# Compound Interest

In compound interest, the interest for every year is calculated on the amount for the previous year. This means, the **amount** for the previous year becomes the **principal** for the current year.

Let’s try and understand this using an example,

\(₹1000\) is invested for \(3\) years at the rate of \(10 \%\) per annum. Find the compound interest.

**For 1 ^{st} year**

\(\begin{align}\text{Principal }(P) &= ₹1000 \\\\ \text{Interest }(I_1) &= 10 \%\text{ of }₹1000 \\ &= ₹100 \\\\ \text{Amount } &= ₹1000 + ₹100 \\&= ₹1100 \end{align}\)

**For 2 ^{nd} year**

\(\begin{align}\text{Principal }(P) &= ₹1100 \\\\ \text{Interest }(I_2) &= 10 \%\text{ of }₹1100 \\ &= ₹110 \\\\ \text{Amount } &= ₹1100 + ₹110 \\&= ₹1210 \end{align}\)

**For 3 ^{rd} year**

\(\begin{align}\text{Principal }(P) &= ₹1210 \\\\ \text{Interest }(I_3) &= 10 \%\text{ of }₹1210 \\ &= ₹121 \\\\ \text{Amount } &= ₹1210 + ₹121 \\&= ₹1331 \end{align}\)

\(\begin{align}\text{Total Interest } &= I_1 + I_2 + I_3 \\ & = ₹100 + ₹110 + ₹121 \\ &=₹331 \end{align}\)

Therefore,

\(\begin{align}\text{Total Amount } &= P + I \\ & = ₹1000 + ₹331 \\ & = ₹1331 \end{align}\)

### Banks, insurance companies, etc. generally levy compound interest.

Interest can be compounded annually, half yearly, quarterly, monthly, etc. The period for which the interest is calculated and added to the principal is called the conversion period.

Aman lends \(₹40000\) to Shikha at an interest rate of \(10 \%\) per annum compounded half-yearly for a period of \(2\) years.

\(\begin{align}\text{Conversion period } &= \text{Half year} \\ \therefore \text{Rate of interest per half year } &= {10 \over 2} \% \\ &= 5 \% \end{align}\)

Time Period |
Principal (₹) |
Interest (₹) |
Amount (₹) |

1^{st} half year |
\(₹40000\) | \(\begin{align}5 \% \text{ of } ₹40000 = ₹2000 \end{align}\) | \(₹42000\) |

2^{nd} half year |
\(₹42000\) | \(\begin{align}5 \% \text{ of } ₹42000 = ₹2100 \end{align}\) | \(₹44100\) |

3^{rd} half year |
\(₹44100\) | \(\begin{align}5 \% \text{ of } ₹44100 = ₹2205 \end{align}\) | \(₹46305\) |

4^{th} half year |
\(₹46305\) | \(\begin{align}5 \% \text{ of } ₹46305 = ₹2315.25 \end{align}\) | \(₹48620.25\) |

\(\begin{align}\text{The total interest to be paid over a period of 2 years } &= ₹2000 + ₹2100 + ₹2205 + ₹2315.25 \\& = ₹8620.25 \end{align} \)

Therefore,

\(\begin{align}\text{Total Amount } &= P + I \\& = ₹40000 + ₹8620.25 \\ & = ₹48620.25 \end{align} \)

## Tips and Tricks

- The rule of \(72\) tells you how long it will take for your money to double. Using the rule of \(72,\) we can find the number of years to double your money by simply dividing \(72\) by the rate of interest. For example, at \(12 \%\) compounded interest rate your money will double in \(72 \div 12 = 6\) years.
- The time duration over which an interest rate is applicable is referred to in many different terms. Sometimes it is called “per annum” or “annual” or “per year”. All of these mean you’ll get the given rate of interest over a period of one year. Semi-annual is \(6\) months. While quarterly is \(3\) months duration.
- Sometimes the question will state an annual rate of interest (say \(12 \%\)). But it will also say that the interest is compounded at an interval which is different from a year (say semi-annually). Here, you must adjust the interest rate as well as time periods accordingly.

In this case, since the time interval is half a year, also take interest as half the annual rate. So, calculate time period in 6-month chunks and interest rate as \(12 \div 2 = 6 \%.\)

The rate of interest and each time period must be of the same duration.

## Test your knowledge

- I lend \(₹1\) lakh to Niraj for a period of \(3\) years, at an interest rate of \(8 \%\) compounded annually. Find the total interest and the total amount that Niraj will pay me after three years.
- On what sum will the interest at \(5 \%\) per annum for \(2\) years, compounded annually, yield a total amount of \(₹22050?\)
- On a certain sum, the interest after \(3\) years compounded annually at the rate of \(10 \%\) per annum is \(₹9930.\) Find the principal.