In a cloth showroom during the festival season, a servant's payment for 12 days is Rs. 400 plus one shirt. The servant leaves after working for 9 days and receives Rs. 280 and a shirt. What is the price of the shirt in rupees?

Introduction / Context:
This is a typical linear equation question from profit and wage distribution problems. The salary package includes both cash and a non cash component, namely a shirt. When the servant leaves early, he still receives a shirt plus some cash. Using the idea that wages are proportional to days worked, we can find the monetary value of the shirt by equating the pay for 9 days to a proportion of the 12 day package.

Given Data / Assumptions:

• Payment for 12 days of work = Rs. 400 plus one shirt.
• Payment for 9 days of work = Rs. 280 plus one shirt.
• Daily wage is assumed to be uniform over the period.
• Value of the shirt is the same in both cases.
• We need to find the price of the shirt.

Concept / Approach:
Think of the combined payment (cash plus shirt) as the total wage for 12 days. The worker should get 9/12 of that total for 9 days of work. The problem gives the 9 day payment explicitly, so we set up an equation relating the proportional wage for 9 days to the actual cash plus shirt he receives. From that equation, we isolate the value of the shirt. This is a direct application of proportional reasoning and simple algebra.

Step-by-Step Solution:
Step 1: Let the price of the shirt be S rupees. Step 2: Total wage for 12 days = Rs. 400 + S. Step 3: Since wages are proportional to days, wage for 9 days should be (9/12) * (400 + S) = (3/4) * (400 + S). Step 4: The servant actually receives Rs. 280 + S for 9 days. Step 5: Set up the equation: (3/4) * (400 + S) = 280 + S. Step 6: Multiply both sides by 4: 3 * (400 + S) = 4 * (280 + S). Step 7: Expand: 1200 + 3S = 1120 + 4S. Step 8: Rearrange: 1200 - 1120 = 4S - 3S, so 80 = S. Step 9: Therefore, price of the shirt = Rs. 80.

Verification / Alternative check:
If the shirt costs Rs. 80, then total wage for 12 days = 400 + 80 = Rs. 480. Daily wage = 480 / 12 = Rs. 40. For 9 days, the worker should receive 9 * 40 = Rs. 360. According to the payment structure, he gets 280 cash and a shirt worth 80, total 360, which matches perfectly. This confirms the correctness of S = 80.

Why Other Options Are Wrong:
If S were Rs. 120, 70, 100 or 60, the 9 day payment would not equal three quarters of the 12 day payment, breaking the proportionality condition. Only Rs. 80 maintains consistent daily wages under both scenarios, so the other options are inconsistent with equal pay per day.

Common Pitfalls:
A common mistake is to treat the Rs. 400 and the shirt separately, forgetting to scale the combined value when moving from 12 to 9 days. Some learners subtract 280 from 400 directly without considering that the shirt appears in both payments. Always express the total wage that includes both cash and non cash benefits before applying proportional reasoning.

Final Answer:
The price of the shirt is Rs. 80.