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)?

Introduction / Context:
This question involves finding the rate of simple interest when the amounts at two different times are given but the principal is not given directly. It is a classic type of problem where you deduce the yearly interest by looking at how much the amount increases over time and then back calculate both the principal and the rate.

Given Data / Assumptions:

• Amount after 2 years, A1 = Rs 918.
• Amount after 3.5 years, A2 = Rs 969.
• Same principal P and same simple interest rate R% per annum for both periods.
• Time periods: T1 = 2 years and T2 = 3.5 years.

Concept / Approach:
Under simple interest, the amount grows linearly with time. The difference between the amounts at two times gives the simple interest for the extra time between those periods. Once we know the yearly interest, we can find the rate by relating it to the principal and total interest formula. The key formulas are:
A = P + SISI = (P * R * T) / 100

Step-by-Step Solution:
Step 1: Find the increase in amount between 2 years and 3.5 years.Increase = A2 - A1 = 969 - 918 = Rs 51Step 2: Determine the extra time corresponding to this increase.Extra time = 3.5 - 2 = 1.5 yearsStep 3: Calculate the interest per year.Interest for 1.5 years = Rs 51Interest per year = 51 / 1.5 = Rs 34Step 4: Find the principal using the 2 year amount.At 2 years, SI2 = 2 * 34 = Rs 68Therefore, P = A1 - SI2 = 918 - 68 = Rs 850Step 5: Use the simple interest formula to find R.For 1 year, SI1 = 3434 = (P * R * 1) / 100 = (850 * R) / 100R = (34 * 100) / 850 = 3400 / 850 = 4So, R = 4% per annum.

Verification / Alternative check:
Check using the principal and rate. With P = 850 and R = 4%: For 2 years, SI2 = (850 * 4 * 2) / 100 = 68, amount A1 = 850 + 68 = 918. For 3.5 years, SI3.5 = (850 * 4 * 3.5) / 100 = 119, amount A2 = 850 + 119 = 969. Both match the given values, confirming that the rate is 4%.

Why Other Options Are Wrong:

• 5%, 6%, and 8% would each yield different amounts at 2 years and 3.5 years that do not match Rs 918 and Rs 969 respectively.
• Only 4% produces the correct pair of amounts for the given times.

Common Pitfalls:
A common mistake is to treat Rs 51 as the interest for just 1 year instead of 1.5 years, which leads to an incorrect calculation of the interest per year. Another error is to attempt solving directly with two equations in two unknowns (P and R) without first using the idea of linear growth, which can make the algebra more cumbersome. Correctly associating the increase in amount with the extra time is the key insight for solving these problems efficiently.

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