# Simple Interest

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## Introduction to Simple Interest

Ever borrowed money from a friend or relative and they said that you would have to return the amount after a year with 10% interest? Well, that means that you will have to return the amount borrowed along with an additional sum of 10% of the borrowed amount after a year. This additional sum is called the interest amount, and here we will explore the concept of simple interest.

## The Big Idea: What is Simple Interest?

• Simple interest is paying a small bonus amount on the fixed original amount. After the passage of each time interval, you get a bonus on the original amount.

• The original fixed amount is also known as the Principal. In other words, the amount of money borrowed or lent for a certain period of time is called the principal. (Denoted as “P”)

• The rate of interest is the percent of interest that is applied on the principal amount. Usually the rate is given as a percentage per year in which case it can be called annual interest rate. (Denoted as “R”)

• There is always a tenure for a loan. In case of Simple Interest, the duration of the loan is generally shorter and is calculated in years. The duration for which the money is borrowed or lent is called time period. (Denoted as “T”)

The interest is calculated using the following formula:

$$I = P\times R\times T$$ where:

$$I$$ is the Simple Interest
$$P$$ is the Principal Amount
$$R$$ is the Rate of Interest
$$T$$ is the Time Period

The total amount (Denoted as "$$A$$" at the end of the time period is the addition of the Principal Amount to the Interest. Hence:

\begin{align}A&=P+[P\times R\times T]\\&=P[1+(R\times T)]\end{align}

Keep in mind that R is in percentage figures, and for the purpose of the calculation, you will need to convert it to a decimal. So, if your loan has an interest rate of 5%, then the R in the calculation will be 0.05.

Using an example to clarify. If you borrowed Rs. 5,000 from a bank with an interest rate of 7% for a period of 8 years, then the Simple Interest would be calculated as:

\begin{align}&\mathrm I=\mathrm P\times\mathrm R\times\mathrm T=5000\times 0.07\times8=\mathrm{Rs}.2800\\&\mathrm{The}\;\mathrm{amount}\;\mathrm{for}\;\mathrm{the}\;\mathrm{above}\;\mathrm{example}\;\mathrm{would}\;\mathrm{be}:\\&\mathrm A=5000+2800=\mathrm{Rs}.7,800\end{align}

## Why Simple Interest? How is it important?

The concept of simple interest is the foundation of the functioning of banks and other financial institutions. Whenever we borrow or lend a certain amount of money, we pay back or are paid back the original sum accompanied with a certain amount of interest on the amount. In real life, Simple Interest is generally used to calculate short term loans (Or money invested) where other external factors do not come into play.

### Be careful:

The rate of interest is usually quoted on an annual basis, but in some cases, interest rates can be at quoted for shorter periods such as quarterly or biannually. In such cases, if the interest of the whole year is asked for, then you will need to multiply the interest with a coinciding number. For example:

\begin{align}&\mathrm{If}\;\mathrm{the}\;\mathrm{interest}\;\mathrm{rate}\;\mathrm{is}\;10\%\;\mathrm{per}\;\mathrm{month},\;\mathrm{SI}=\frac{12 \;(\mathrm P\times10\times\mathrm T)}{100}\\&\mathrm{If}\;\mathrm{the}\;\mathrm{interest}\;\mathrm{rate}\;\mathrm{is}\;10\%\;\mathrm{per}\;\mathrm{quarter},\;\mathrm{SI}=\frac{4\;(\mathrm P\times10\times\mathrm T)}{100}\\&\mathrm{If}\;\mathrm{the}\;\mathrm{interest}\;\mathrm{rate}\;\mathrm{is}\;10\%\;\mathrm{biannually},\;\mathrm{SI}=\frac{2\;(\mathrm P\times10\times\mathrm T)}{100}\end{align}

Formulae and shortcuts to keep in mind while doing simple interest problems:

• \begin {align}\mathrm R=\frac{\mathrm S.\mathrm I\times100}{\mathrm P\times\mathrm T}\end {align}(The 100 represents the conversion of R into a decimal)

• \begin {align}\mathrm P=\frac{\mathrm S.\mathrm I\times100}{\mathrm R\times\mathrm T}\end {align}(The 100 represents the conversion of R into a decimal)

• \begin {align}\mathrm T=\frac{\mathrm S.\mathrm I\times100}{\mathrm P\times\mathrm R}\end {align}(The 100 represents the conversion of R into a decimal)

• If rate to interest is R1% for T1 years, R2% for T2 years…. Rn for Tn years for an investment. And if the Simple Interest obtained is ₹I on the investment. Then the principal amount is given by

\begin {align}\mathrm P=\frac{\mathrm I\times100}{({\mathrm R}_1{\mathrm T}_1\;+\;{\mathrm R}_2{\mathrm T}_2\;+\;\dots+\;{\mathrm R}_{\mathrm n}{\mathrm T}_{\mathrm n})}\;\end {align}

• If a person deposits sum of ₹A at R1% p.a. and sum of ₹B at R2% p.a. then the rate of interest for whole sum is

\begin {align}\mathrm R=\frac{({\mathrm{AR}}_1+{\mathrm{BR}}_2)}{(\mathrm A+\mathrm B)}\end {align}

• If a sum of money becomes “N” times in “T years” at Simple Interest, then the rate of interest p.a. is

\begin {align}\mathrm R=\frac{100(\mathrm N-1)}{\mathrm T}\end {align}

• If a certain sum of money is lent out in N parts in such a manner that equal sum of money is obtained at simple interest on each part where interest rates are R1, R2,…, RN respectively and time periods are T1, T2,…, TN respectively, then the ratio in which the sum will be divided in n parts can be given by

\begin {align}\frac1{{\mathrm R}_1{\mathrm T}_1}:\;\frac1{{\mathrm R}_2{\mathrm T}_2}:\,...:\frac1{{\mathrm R}_{\mathrm N}{\mathrm T}_{\mathrm N}}\end {align}

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