A and B together can do a piece of work in 16 days. B and C together can do the same work in 32 days. C can complete the same work in 80 days. After A has worked for 4 days, B for 12 days, time taken by C to complete the remaining job is x days.
P, Q and R take (x - 28) days, (x - 18) days and (x - 8) days respectively to complete an another job. The three work in a rotation to complete the job with only 1 person working on a day. Who should start the job so that the job is completed in the least possible time?

Options

A. P
B. Q
C. R
D. Any one of the three

Answer

To solve the problem step-by-step, we will break it down into manageable parts.

### Step 1: Determine the efficiencies of A, B, and C

1. **A and B together can do the work in 16 days.**
- Efficiency of A + B = Total Work / Time = 160 / 16 = 10

2. **B and C together can do the work in 32 days.**
- Efficiency of B + C = Total Work / Time = 160 / 32 = 5

3. **C can complete the work in 80 days.**
- Efficiency of C = Total Work / Time = 160 / 80 = 2

Now we have:
- Efficiency of A + B = 10
- Efficiency of B + C = 5
- Efficiency of C = 2

### Step 2: Find the efficiency of A and B separately

Using the equations:
- From A + B = 10 and C = 2, we can find B:
- B + C = 5
- B + 2 = 5 → B = 3

Now substituting B into A + B:
- A + 3 = 10 → A = 7

So, we have:
- Efficiency of A = 7
- Efficiency of B = 3
- Efficiency of C = 2

### Step 3: Calculate the work done by A, B, and C

1. **A works for 4 days:**
- Work done by A = Efficiency of A × Days worked = 7 × 4 = 28

2. **B works for 12 days:**
- Work done by B = Efficiency of B × Days worked = 3 × 12 = 36

3. **Total work done by A and B:**
- Total work done = 28 + 36 = 64

4. **Remaining work:**
- Remaining work = Total Work - Work done = 160 - 64 = 96

### Step 4: Calculate time taken by C to complete the remaining work

1. **C's efficiency is 2:**
- Time taken by C = Remaining Work / Efficiency of C = 96 / 2 = 48 days

Thus, \( x = 48 \).

### Step 5: Determine the time taken by P, Q, and R

1. **P takes (x - 28) days:**
- P = 48 - 28 = 20 days

2. **Q takes (x - 18) days:**
- Q = 48 - 18 = 30 days

3. **R takes (x - 8) days:**
- R = 48 - 8 = 40 days

### Step 6: Calculate efficiencies of P, Q, and R

1. **Efficiency of P:**
- Efficiency of P = Total Work / Time = 160 / 20 = 8

2. **Efficiency of Q:**
- Efficiency of Q = Total Work / Time = 160 / 30 = 5.33

3. **Efficiency of R:**
- Efficiency of R = Total Work / Time = 160 / 40 = 4

### Step 7: Determine the optimal starting person

1. **Total work done in one cycle (3 days):**
- Work done in 3 days = Efficiency of P + Efficiency of Q + Efficiency of R = 8 + 5.33 + 4 = 17.33

2. **Total cycles needed:**
- Total work = 160
- Cycles = Total work / Work done in 3 days = 160 / 17.33 ≈ 9.23 cycles

3. **Remaining work after 9 cycles:**
- Work done in 9 cycles = 9 × 17.33 = 156
- Remaining work = 160 - 156 = 4

4. **Determine who should do the remaining work:**
- If P starts, he can do 8 work in a day.
- If Q starts, he can do 5.33 work in a day.
- If R starts, he can do 4 work in a day.

Since P has the highest efficiency, he should start the job to complete it in the least possible time.

### Final Answer:
**P should start the job.**

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