On simple interest, an amount becomes Rs. 812 in 2 years and Rs. 924 in 4 years. What is the annual rate of simple interest on the original principal?

Introduction / Context:
This question asks you to find the rate of simple interest when you know the total amount after different time periods but not the principal directly. Since the interest under simple interest increases linearly with time, the difference between the amounts at different times reveals the yearly interest.

Given Data / Assumptions:

• Amount after 2 years (A1) = Rs. 812.
• Amount after 4 years (A2) = Rs. 924.
• Principal (P) is the same in both cases and earns simple interest.
• Time difference between the amounts = 2 years.
• The interest is simple, not compound.

Concept / Approach:
Under simple interest, the amount at time T is A = P + T * I, where I is the interest earned per year. Thus, the difference between the amounts at 4 years and 2 years is equal to the interest for the extra 2 years. Once the yearly interest is known, you can find the principal and finally compute the rate R% using R = (I * 100) / P.

Step-by-Step Solution:
Step 1: Interest for the additional 2 years = A2 - A1 = 924 - 812 = Rs. 112. Step 2: This Rs. 112 is the simple interest for 2 years at the same rate on the same principal. Step 3: Hence, yearly interest I per year = 112 / 2 = Rs. 56. Step 4: Amount after 2 years is A1 = P + 2I = P + 2 * 56. Step 5: So 812 = P + 112, giving P = 812 - 112 = Rs. 700. Step 6: Rate R% is given by R = (I * 100) / P = (56 * 100) / 700. Step 7: Compute (56 * 100) / 700 = 5600 / 700 = 8% per annum.

Verification / Alternative check:
Check the values by recomputing the amounts. For P = 700 and R = 8%: yearly interest I = (700 * 8) / 100 = Rs. 56. Amount after 2 years = 700 + 2 * 56 = 812. Amount after 4 years = 700 + 4 * 56 = 924. Both match the given amounts, confirming that the rate is correct.

Why Other Options Are Wrong:
7.2%, 8.5%, and 9.3% would produce yearly interest values that do not match Rs. 56 for a reasonable principal and would therefore not reconstruct both given amounts consistently.

Common Pitfalls:
A common error is to assume that the difference between the two amounts directly gives the rate, instead of using it to find the yearly interest first. Another mistake is to forget that the principal remains the same and to misinterpret the amounts as separate investments. Remember that under simple interest, equal increments in time produce equal increments in interest.

Final Answer:
The annual rate of simple interest on the amount is 8%.