The true discount on Rs 2562 due 4 months hence is Rs 122. Assuming simple interest, what is the annual rate of interest (in percent) used to calculate this true discount?

Introduction / Context:
This question is about finding the rate of simple interest when we know the true discount on a bill, the amount of the bill, and the time until it is due. The true discount connects the amount due, the present value and the interest rate. Identifying the correct rate is important for understanding financial discounting in commercial mathematics.

Given Data / Assumptions:
- Amount due A = Rs 2562.
- True discount TD = Rs 122.
- Time until due = 4 months, which is 1/3 of a year.
- Rate of interest r percent per annum is unknown and must be found.
- Simple interest is assumed throughout.

Concept / Approach:
For simple interest, true discount TD on a sum A due after time t years at rate r percent per annum is given by TD = A * r * t / (100 + r * t). Here A and TD are known, t = 1/3 year, and r is the unknown. We substitute values into the formula and solve the resulting equation for r. This will give the annual rate of interest.

Step-by-Step Solution:
Step 1: Given TD = 122, A = 2562, time t = 4 months = 1/3 year. Step 2: True discount formula under simple interest: TD = A * r * t / (100 + r * t). Step 3: Substitute values: 122 = 2562 * r * (1/3) / (100 + r * (1/3)). Step 4: Simplify 2562 * (1/3) to 854, so 122 = 854 * r / (100 + r/3). Step 5: Cross multiply: 122 * (100 + r/3) = 854 * r. Step 6: Expand left side: 12200 + 122 * r / 3 = 854 r. Step 7: Multiply entire equation by 3 to clear the denominator: 36600 + 122 r = 2562 r. Step 8: Rearranging: 36600 = 2562 r − 122 r = 2440 r. Step 9: Thus r = 36600 / 2440 = 15 percent per annum.

Verification / Alternative check:
Check with r = 15 percent. Compute TD using the formula: TD = 2562 * 15 * (1/3) / (100 + 15 * (1/3)). The numerator becomes 2562 * 5 = 12810. The denominator becomes 100 + 5 = 105. So TD = 12810 / 105 = 122 rupees, which matches the given value. This confirms that the rate is correctly found as 15 percent per annum.

Why Other Options Are Wrong:
Using 12, 13 or 14 percent in the true discount formula does not produce a discount of Rs 122. Instead, the computed true discount is either too small or too large compared with 122. Only a rate of 15 percent provides the exact discount specified in the problem.

Common Pitfalls:
Some learners treat 4 months as 4/10 of a year instead of 4/12, or they mistakenly use the simple interest formula A * r * t / 100 rather than the true discount formula. Errors also arise when clearing denominators. Careful algebra and correct conversion of months to years are essential for answering such questions correctly.

Final Answer:
The annual rate of interest is 15 percent.