A certain sum of money is lent out at simple interest. If that money becomes Rs. 24,000 in 2 years and Rs. 32,000 in 4 years, the annual rate of interest is:  

Given:

The amount becomes Rs. 24,000 in 2 years.

The amount becomes Rs. 32,000 in 4 years.

Formula Used:

Simple interest = Amount after time - Principal

Interest for 2 years = Amount after 2 years - Principal

Interest for 4 years = Amount after 4 years - Principal

Annual interest = (Interest for 4 years - Interest for 2 years) / 2

Calculation:

Let the principal be P and the annual interest rate be R%.

Interest for 2 years = Rs. 24,000 - P

Interest for 4 years = Rs. 32,000 - P

The interest earned in 2 years = Interest for 4 years - Interest for 2 years

Interest for 4 years - Interest for 2 years = (Rs. 32,000 - P) - (Rs. 24,000 - P)

Interest for 2 years = Rs. 32,000 - Rs. 24,000

Interest for 2 years = Rs. 8,000

Annual interest = Rs. 8,000 / 2 = Rs. 4,000

Since the interest for 2 years is Rs. 8,000, the annual interest is Rs. 4,000.

Interest = Principal × Rate × Time / 100

8,000 = P × R × 2 / 100

R = (8,000 × 100) / (P × 2)

To find P, use the amount after 2 years:

P + 8,000 = Rs. 24,000

P = Rs. 24,000 - Rs. 8,000

P = Rs. 16,000

Substitute P into the formula for R:

R = (8,000 × 100) / (16,000 × 2)

R = 500 / 2

R = 25%

The annual rate of interest is 25%.