If the true discount on a sum due 2 years hence at 14% simple interest per annum is Rs. 168, what is the sum that will be due at the end of 2 years?

Introduction / Context:
Here we are asked to find the face value of a sum due after 2 years when we know the true discount allowed today and the simple interest rate. This tests the relation between true discount, sum due, interest rate, and time, and gives practice with rearranging the true discount formula to obtain the principal amount.

Given Data / Assumptions:

• True discount TD = Rs. 168.
• Time t = 2 years.
• Rate of simple interest r = 14 percent per annum.
• Sum due after 2 years, S, is unknown.
• Standard true discount formula under simple interest holds.

Concept / Approach:
For a sum S due after time t at simple interest rate r, the true discount is: TD = S * r * t / (100 + r * t) We know TD, r, and t. Let k = r * t = 14 * 2 = 28. Substitute TD and k into the formula, solve for S, and then identify the correct option.

Step-by-Step Solution:
Step 1: Compute k = r * t = 14 * 2 = 28. Step 2: Write TD formula: TD = S * k / (100 + k). Step 3: Substitute TD = 168, k = 28. 168 = S * 28 / (100 + 28) = S * 28 / 128. Step 4: Rearrange to solve for S. S = 168 * 128 / 28. Step 5: Simplify 168 / 28 = 6, so S = 6 * 128 = 768. Thus the sum due after 2 years is Rs. 768.

Verification / Alternative check:
Check by computing the present worth PW and confirming the true discount. With S = 768, k = 28: PW = S * 100 / (100 + k) = 768 * 100 / 128 = 76800 / 128 = 600. TD = S − PW = 768 − 600 = 168. This matches the given true discount, so S = 768 is correct.

Why Other Options Are Wrong:
Values such as Rs. 600, Rs. 668, Rs. 700, or Rs. 878 would not produce a true discount of exactly Rs. 168 when used with the same rate and time. For example, if S were Rs. 600, the true discount would be lower and would not match the given 168.

Common Pitfalls:
Students sometimes confuse true discount with simple interest and use SI = S * r * t / 100 directly, which would give a different result. Another frequent error is forgetting to add r * t to 100 in the denominator. Always compute k = r * t first, write TD = S * k / (100 + k), and then solve carefully for S, reducing fractions step by step to avoid arithmetic mistakes.

Final Answer:
The sum due after 2 years is Rs. 768.