## Compound interest

In compound interest, the principal does not remain the same but increases at the end of each interest period. In other words, compound interest is calculated on interest as well as on principal. Therefore, compound interest is sometimes described as ‘interest on interest’.

**Some important definitions**

**Compound Interest – **It is the difference between the compound amount and original principal amount. so that we can find the compound interest by using the formula- CI = Amount – Principle

**Compound Amount – **It is the total amount due at the end of the last period.

**Frequency of Compounding – **It indicates the number of times the interest is compounded in one year.

**Compounding Period – **The time period between two consecutive points in time at which interest is compounded.

Let-

P- Principal

A – Amount

i = interest on re. 1 for a year

n = interest period

The amount can be calculated as below

And

**Note-** By using algorithm the above formula can be written as below-

**Note- **if the compound interest is paid half-yearly, quarterly, monthly instead of a year there will be different formulae as given in the table below-

Time | Amount |

Annually | |

Half-yearly | |

Quarterly |

**Example:** **Aman invests 1000 rupees at 5 percent p.a for four years then find the compound interest on it.**

**Solution**

Here P = 1000, i = 0.05 and n = 4

As we know that-

So that on taking log, we get-

So that the compound interest will be-

C.I = 1215 – 1000 = 215

This is the required answer.

**Example: A sum of money invested at C.I. payable yearly amounts to 10, 816 Rs. at the end of the second year and to 11,248.64 Rs. at the end of the third year. Find the rate of interest and the sum.**

**Solution**

Here A1 = 10,816, n = 2, and A2 = 11,248.64, n = 3

We know that

A = P (1 + i)n hence we get,

Here on dividing equation (2) by (1)-

Therefore we get

And

Hence the rate is 4 percent.

From first equation-

Or

Now

P = antilog 4.000 = 10,000

**Nominal and effective rate of interest**

Nominal rate of interest is the rate of interest per annum which is compounded yearly, half yearly, quarterly, monthly, n times in a year or continuously.

Effective rate of interest is the rate of interest per annum compounded only once in a year.

There is a relationships between nominal and effective rate of interest under two different conditions-

1. If compounding is ‘n’ times in a year-

Where ‘r’ is nominal rate and ‘R’ is effective rate.

1. If compounding is continuous-

Or

**Relationship between two nominal rates**

If interest is compounded quarterly at r1 percent and the interest is compounded half yearly at r2 percent,

the relationship between the two is-

**Example: Ronak deposited Rs. 10,000 in a bank for 3 years** and **Bank gives two offers either 10 percent compounded quarterly or 8% compounded continuously, then which offer is preferable for Ronak?**

**Solution**

Balance after three years under first offer-

Balance after 3 years under second offer-

Hence the first offer is preferable.