The present worth of a bill due sometime hence is Rs 1100, and the true discount on the bill is Rs 110. Find the banker's discount and the banker's gain for this transaction.

Introduction:
This problem gives the present worth (PW) of a bill and the true discount (TD), and asks us to find the banker's discount (BD) and the banker's gain (BG). Using the relationships among PW, TD, face value, and the rate-time product r * t, we can determine BD and BG without knowing the exact rate or time individually.

Given Data / Assumptions:
Present worth PW = Rs 1100. True discount TD = Rs 110. Face value P = PW + TD (since P = present worth + true discount). We must find BD and BG.

Concept / Approach:
First, compute the face value: P = PW + TD. Next, recall the relationship for true discount: TD / PW = r * t / 100. This follows because: PW = P * 100 / (100 + r * t) and TD = P − PW. Once we know r * t, we can compute BD using: BD = P * r * t / 100 = P * (r * t) / 100. Finally, BG = BD − TD.

Step-by-Step Solution:
Step 1: Find face value P. P = PW + TD = 1100 + 110 = Rs 1210. Step 2: Use TD / PW to find r * t. TD / PW = 110 / 1100 = 1 / 10. But TD / PW = r * t / 100. So r * t / 100 = 1 / 10 ⇒ r * t = 10. Step 3: Compute BD. BD = P * r * t / 100 = 1210 * 10 / 100. BD = 1210 / 10 = Rs 121. Step 4: Compute BG. BG = BD − TD = 121 − 110 = Rs 11.

Verification / Alternative check:
We can check TD using P = 1210 and r * t = 10: TD = P * r * t / (100 + r * t) = 1210 * 10 / (100 + 10) = 12100 / 110 = Rs 110. This matches the given TD. Then BG = BD − TD = 121 − 110 = 11, consistent with our earlier calculation.

Why Other Options Are Wrong:
All other pairs (BD, BG) shown would either result in a different TD using the standard formulas or yield a different present worth when recomputed. Only BD = Rs 121 and BG = Rs 11 preserve both the given PW and TD.

Common Pitfalls:
Learners sometimes treat PW as equal to the face value P, which is incorrect; the face value is always PW + TD. Another error is to ignore the useful relation TD / PW = r * t / 100, which directly gives the rate-time product and greatly simplifies the calculations.

Final Answer:
The correct values are banker's discount = Rs 121 and banker's gain = Rs 11.