Question

On a certain sum of money invested at the rate of 4% per annum compounded annually, the difference between the interest of third year and first year is ₹ 81.60. Find the sum.

Answer 1

Given:

• Rate of interest (R) = 4% per annum (compounded annually)
• Difference between compound interest (CI) of 3rd year and 1st year = ₹ 81.60
• Need to find the principal sum (P).

Step-wise calculation:

1. Let the principal be ₹ P.

2. Compound interest is calculated on the amount at the beginning of each year.

Amount at the end of 1st year 

= `P(1 + R/100)`

= `P(1 + 4/100)`

= P × 1.04

Interest for 1st year

= Amount – Principal 

= P × 1.04 – P

= P × 0.04

Amount at the end of 2nd year = P × 1.04²

Amount at the end of 3rd year = P × 1.04³

Interest for 3rd year = Amount at end of 3rd year – Amount at end of 2nd year

= P × 1.04³ – P × 1.04²

= P × 1.04²(1.04 – 1)

= P × 1.04² × 0.04

3. Difference between the interest of 3rd year and 1st year:

= Interest for 3rd year – Interest for 1st year

= P × 1.04² × 0.04 – P × 0.04

= P × 0.04(1.04² – 1)

4. Given the difference is ₹ 81.60:

P × 0.04 × (1.0816 – 1) = 81.60

Since (1.04² = 1.0816)

P × 0.04 × 0.0816 = 81.60

5. Calculate P:

`P = 81.60/(0.04 xx 0.0816)`

`P = 81.60/0.003264`

P = 25000