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

Introduction / Context:
This question provides both the banker's discount and the true discount on a sum due after 8 months and asks us to find the underlying face value. Using the standard formulas and the given pair of discounts, we can eliminate the interest rate and time by working with ratios and then solve for the face value without being told the rate explicitly.

Given Data / Assumptions:
- Banker's discount BD = Rs 120
- True discount TD = Rs 110
- Time until the sum is due = 8 months = 2 / 3 year
- Rate of simple interest r is unknown but same for both BD and TD
- Let face value of the bill be F

Concept / Approach:
Banker's discount is given by BD = F × r × t / 100, while true discount is TD = F × r × t / (100 + r × t). The ratio BD / TD equals (100 + r t) / 100. We can also use the true discount equation to link F, r t and TD. By expressing everything in terms of the single product r t, we can first find r t from one equation and then solve for F using either the BD or TD expression.

Step-by-Step Solution:
Step 1: Let x = r × t. Then BD = F × x / 100 = 120 and TD = F × x / (100 + x) = 110.Step 2: From BD we get F = 120 × 100 / x = 12000 / x.Step 3: Substitute this into the TD formula: 110 = (12000 / x) × x / (100 + x) = 12000 / (100 + x).Step 4: Rearranging, 12000 = 110 × (100 + x) = 11000 + 110x.Step 5: So 12000 − 11000 = 110x and 1000 = 110x, giving x = 1000 ÷ 110 = 100 ÷ 11.Step 6: Substitute x back into F = 12000 / x: F = 12000 × 11 / 100 = 1320.

Verification / Alternative check:
We can verify the answer by recomputing BD and TD using F = 1320. Since x = r t = 100 / 11, BD = F × x / 100 = 1320 × (100 / 11) / 100 = 1320 ÷ 11 = 120, which matches the given BD. True discount TD = F × x / (100 + x) = 1320 × (100 / 11) / (100 + 100 / 11), which simplifies to 1320 × (100 / 11) × 11 / 120 = 1320 × 100 / 120 = 132000 / 120 = 110, matching the given TD. This confirms that F = 1320 is correct.

Why Other Options Are Wrong:
Values like 1140, 1260 or 1457 would lead to BD and TD that do not equal 120 and 110 when calculated using any positive simple interest rate over 8 months. The option 1540 is also incompatible with the exact ratio between BD and TD implied by the problem. Only 1320 satisfies both discount equations simultaneously.

Common Pitfalls:
Students may try to assume a rate r and then work numerically, which is unnecessary and can introduce error. Others confuse BD − TD with the sum or use simple interest formulas incorrectly. Working with the product r t and keeping equations symbolic until the final substitution is a safer and more general method.

Final Answer:
The face value of the bill is 1320 rupees.