The true discount on a bill due 9 months hence at 16 percent per annum simple interest is Rs 189. What is the amount (face value) of the bill?

Introduction / Context:
Here we have a true discount question with the amount of the discount given, the rate of simple interest, and the time until the bill is due. We are asked to find the face value (amount) of the bill. This is a typical problem in commercial mathematics where present value and true discount concepts are applied to find unknown amounts.

Given Data / Assumptions:
- True discount TD = Rs 189.
- Time until due = 9 months = 3/4 of a year.
- Rate of interest r = 16 percent per annum simple interest.
- Let the amount of the bill be A rupees.

Concept / Approach:
Under simple interest, the true discount TD for a sum A due after time t years at rate r percent per annum is TD = A * r * t / (100 + r * t). We know TD, r and t, so we can substitute and solve for A. Here t = 3/4 year and r = 16 percent, which makes r * t equal to 12. After writing down the formula with these values, we solve the resulting linear equation for A.

Step-by-Step Solution:
Step 1: Let amount of the bill be A. Step 2: Given TD = 189, r = 16 percent, time t = 9 months = 3/4 year. Step 3: True discount formula: TD = A * r * t / (100 + r * t). Step 4: Compute r * t = 16 * 3/4 = 12. Therefore TD = A * 16 * 3/4 / (100 + 12). Step 5: Simplify numerator: 16 * 3/4 = 12. Denominator = 112. Step 6: So TD = A * 12 / 112 = A * 3 / 28. Step 7: Given TD = 189, so 189 = 3A / 28. Step 8: Solve for A: A = 189 * 28 / 3. Step 9: Compute 189 / 3 = 63 and 63 * 28 = 1764. So A = Rs 1764.

Verification / Alternative check:
Verify by computing the true discount from the found amount. Present value P is A * 100 / (100 + r * t) = 1764 * 100 / 112 = 1764 * 25 / 28. Compute 1764 / 28 = 63, so P = 63 * 25 = 1575. True discount TD = A − P = 1764 − 1575 = 189, which matches the given discount. This confirms that A has been correctly determined.

Why Other Options Are Wrong:
Options 1200, 1600 and 1354 do not satisfy the true discount formula when used as A. For each of these, the calculated true discount does not equal Rs 189, so they cannot be the correct face value of the bill.

Common Pitfalls:
Learners sometimes misconvert 9 months into a fraction of a year, or they use the simple interest formula A * r * t / 100 instead of the correct true discount formula with 100 + r * t in the denominator. Errors in simplifying fractions also frequently lead to wrong answers. Careful handling of r * t and the algebraic steps is necessary.

Final Answer:
The amount of the bill is Rs 1764.