Question

Bob can finish a job in 40 days,if he works alone. Alex is twice as fast as Bob and thrice as fast as Cole in the same job.Suppose Alex and Bob work together on the first day,Bob and Cole work together on the second day,Cole and Alex work together on the third day and then,they continue the work by repeating this three-day roster,with Alex and Bob working together on the fourth day and so on.Then,the total number of days Alex would have worked when the job gets finished,is

Answer 1

Three people – Bob, Alex, and Cole – work on a task. Their individual efficiencies (units per day) follow a pattern and the goal is to determine how many days **Alex** works until the task is completed.

Step 1: Determine Efficiencies

Let Bob’s efficiency = 3 units/day. Then:

• Alex’s efficiency = 6 units/day
• Cole’s efficiency = 2 units/day

Bob alone finishes the task in 40 days: \[ \text{Total work} = 40 \times 3 = 120 \text{ units} \]

Step 2: Work Cycle (3-Day Repeating Pattern)

Let’s define the work done in each of the first three days (cycle):

1. Day 1: Alex + Bob = \(6 + 3 = 9\) units
2. Day 2: Bob + Cole = \(3 + 2 = 5\) units
3. Day 3: Alex + Cole = \(6 + 2 = 8\) units

Total work in one 3-day cycle: \[ 9 + 5 + 8 = 22 \text{ units} \] So, every 3 days, 22 units of work is completed.

Step 3: Work Completed in First 15 Days

\[ \text{15 days} = 5 \text{ cycles} \Rightarrow 5 \times 22 = 110 \text{ units} \] Remaining work = \(120 - 110 = 10\) units

Step 4: Day-by-Day After 15 Days

Day 16: Alex + Bob = 6 + 3 = 9 units ⇒ total = 119 units
Day 17: Bob + Cole = 3 + 2 = 5 units ⇒ overshoot, but work completed Hence, task completed on Day 17.

Step 5: Count How Many Days Alex Worked

Alex works on:

• Days 1, 3, 4, 6, 7, 9, 10, 12, 13, 15 (i.e., 10 of first 15 days)
• Day 16 (Alex + Bob)

\[ \boxed{\text{Alex worked for 11 days in total}} \]

Final Answer:

\[ \boxed{11} \]