A certain sum of money amounts to Rs 918 in 2 years and Rs 969 in 3.5 years at simple interest; what is the annual rate of simple interest in percent per annum for this sum?

Introduction / Context:
This question provides the total amounts at two different times for the same principal under simple interest. The aim is to use the difference between the amounts to work out the yearly interest and then derive the annual rate. It tests understanding that under simple interest, the interest grows linearly with time and that amount is principal plus interest.

Given Data / Assumptions:
- Amount after 2 years, A2 = Rs 918. - Amount after 3.5 years, A3.5 = Rs 969. - The principal P is the same in both cases. - Interest is calculated using simple interest. - Required quantity is the annual simple interest rate R in percent per annum.

Concept / Approach:
Under simple interest, amount A = P + SI, where SI = (P * R * T) / 100. The difference between amounts at two times represents the interest earned during the additional time interval. Here, the difference between 3.5 years and 2 years is 1.5 years. By subtracting the two amounts, we directly compute the interest for 1.5 years. Dividing this interest by 1.5 gives the interest per year. Then we use the yearly interest and principal to find the annual rate.

Step-by-Step Solution:
Step 1: Compute the difference between the two amounts: A3.5 − A2 = 969 − 918. Step 2: 969 − 918 = 51, so interest for the extra 1.5 years is Rs 51. Step 3: Interest per year = 51 / 1.5. Step 4: Compute 51 / 1.5 = 34, so yearly simple interest is Rs 34. Step 5: Use A2 to find principal: A2 = P + SI for 2 years. Step 6: SI for 2 years = 2 * yearly interest = 2 * 34 = Rs 68. Step 7: Therefore P = A2 − SI2 = 918 − 68 = Rs 850. Step 8: Now yearly interest is Rs 34 on principal Rs 850, so use R = (SI per year * 100) / P. Step 9: R = (34 * 100) / 850 = 3,400 / 850 = 4. Step 10: Thus R = 4 percent per annum.

Verification / Alternative check:
Check with P = 850 and R = 4 percent. For 2 years, SI2 = (850 * 4 * 2) / 100 = (850 * 8) / 100 = 6,800 / 100 = Rs 68, so A2 = 850 + 68 = 918, which matches the first amount. For 3.5 years, SI3.5 = (850 * 4 * 3.5) / 100 = (850 * 14) / 100 = 11,900 / 100 = Rs 119, giving A3.5 = 850 + 119 = 969, matching the second amount. This confirms that 4 percent is correct.

Why Other Options Are Wrong:
If R were 5 percent, the yearly interest would be (850 * 5) / 100 = 42.50, which does not fit the observed increase of Rs 51 over 1.5 years. At 6 percent or 8 percent, the interest per year is even larger and would produce amounts that do not match Rs 918 and Rs 969. Only 4 percent leads to both amounts being consistent with simple interest growth.

Common Pitfalls:
Some learners incorrectly treat the difference between amounts as the yearly interest instead of interest over the extra time. Others forget that 3.5 years is 1.5 years more than 2 years and miscalculate the interval. Carefully computing interest over the additional time and then converting it to an annual figure avoids these mistakes.

Final Answer:
The annual rate of simple interest is 4% per annum.