Question: A can do \displaystyle \frac{1}{2} of a piece of a work in 5 days, B can do \displaystyle \frac{3}{5} of the same work in 9 days and C can do \displaystyle \frac{2}{3} of the work in 8 days. In how many days can three of them together do the work? (a) 3 days (b) 5 days (c) \displaystyle 4\frac{1}{2} days (d) 4 days

Ans: (d) 4 days

Explanation:

Step 1: Determine the total time taken by A, B, and C to complete the work individually

Total Work = 1 part

A can do \displaystyle\frac{1}{2}​ of the work in 5 days.

Total time for A to complete the work = 5 \times 2 = 10 days.

B can do \displaystyle\frac{3}{5}​ of the work in 9 days.

Total time for B to complete the work = 9 \times \displaystyle\frac{5}{3} = 15 days.

C can do \displaystyle\frac{2}{3}​ of the work in 8 days.

Total time for C to complete the work = 8 \times \displaystyle\frac{3}{2} = 12 days.

Step 2: Total Work

Let the total work = LCM of 10,15 and 12 = 60 units.

Step 3: Calculate the 1-day work of A, B, and C

A’s Efficiency = \displaystyle\frac{60}{10} = 6 units/day.

B’s Efficiency = \displaystyle\frac{60}{15} = 4 units/day.

C’s Efficiency = \displaystyle\frac{60}{12} = 5 units/day.

Step 4: Combined work rate of A, B, and C

Combined Efficiency = 6 + 4 + 5 = 15 units/day.

Step 5: Time taken to complete the work together

The total time required to complete 60 units is:

\text{Time} = \displaystyle\frac{\text{Total work}}{\text{Combined work rate}} = \displaystyle\frac{60}{15} = 4 \, \text{days}.

Final Answer:

The three of them together can complete the work in 4 days.
Option (d): 4 days.