Three Workers with Different Efficiencies Sunil completes the work in 4 days, Dinesh in 6 days, and Ramesh works 1.5 times as fast as Sunil. If all three work together from the start, how many days are required to finish the work?

Introduction / Context:
This problem features three different rates, including one given as a multiple of another's efficiency. We must convert all to daily work rates, add them, and invert to get the combined time. It highlights the importance of working with rates rather than times.

Given Data / Assumptions:

• Sunil time = 4 days ⇒ rate = 1/4.
• Dinesh time = 6 days ⇒ rate = 1/6.
• Ramesh is 1.5 times as fast as Sunil ⇒ rate = 1.5*(1/4) = 3/8.
• All three work together without interruption.

Concept / Approach:
Sum the rates: r_total = r_Sunil + r_Dinesh + r_Ramesh. Then total time = 1 / r_total. Keep fractions exact to avoid rounding issues.

Step-by-Step Solution:
r_total = 1/4 + 1/6 + 3/8LCM 24: 1/4 = 6/24, 1/6 = 4/24, 3/8 = 9/24r_total = (6 + 4 + 9)/24 = 19/24 job/dayTime = 1 / (19/24) = 24/19 days ≈ 1.263 days

Verification / Alternative check:
Decimal check: 0.25 + 0.1667 + 0.375 = 0.7917 job/day; inverse ≈ 1.263 days, matching 24/19.

Why Other Options Are Wrong:
7/12, 15/12, and 15/7 do not equal 24/19. Hence the correct selection is “None of these.”

Common Pitfalls:
Using 1.5 days instead of 1.5 times rate; remember “times as fast” multiplies the rate, not the time.

Final Answer:
None of these