Question

A certain sum of money amounts to ₹ 4840 in 2 years and ₹ 5856.40 in 4 years at compound interest. Find the sum and the rate of interest.

Answer 1

Given:

• Amount after 2 years = ₹ 4840
• Amount after 4 years = ₹ 5856.40
• Compound interest applied

To find:

• Principal sum (P)
• Rate of interest (r)

Step-wise calculation:

1. Let the principal sum be P and the rate of interest be r% per annum compounded annually.

2. Using compound interest formula,

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

where (A) is the amount after (n) years.

3. For 2 years: `4840 = P(1 + r/100)^2`

4. For 4 years: `5856.40 = P(1 + r/100)^4`

5. Divide equation (4) by equation (3) to eliminate P:

`5856.40/4840 = (P(1 + r/100)^4)/(P(1 + r/100)^2)`

`5856.40/4840 = (1 + r/100)^(4 - 2)`

`5856.40/4840 = (1 + r/100)^2`

6. Calculate the ratio:

`5856.40/4840 = 1.21`

Hence, `(1 + r/100)^2 = 1.21`

7. Taking the square root:

`1 + r/100 = sqrt(1.21)`

`1 + r/100 = 1.1`

8. Calculate r:

`r/100 = 1.1 - 1`

`r/100 = 0.1`

⇒ r = 10

9. Now substitute r = 10% back into equation (3) to find P:

`4840 = P(1 + 10/100)^2`

4840 = P(1.1)²

4840 = P × 1.21

`P = 4840/1.21`

P = 4000