A sum becomes Rs. 1392 in 2 years and Rs. 1488 in 3 years at simple interest. What is the annual rate of simple interest (in percent per annum)?

Introduction / Context:
This question shows how to use successive amounts at different times under simple interest to find both the yearly interest and the rate. Because simple interest adds the same amount of interest every year, the difference between the amounts at 2 years and 3 years is exactly the interest earned in one additional year. Once we know the interest per year, we can work backwards to find the principal and then compute the annual rate of interest in percent per annum.

Given Data / Assumptions:

• Amount after 2 years, A2 = Rs. 1392.
• Amount after 3 years, A3 = Rs. 1488.
• Interest is calculated using simple interest.
• The rate remains constant over the entire period.

Concept / Approach:
For simple interest, the amount after T years is A = P + SI, and SI is equal to P * R * T / 100. The change in amount from year to year is constant and equal to the yearly simple interest. Here, A3 - A2 gives the interest for exactly one year. Once we find that yearly interest, we can subtract two years of interest from A2 to find principal P, then apply the simple interest formula SI = P * R * 1 / 100 to determine R. This method is simpler and faster than setting up two full equations and solving them simultaneously.

Step-by-Step Solution:
Step 1: Compute yearly interest by subtracting the two amounts: A3 - A2 = 1488 - 1392. Step 2: 1488 - 1392 = 96, so yearly simple interest = Rs. 96. Step 3: Let the principal be P. Amount after 2 years: A2 = P + 2 * 96. Step 4: So 1392 = P + 192. Step 5: Therefore P = 1392 - 192 = 1200. Step 6: Yearly interest is SI1 = Rs. 96 for 1 year on P = 1200. Step 7: Use SI = P * R * T / 100 for T = 1 year: 96 = 1200 * R * 1 / 100. Step 8: Simplify: 96 = 1200 * R / 100 = 12R. Step 9: So R = 96 / 12 = 8. Step 10: The annual simple interest rate is 8% per annum.

Verification / Alternative check:
Check using P = 1200 and R = 8%. Amount after 2 years: SI2 = 1200 * 8 * 2 / 100 = 192, so A2 = 1200 + 192 = 1392. After 3 years: SI3 = 1200 * 8 * 3 / 100 = 288, so A3 = 1200 + 288 = 1488. Both values match the given amounts exactly, confirming that the rate is indeed 8% per annum.

Why Other Options Are Wrong:
If R = 10%, yearly interest would be 120 rupees, leading to A2 = 1440 and A3 = 1560, not matching the problem. If R = 12%, yearly interest would be 144 rupees, giving A2 = 1488 after 2 years, which does not match 1392. If R = 8.5%, yearly interest would be 1200 * 8.5 / 100 = 102, which does not match the difference of 96. Only R = 8% fits all conditions exactly.

Common Pitfalls:
Students sometimes incorrectly assume compound interest and attempt to use powers, which is unnecessary here. Another mistake is to miscalculate the difference between A3 and A2 or forget that this difference represents exactly one year of simple interest. Some also overlook the straightforward method and instead set up two equations, increasing chances of algebraic errors. Always look for such yearly differences in simple interest problems to simplify your work.

Final Answer:
The annual simple interest rate is 8% per annum.