A, B, C can finish alone in 8, 12, 16 days; total payment ₹1440: With D's help, they finish in 3 days. Assuming proportional sharing, how much does D get?

Introduction / Context:
When several workers collaborate, total daily rate is the sum of individual rates. Once the group completion time is known, each worker’s contribution equals their rate multiplied by total days. Payment is then split by contribution fraction.

Given Data / Assumptions:

• A alone: 8 days ⇒ rate(A) = 1/8.
• B alone: 12 days ⇒ rate(B) = 1/12.
• C alone: 16 days ⇒ rate(C) = 1/16.
• All four finish in 3 days ⇒ group rate = 1/3.
• Total payment = ₹1440.

Concept / Approach:
First find rate(A+B+C). Subtract from group rate to get rate(D). Multiply rate(D) by 3 days to get D’s share of total work, and then D’s share of money.

Step-by-Step Solution:
rate(A) + rate(B) + rate(C) = 1/8 + 1/12 + 1/16.Use denominator 48: 6/48 + 4/48 + 3/48 = 13/48.Group rate = 1/3 = 16/48 ⇒ rate(D) = 16/48 − 13/48 = 3/48 = 1/16.D’s share of work in 3 days = 3 * (1/16) = 3/16.D’s payment = (3/16) * 1440 = 270.

Verification / Alternative check:
A, B, C contribute 13/48 per day; D contributes 1/16 per day. Over 3 days, total work is (13/48 + 1/16) * 3 = (13/48 + 3/48) * 3 = (16/48) * 3 = 1, as required.

Why Other Options Are Wrong:

• ₹265 and ₹300 do not match the exact 3/16 share of ₹1440.
• “None of these” is unnecessary since a precise value exists.

Common Pitfalls:

• Miscalculating common denominators or forgetting to multiply by the total number of days.

Final Answer:
₹ 270