Question

A certain sum of money amounts to ₹ 26460 in 2 years and ₹ 29172.15 in 4 years at compound interest. Find the sum and the rate of interest.

Answer 1

Given:

• Amount after 2 years, (A₂ = ₹ 26460)
• Amount after 4 years, (A₄ = ₹ 29172.15)
• Interest is compounded annually.
• Principal (P) and rate of interest (r%) are to be found.

Step 1: Use compound interest formula

`A = P(1 + r/100)^n`

For 2 years,

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

A₂ = 26460

For 4 years,

`A_4 = P(1 + r/100)^4`

A₄ = 29172.15

Step 2: Find ratio of amounts to eliminate (P)

`A_4/A_2 = (P(1 + r/100)^4)/(P(1 + r/100)^2)`

`A_4/A_2 = (1 + r/100)^2`

`A_4/A_2 = 29172.15/26460`

Calculate the ratio:

`(1 + r/100)^2 = 29172.15/26460`

`(1 + r/100)^2 = 1.1025`

Step 3: Solve for the rate (r)

Taking square root on both sides,

`1 + r/100 = sqrt(1.1025)`

`1 + r/100 = 1.05`

⇒ `r/100 = 1.05 - 1`

⇒ `r/100 = 0.05`

r = 5%

Step 4: Find the principal (P)

Using the amount for 2 years:

26460 = P(1.05)²

26460 = P × 1.1025

`P = 26460/1.1025`

P = 24000