A sum of money becomes ₹34,800 in 2 years and ₹42,000 in 5 years at a rate of…

Concept. Under simple interest the interest earned in every year is the same amount, equal to P × R / 100, where P is the principal (the sum lent) and R is the annual rate. The amount therefore grows in equal yearly steps, A = P + (P × R × T) / 100. A direct consequence is that the difference between the amounts at two different times is exactly the interest earned over the years in between, and nothing else.

Application.

1. Interest earned over the 3 years between year 2 and year 5 = ₹42,000 − ₹34,800 = ₹7,200.

2. Because equal interest accrues each year, the interest for one year = ₹7,200 ÷ 3 = ₹2,400.

3. The amount at 2 years already contains 2 years of interest, so the principal P = ₹34,800 − (2 × ₹2,400) = ₹34,800 − ₹4,800 = ₹30,000.

4. The annual rate R = (interest for one year × 100) / P = (2,400 × 100) / 30,000 = 8 per cent per annum.

Cross-check.

• Interest for 2 years = (30,000 × 8 × 2) / 100 = ₹4,800, so the amount is ₹30,000 + ₹4,800 = ₹34,800, matching the value given for 2 years.

• Interest for 5 years = (30,000 × 8 × 5) / 100 = ₹12,000, so the amount is ₹30,000 + ₹12,000 = ₹42,000, matching the value given for 5 years.

Contrast. Had the money grown under compound interest, the yearly increases would not have been equal, so the ₹7,200 rise across three years could not simply be divided by 3. The equal-step property of simple interest is exactly what licenses that division. Hence the rate is 8 per cent per annum and the sum of money is ₹30,000.