The true discount on a certain sum of money due 3 years hence is Rs. 250, and the simple interest on the same sum for the same time and at the same rate is Rs. 375. What is the annual rate of interest?

Introduction / Context:
This question is similar in pattern to an earlier one but with larger numbers. Both true discount and simple interest are given for the same sum, same time, and same rate. The relationship between SI and TD helps us find the effective rate time product and then the annual rate. It strengthens understanding of the connection between discounted values and interest over time.

Given Data / Assumptions:

• True discount TD = Rs. 250.
• Simple interest SI = Rs. 375.
• Time t = 3 years.
• Rate of interest r percent per annum is unknown.
• Both TD and SI are calculated on the same sum S.

Concept / Approach:
For sum S, time t, and rate r, the formulas are: SI = S * r * t / 100 TD = S * r * t / (100 + r * t) The ratio SI / TD simplifies to: SI / TD = (100 + r * t) / 100. We can therefore compute r * t from the known numerical ratio and then divide by t = 3 to get r.

Step-by-Step Solution:
Step 1: Compute SI / TD from the data. SI / TD = 375 / 250 = 3 / 2 = 1.5. Step 2: Set this equal to (100 + r * t) / 100. 3 / 2 = (100 + r * t) / 100. Step 3: Cross multiply: 3 * 100 = 2 * (100 + r * t). 300 = 200 + 2 * r * t. Step 4: 2 * r * t = 100, so r * t = 50. Step 5: Time t = 3 years, so r = (r * t) / t = 50 / 3. Step 6: 50 / 3 = 16 whole and 2 thirds percent, which is (16 + 2/3)%.

Verification / Alternative check:
We may find S and check. Using SI = S * r * t / 100: 375 = S * 50 / 100, so S = 375 * 100 / 50 = 750. Now compute TD with S = 750 and r * t = 50: TD = S * r * t / (100 + r * t) = 750 * 50 / 150 = 750 * 1 / 3 = 250. This matches the given true discount, confirming that r = 50 / 3 percent per year, that is (16 + 2/3)% is correct.

Why Other Options Are Wrong:
Rates such as (16 + 1/3)% or (16 + 4/3)% correspond to r * t values less than or greater than 50 respectively. They would not preserve the ratio SI / TD = 3 / 2 and therefore cannot match the problem data. The 20 percent option is also incorrect, as it gives r * t = 60, not 50.

Common Pitfalls:
Mistakes often occur when simplifying the ratio 375 / 250 or when solving the cross multiplication step. Some students also forget that both TD and SI refer to the same sum, which is why the ratio eliminates S and simplifies the problem. Carefully handling the algebra and keeping track of the effective product r * t avoids these errors.

Final Answer:
The annual rate of interest is (16 + 2/3)%.