# Compound Interest - Basic Concepts

Sometimes the borrower and the lender agree to add the simple interest earned to the principal. The sum becomes the new principal. The next instalment of interest is calculated on the increased amount and hence there is a increase in interest earned every time. This is called compounding.

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**Formulae:**

1. Let principal = Rs P, Time = n years and Rate = R% p.a.

**Case I:** When interest is compounded Annually: Amount = P (1 + R/100)^{n}.

(Note that the above formula becomes quite cumbersome if the number of years is great. It may not be easy to calculate (1.04)^{7} if we are to calculate compound interest @4% for 7 years. Students should therefore try to learn compounding simple interest and arrive at the answer. See the next section for an explanation of this method.)

**Case II:** When interest is compounded Half-yearly:

Amount = P (1 + (Â½ R)/100)^{2n}

**Case III:** When time is compounded Quarterly:

Amount = P (1 + (Â¼ R)/100)^{4n}

**Case IV:** When time is fraction of a year, say 3 1/5 years, then

Amount = P (1 + R/100)^{3} Ã— (1 + (1/5 R)/100).

**Case V:** When rate of interest is R_{1} %, R_{2}% Âand R_{3}% for 1^{st} year, 2^{nd} year and 3^{rd} year respectively, then

Amount = P (1 + R_{1}/100) Ã— (1 + R_{2}/100) Ã— (1 + R_{3}/100).

2. Present Worth of a sum of Rs. x due n years hence is given by: Present Worth = (x/[1 + R/100]^{n}).

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**Illustration**:

What sum of money lent out at compound interest will amount to Rs 968 in 2 years at 10% p.a., interest being charged annually?

An amount of x will become 1.1x in 1 year at 10% interest.

In 2 years time it will become 1.1Ã—1.1 x = 1.21x.

So, 1.21 x = 968.

Therefore, x = 968/1.21 = 800.

# Calculation of Interest for Part of An Year

365 days are taken to a year while counting days for interest.

When the period is given in months & days, 12 months are taken to a year and 30 days are taken to a month. February of a leap year has 29 days.

The time (T) becomes (no. of days Ã· 365).

For interest to be paid every quarter or six-monthly, the rate is divided by 4 or 2 respectively.

# Compounding Interest Or Population

Compound interest changes every year. The interest earned in a period is added to the principal and the interest for the next year is calculated on the new amount. A sum of Rs 100 at 10% compound interest per annum will earn Rs 10 in the first year but in the second will earn interest on Rs 110, which is the new amount. The formula for compound interest is: A=P(1+R/100)^{n}, where A is the amount and n is the number of years. If the interest is compounded quarterly, the number of terms will be 4n whereas the rate per quarter will be R/4. The formula will change to A=P(1+R/400)^{4n} in the case of quarterly compounding.

It is usually difficult to solve compound interest sums if the number of years is very high, so one should look for simpler methods of solving sums.

For instance to calculate CI on Rs 10,000 @5% for 10 years means that the student has to do: 10,000 Ã— (1.05)^{10}. As this is difficult students should learn approximation techniques and also understand the concept of â€˜interest on interestâ€™.