The banker's discount and the true discount on a sum of money due 8 months hence are Rs 120 and Rs 110 respectively. Find the face value (sum) of the bill.

Introduction:
This question provides both the banker's discount (BD) and the true discount (TD) for a bill due after 8 months and asks us to determine the sum (face value). We do not know the interest rate directly, but the ratio BD / TD helps us eliminate it and solve for the face value.

Given Data / Assumptions:
Banker's discount BD = Rs 120. True discount TD = Rs 110. Time t = 8 months = 8/12 = 2/3 year. Rate r (unknown, but same for both discounts). We must find the sum P (face value).

Concept / Approach:
Let x = r * t. Then: BD = P * x / 100. TD = P * x / (100 + x). The ratio BD / TD becomes: BD / TD = (100 + x) / 100. We use the given BD and TD to find x, then use TD = P * x / (100 + x) to solve for P directly.

Step-by-Step Solution:
Step 1: Use BD / TD to find x. BD / TD = 120 / 110 = 12 / 11. But BD / TD = (100 + x) / 100. (100 + x) / 100 = 12 / 11. 100 + x = 100 * 12 / 11 = 1200 / 11. x = 1200 / 11 − 100 = 1200 / 11 − 1100 / 11 = 100 / 11. So x = r * t = 100 / 11. Step 2: Use TD formula to find P. TD = P * x / (100 + x). Here 100 + x = 1200 / 11. 110 = P * (100 / 11) / (1200 / 11). 110 = P * (100 / 11) * (11 / 1200) = P * 100 / 1200 = P / 12. Therefore P / 12 = 110 ⇒ P = 110 * 12 = Rs 1320.

Verification / Alternative check:
With P = 1320 and x = 100 / 11: BD = P * x / 100 = 1320 * (100 / 11) / 100 = 1320 / 11 = Rs 120. TD = P * x / (100 + x) = 1320 * (100 / 11) / (1200 / 11) = 1320 * 100 / 1200 = Rs 110. This matches the given values.

Why Other Options Are Wrong:
Rs 1457, Rs 1140, Rs 1260, Rs 1200: None of these sums produce BD = 120 and TD = 110 simultaneously under the standard formulas, so they are inconsistent with the given data.

Common Pitfalls:
A typical mistake is trying to assume a rate and then solving for time or vice versa, which is unnecessary. The cleanest method is to treat x = r * t as a single unknown and use the relationships involving BD, TD, and x to determine P directly.

Final Answer:
The face value (sum) of the bill is Rs 1320.