A sum invested at simple interest amounts to Rs. 1392 in 2 years and to Rs. 1488 in 3 years. Using these two successive amounts, what is the rate of interest per annum (in percentage) being applied?

Introduction / Context:
This question uses the property of simple interest that the amount increases by the same fixed interest amount each year. We are given amounts after 2 and 3 years and must determine the annual rate. The difference between the two amounts directly tells us the interest for one year, from which we can find the rate as a percentage of the principal.

Given Data / Assumptions:

• Amount after 2 years A2 = Rs. 1392.
• Amount after 3 years A3 = Rs. 1488.
• Principal P is the same in both cases.
• Simple interest is used, so annual interest is constant.
• Relation: A = P + SI = P * (1 + R * T / 100).

Concept / Approach:
The extra amount from year 2 to year 3 is just one year’s simple interest. Once we know the yearly interest, we subtract 2 years’ interest from A2 to find P, and then compute the rate R using SI = (P * R * T) / 100 with T = 1 year.

Step-by-Step Solution:
Difference between A3 and A2: 1488 - 1392 = Rs. 96. This Rs. 96 is the interest for 1 additional year. So yearly interest I_year = Rs. 96. Interest for 2 years = 2 * 96 = Rs. 192. Principal P = A2 - interest for 2 years = 1392 - 192 = Rs. 1200. Now, for 1 year: SI = 96 = (P * R * 1) / 100 = 1200 * R / 100. So 96 = 12R, giving R = 96 / 12 = 8% per annum.
Verification / Alternative check:
Check using P = 1200 and R = 8%. Amount after 2 years: A2 = 1200 * (1 + 8 * 2 / 100) = 1200 * (1 + 0.16) = 1200 * 1.16 = 1392. Amount after 3 years: A3 = 1200 * (1 + 8 * 3 / 100) = 1200 * (1.24) = 1488. Both match the given values, confirming R = 8%.

Why Other Options Are Wrong:
8.5%, 9%, 10%, and 12% produce different yearly interest amounts and thus lead to different A2 and A3 values. When substituted, they do not maintain the difference of exactly Rs. 96 between the second and third year while also matching the given amounts.

Common Pitfalls:
Students sometimes divide the total 3-year amount difference by 3 or misidentify which difference corresponds to one year. Another common issue is trying to find P and R simultaneously with unnecessary algebra instead of first extracting the yearly interest from the consecutive amounts.

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