If the true discount on Rs. 2562 due 4 months hence is Rs. 122, what is the annual rate of simple interest?

Introduction / Context:
This problem gives the true discount on a sum due after 4 months and asks for the annual rate of interest. It requires careful handling of time in years and the true discount formula. The effective rate time product must be inferred using algebra, then converted to an annual percentage rate.

Given Data / Assumptions:

• Sum due, S = Rs. 2562.
• True discount TD = Rs. 122.
• Time t = 4 months = 4 / 12 year = 1 / 3 year.
• Rate of simple interest r percent per annum is unknown.
• True discount is calculated under simple interest assumptions.

Concept / Approach:
We use the true discount formula: TD = S * r * t / (100 + r * t) Let r be the annual rate. Time t is 1 / 3 year, so r * t = r / 3. Substituting the known values leads to an equation in r, which can be solved algebraically. After finding r, we interpret it as an annual percentage rate and choose the matching option.

Step-by-Step Solution:
Step 1: Substitute S = 2562, TD = 122, t = 1 / 3 into the formula. 122 = 2562 * r * (1 / 3) / (100 + r / 3). Step 2: Simplify numerator: 2562 * r / 3. So 122 = (2562 * r / 3) / (100 + r / 3). Step 3: Multiply both sides by (100 + r / 3). 122 * (100 + r / 3) = 2562 * r / 3. Step 4: Multiply both sides by 3 to clear the denominator. 366 * (100 + r / 3) = 2562 * r. Step 5: Expand left side: 36600 + 366 * r / 3. 366 * r / 3 = 122r. So 36600 + 122r = 2562r. Step 6: Move terms: 2562r − 122r = 2440r = 36600. Step 7: Solve r = 36600 / 2440 = 15. Thus r = 15 percent per annum.

Verification / Alternative check:
Check by recomputing true discount with r = 15 percent and t = 1 / 3 year. Then r * t = 15 / 3 = 5: TD = 2562 * 5 / (100 + 5) = 2562 * 5 / 105. Multiply: 2562 * 5 = 12810. Divide: 12810 / 105 = 122. This matches the given true discount, confirming the rate is indeed 15 percent per annum.

Why Other Options Are Wrong:
Rates such as 12 percent, 13 and one third percent, 14 percent, or 10 percent produce different true discounts when substituted into the formula, not Rs. 122. Only 15 percent yields exactly the stated true discount for the given sum and time.

Common Pitfalls:
Students often forget to convert 4 months into 1 / 3 year and instead use t = 4, which leads to unrealistic values. Another pitfall is mishandling the algebraic step when clearing denominators or expanding 366 * (100 + r / 3). Careful manipulation of fractions and verifying the result by recomputing the true discount helps avoid such errors.

Final Answer:
The annual rate of simple interest is 15%.