Introduction / Context:This is a straightforward simple interest comparison question. The same principal is invested at two different rates for the same time period. We are told the difference in the interests obtained and asked to find the difference in the rates. It illustrates how small changes in rate translate into interest differences for a fixed principal and time.Given Data / Assumptions:
- Principal (P) = Rs. 1,500
- Time (T) = 3 years
- Difference in simple interests from two sources = Rs. 13.50
- Let the two rates be R1 and R2 percent per annum
- We must find |R1 - R2|
Concept / Approach:Simple interest for a given principal and time is directly proportional to the rate. Therefore, if we denote the difference in rates by d percent, the difference in interests can be written as:
Difference in SI = (P * d * T) / 100We can set this equal to Rs. 13.50 and solve for d.
Step-by-Step Solution:Let d = |R1 - R2|Difference in SI = (1500 * d * 3) / 100This must equal 13.50So (1500 * 3 * d) / 100 = 13.50(4500 * d) / 100 = 13.5045d = 13.50d = 13.50 / 45d = 0.3Thus the difference between the two rates is 0.3% per annumVerification / Alternative check:If the difference in rates is 0.3%, then the difference in interest for each year is:
(1500 * 0.3) / 100 = Rs. 4.50 per yearOver 3 years this becomes 4.50 * 3 = Rs. 13.50, which matches the given difference. This confirms that the calculated rate difference is correct.
Why Other Options Are Wrong:If the difference were 0.1%, the interest difference over 3 years would be 1500 * 0.1 * 3 / 100 = Rs. 4.50, not 13.50. For 0.2%, it would be Rs. 9. For 0.4%, it would be Rs. 18. Only 0.3% produces a difference of Rs. 13.50.
Common Pitfalls:Sometimes students confuse the absolute rates with their difference and try to assign arbitrary values to R1 and R2. There is no need to do this. It is enough to work with the difference in rates as a single variable. Another error is mishandling decimal arithmetic when dividing 13.50 by 45.
Final Answer:The difference between the two rates of interest is
0.3% per annum.
