The simple interest on Rs. 750 for 2 years is the same as the true discount on Rs. 960 due 2 years hence. If the rate of interest is the same in both cases, what is the annual rate of interest?

Introduction / Context:
This is a conceptual question connecting simple interest and true discount. The same rate and time apply to two different sums but through different formulas. The simple interest on one sum equals the true discount on another. The task is to find the common rate of interest, which requires equating the algebraic expressions for simple interest and true discount and solving for the rate.

Given Data / Assumptions:

• Principal for simple interest, P1 = Rs. 750.
• Time for simple interest, t1 = 2 years.
• Bill amount for true discount, S2 = Rs. 960, due after t2 = 2 years.
• Same rate of interest r percent per annum in both cases.
• Simple interest on Rs. 750 for 2 years equals true discount on Rs. 960 for 2 years.

Concept / Approach:
Simple interest SI is given by: SI = P * r * t / 100 True discount TD for sum S due after time t is: TD = S * r * t / (100 + r * t) We are told: SI on 750 for 2 years = TD on 960 for 2 years. We set up the equations in terms of r and solve for r. Since time is 2 years in both cases, we can simplify the algebra.

Step-by-Step Solution:
Step 1: Compute SI expression. SI = 750 * r * 2 / 100 = 1500r / 100 = 15r. Step 2: Write TD expression for S2 = 960 and t2 = 2 years. TD = 960 * r * 2 / (100 + 2r) = 1920r / (100 + 2r). Step 3: Equate SI and TD. 15r = 1920r / (100 + 2r). Step 4: Since r is non zero, divide both sides by r. 15 = 1920 / (100 + 2r). Step 5: Cross multiply: 15 * (100 + 2r) = 1920. 1500 + 30r = 1920. Step 6: 30r = 1920 − 1500 = 420. Step 7: r = 420 / 30 = 14. So the annual rate of interest is 14 percent.

Verification / Alternative check:
Compute SI at 14 percent: SI = 15 * 14 = Rs. 210. Compute TD on Rs. 960 at the same rate and time. Here r * t = 28: TD = 960 * 28 / (100 + 28) = 960 * 28 / 128. Simplify 960 / 128 = 7.5, so TD = 7.5 * 28 = 210. Both SI and TD equal Rs. 210, which confirms the value r = 14 percent.

Why Other Options Are Wrong:
Rates like 12 percent, 15 percent, 16 percent, or 10 percent do not satisfy the equality condition between SI and TD when substituted into the formulas. They give mismatched values for simple interest and true discount and therefore cannot be correct.

Common Pitfalls:
A common mistake is to directly treat SI and TD as both equal to P * r * t / 100 and ignore the denominator 100 + r * t in the true discount formula. This leads to a trivial equality and no solution. Students may also forget to divide through by r or mishandle cross multiplication. Carefully writing both expressions and keeping the denominator intact is key to solving such problems correctly.

Final Answer:
The annual rate of interest is 14%.