Question

On a sum of money, the compound interest calculated yearly is ₹ 880 for the second year and ₹ 968 for the third year. Calculate the rate of interest and the sum of money.

Answer 1

We are given that the compound interest for the second year is ₹ 880 and for the third year is ₹ 968. We need to calculate the rate of interest and the sum of money.

Let the principal be P and the rate of interest be r.

Step-by-step solution:

1. Understanding the compound interest for each year:

The compound interest for any year can be calculated as the difference between the amount at the end of that year and the amount at the end of the previous year.

Let the amount at the end of the first year be A₁, at the end of the second year be A₂, and at the end of the third year be A₃.

2. Formulas for compound interest:

The amount at the end of the first year is:

`A_1 = P(1 + r/100)`

The amount at the end of the second year is:

`A_2 = P(1 + r/100)^2`

The amount at the end of the third year is:

`A_3 = P(1 + r/100)^3`

3. Given information:

• The compound interest for the second year is ₹ 880. This means:

CI for second year = A₂ – A₁ = 880

Substituting the formulas for A₂ and A₁:

`P(1 + r/100)^2 - P(1 + r/100) = 880`

Factor out P:

`P[(1 + r/100)^2 - (1 + r/100)] = 880`

Simplifying:

`P[(1 + r/100) - 1](1 + r/100) = 880`

`P xx r/100 xx (1 + r/100) = 880`

This gives us the first equation:

`P xx r/100 xx (1 + r/100) = 880`  ...(1)

• The compound interest for the third year is ₹ 968. This means:

CI for third year = A₃ – A₂ = 968

Substituting the formulas for A₃ and A₂:

`P(1 + r/100)^3 - P(1 + r/100)^2 = 968`

Factor out P:

`P[(1 + r/100)^3 - (1 + r/100)^2] = 968`

Simplifying:

`P xx [(1 + r/100)^2 - 1] xx (1 + r/100) = 968`

`P xx r/100 xx (1 + r/100)^2 = 968`

This gives us the second equation:

`P xx r/100 xx (1 + r/100)^2 = 968`  ...(2)

4. Solving the system of equations:

To solve the system of equations, divide equation (2) by equation (1):

`(P xx r/100 xx (1 + r/100)^2)/(P xx r/100 xx (1 + r/100)) = 968/880`

Simplifying:

`((1 + r/100))/1 = 968/880`

`1 + r/100 = 968/880`

`1 + r/100 = 1.1`

`r/100 = 0.1`

r = 10

So, the rate of interest is 10%.

5. Finding the principal P:

Substitute r = 10 into equation (1):

`P xx 10/100 xx (1 + 10/100) = 880`

`P xx 0.1 xx 1.1 = 880`

`P xx 0.11 = 880`

`P = 880/0.11 = 8000`