Question

On a certain sum of money invested at the rate of 11% per annum compounded annually, the difference between the interest of the third year and first year is ₹ 255.31. Find the sum.

Answer 1

Step 1: Write down the formulas for the interest of the first and third year.

The formula for the interest of the first year (I₁) on a principal sum P at a rate r compounded annually is the same as simple interest:

I₁ = P × r

The formula for the interest of the third year (I₃) is the difference between the total amount after three years and the total amount after two years.

I₃ = P(1 + r)³ – P(1 + r)²

Factoring, this can be simplified to:

I₃ = P(1 + r)² × r

Step 2: Set up the equation based on the given information.

Given that the rate r = 11% = 0.11 and the difference between the interest of the third year and first year is ₹ 255.31, we can write the equation:

I₃ – I₁ = 255.31

P(1 + r)² × r – P × r = 255.31

P × r[(1 + r)² – 1] = 255.31

Step 3: Substitute the value of the rate and solve for the principal sum, P.

Substitute r = 0.11 into the equation:

P × (0.11)[(1 + 0.11)² – 1] = 255.31

P × (0.11)(1.11)² – 1] = 255.31

P × (0.11)[1.2321 – 1] = 255.31

P × (0.11)[0.2321] = 255.31

P × (0.025531) = 255.31

`P = 255.31/0.025531`

P = 10000