A bill of Rs. 1764 is due 2 years hence. If the rate of compound interest is 5% per annum, what is the present worth of this bill?

Introduction / Context:This question deals with present worth under compound interest. You are given a future amount (the sum due after a certain time) and the rate of compound interest, and asked to find its current value. This is a standard type of time value of money calculation, often used in finance and competitive exams to understand how future sums are discounted back to the present.

Given Data / Assumptions:

• Future value (sum due) S = Rs. 1764.
• Rate of compound interest R = 5% per annum.
• Time n = 2 years.
• Compounding is annual.

Concept / Approach:For compound interest, the relationship between present value (P), rate (R), time (n) and future value (S) is:

• S = P * (1 + R/100)^n.

To find the present worth P when S is known, we rearrange the formula as:

• P = S / (1 + R/100)^n.

We simply substitute the numerical values and perform the calculation.

Step-by-Step Solution:Step 1: Write the formula for present worth under compound interest: P = S / (1 + R/100)^n.Step 2: Here S = 1764, R = 5% and n = 2 years.Step 3: Compute the compound factor: (1 + R/100) = 1 + 5/100 = 1.05.Step 4: Now (1.05)^2 = 1.05 * 1.05 = 1.1025.Step 5: Present worth P = 1764 / 1.1025.Step 6: Perform the division: 1764 ÷ 1.1025 = 1600.Step 7: Therefore, the present worth of the bill is Rs. 1600.

Verification / Alternative check:We can verify by moving forward from P to S. If P = 1600 and R = 5%, then after 1 year the amount becomes 1600 * 1.05 = 1680. After another year, it becomes 1680 * 1.05 = 1764. This matches the given future value, confirming that P = 1600 is the correct present worth.

Why Other Options Are Wrong:Rs. 1200 and Rs. 1400 are too low; if used as present worth, compounding at 5% for 2 years would give amounts significantly below 1764. Rs. 1800 and Rs. 1500 are either too high or too low and do not yield exactly 1764 when grown at 5% for 2 years. Only Rs. 1600 produces the correct future value under the given conditions.

Common Pitfalls:

• Using simple interest formula instead of compound interest.
• Forgetting to square the factor (1.05) when the time is 2 years.
• Rounding too early, which can introduce errors into the final answer.
• Confusing present worth with discount calculated in a banker's discount sense.

Final Answer:The present worth of the bill is Rs. 1600.