## Compound Interest

### Important Formulae

Let Principal = P, Rate = R% per annum, Time = n years

1. When interest is compound Annually:

Amount = P(1+R/100)^n

2. When interest is compounded Half-yearly:

Amount = P[1+(R/2)/100]^2n

3. When interest is compounded Quarterly:

Amount = P[1+(R/4)/100]^4n

4. When interest is compounded Annually but time is in fraction, say 3(2/5) years

Amount = {P[1+R/100]^3}*{[1+(2/5)R/100]}

5. When Rates are different for different years, say R1%, R2%, R3% for 1st, 2nd and 3rd year respectively

Then, Amount = P(1+R1/100)(1+R2/100)(1+R3/100)

6.Present worth of Rs. x due n years hence is given by:

Present Worth =x/(1+R/100)

### General Tips

Example 1:

If Sarah invests $5,000 at an interest rate of 5%, compounded semi-annually, how much money will she have at the end of a year?

Solution

The word semiannually means “twice per year”, the bank will pay Sarah half of her interest after 6 months, and the other half at the end of the year. What’s cool about that is that Sarah will actually be earning interest on her interest, which means that she actually earns more money than under a simple interest situation

The important part is to understand that when you’re dealing with compound interest, you simply need to divide the interest rate by the total number of compounding periods before doing the math. In this case, semiannually means two (2) compounding periods. So, we need to divide 5% by 2, which is 2.5%

Step 1: In the first time period (6 months), Sarah’s principle becomes $5,000 * 1.025 = $5,125

Step 2: In the second period (next 6 months), that new principle balance of $5,125 earns an additional 2.5% interest, or $5,125 * 1.025 = $5,253.13

Example 2:

If the simple interest on a sum of money for 2 years at 5% per annum is Rs.50, what is the compound interest on the same at the same rate and for the same time?

Solution

Sum =Rs.(50 x 100/2x5)

= Rs. 500

Amount =[Rs.(500 x(1+5/100)^2]

= Rs(500x21/20x21/20)

=Rs. 551.25

C.I = Rs. (551.25 - 500)

= Rs. 51.25

Example 3:

Simple interest on a certain sum of money for 3 years at 8% per annum is half the compound interest on Rs. 4000 for 2 years at 10% per annum. The sum placed on simple interest is

Solution

C.I. =Rs[4000x(1+10/100)^2-4000]

Rs.(4000x11/10x11/10-4000) = Rs.940.

Sum =Rs. [420x100 /3x8]

= Rs.1750.>