If the true discount on a sum due 2 years hence at 14 percent per annum simple interest is Rs 168, what is the sum (amount due at the end of 2 years)?

Introduction / Context:
This is a direct application of the true discount formula in simple interest. The true discount on a sum due after a certain period and at a specified interest rate is given, and we must find the sum due at maturity. Problems like this help in understanding how present value and true discount relate to the face value of a bill.

Given Data / Assumptions:
- True discount TD = Rs 168.
- Time until due = 2 years.
- Rate of interest r = 14 percent per annum simple interest.
- Let the sum due (face value) be A rupees.

Concept / Approach:
For simple interest, the relationship between amount A, true discount TD, rate r and time t is TD = A * r * t / (100 + r * t). This formula arises from the difference between the amount and its present value. Given TD, r and t, we can plug into this expression and solve for A directly. Here, t = 2 years and r = 14 percent, so r * t = 28. The calculation then becomes straightforward.

Step-by-Step Solution:
Step 1: Let sum due be A rupees. Step 2: Given true discount TD = Rs 168, rate r = 14 percent, time t = 2 years. Step 3: Under simple interest, TD = A * r * t / (100 + r * t) = A * 14 * 2 / (100 + 28). Step 4: Simplify numerator: 14 * 2 = 28. Denominator: 100 + 28 = 128. Step 5: So TD = A * 28 / 128. Step 6: We are given TD = 168, so 168 = A * 28 / 128. Step 7: Rearranging, A = 168 * 128 / 28. Step 8: Compute 168 / 28 = 6, so A = 6 * 128 = Rs 768.

Verification / Alternative check:
Find the present value P and verify. Under simple interest, the present value P of A due after time t at rate r is P = A * 100 / (100 + r * t). Here P = 768 * 100 / 128 = 600 rupees. Simple interest on P for 2 years at 14 percent is 600 * 14 * 2 / 100 = 168 rupees, which equals the given true discount. This confirms that the sum due A is correctly found as Rs 768.

Why Other Options Are Wrong:
Options Rs 948, Rs 876 and Rs 658 each produce a different true discount when used in the formula, and none of those values equals Rs 168. Only A = Rs 768 satisfies the condition TD = 168 at 14 percent per annum for 2 years.

Common Pitfalls:
A common mistake is to treat 168 as the simple interest on the sum due directly or to use the simple interest formula TD = A * r * t / 100, which is incorrect for true discount. Another error is forgetting to add r * t in the denominator. Always remember that TD is calculated with the expression TD = A * r * t / (100 + r * t) under simple interest.

Final Answer:
The sum due at the end of 2 years is Rs 768.