A working alone & B working alone can complete a piece of work in 15 days & 9 days respectively. B & C working together can complete same work in 6 days. If A & C starts working together on same work, then find in how many days the work will be completed?

Options

A. A)`6(1)/(2)`days
B. B)`7(2)/(3)`days
C. C)`5(3)/(4)`days
D. D)`8(2)/(11)`days

Answer

To solve the problem step by step, we will first determine the work efficiencies of A, B, and C based on the information provided.

### Step 1: Determine the work efficiencies of A and B
- A can complete the work in 15 days.
- B can complete the work in 9 days.

To find their efficiencies, we can calculate the amount of work they do in one day:
- Work done by A in one day = 1/15 of the work
- Work done by B in one day = 1/9 of the work

### Step 2: Calculate the total work in terms of units
To make calculations easier, we can assume the total work is a common multiple of their completion days. The least common multiple (LCM) of 15 and 9 is 45. Therefore:
- Total work = 45 units

Now we can find the efficiencies:
- Efficiency of A = Total work / Days taken by A = 45 / 15 = 3 units/day
- Efficiency of B = Total work / Days taken by B = 45 / 9 = 5 units/day

### Step 3: Determine the efficiency of B and C working together
- B and C together can complete the work in 6 days.
- Work done by B and C together in one day = 1/6 of the work.

Thus, their combined efficiency:
- Efficiency of B and C = Total work / Days taken by B and C = 45 / 6 = 7.5 units/day

### Step 4: Calculate the efficiency of C
We already know the efficiency of B:
- Efficiency of B = 5 units/day

Now, we can find the efficiency of C:
- Efficiency of C = Efficiency of B and C - Efficiency of B
- Efficiency of C = 7.5 - 5 = 2.5 units/day

### Step 5: Calculate the combined efficiency of A and C
Now that we have the efficiencies of A and C, we can find their combined efficiency:
- Efficiency of A = 3 units/day
- Efficiency of C = 2.5 units/day

Combined efficiency of A and C:
- Combined efficiency = Efficiency of A + Efficiency of C = 3 + 2.5 = 5.5 units/day

### Step 6: Calculate the time taken by A and C to complete the work together
Now we can find the time taken for A and C to complete the work together:
- Time = Total work / Combined efficiency
- Time = 45 units / 5.5 units/day = 45 / 5.5 = 8.18 days (approximately)

### Final Answer
A and C together will complete the work in approximately **8.18 days** or **8 2/11 days**.