’) is the rate of interest.

We hope that you found this article about Calculation of Interest helpful. If you have any queries about Calculation of Interest or Mathematics then please drop a comment below and we will reach out to you as soon as possible.