A and B. working together, can complete a work in 16 days, C and A together can complete it in 32 days, B and C together can complete it in 24 days. They worked together for 12 days. In how many days will C alone complete the remaining work ?

Options

A. 40
B. 36
C. 45
D. 32

Answer

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

### Step 1: Determine the total work
Let’s assume the total work is 96 units (this is chosen as it is the LCM of the days given).

### Step 2: Calculate the work efficiencies
1. **A and B together can complete the work in 16 days:**
\[ \text{Efficiency of A + B} = \frac{96}{16} = 6 \text{ units/day} \]

2. **C and A together can complete the work in 32 days:**
\[ \text{Efficiency of C + A} = \frac{96}{32} = 3 \text{ units/day} \]

3. **B and C together can complete the work in 24 days:**
\[ \text{Efficiency of B + C} = \frac{96}{24} = 4 \text{ units/day} \]

### Step 3: Set up equations
From the efficiencies calculated, we can set up the following equations:
1. \( A + B = 6 \)
2. \( C + A = 3 \)
3. \( B + C = 4 \)

### Step 4: Solve the equations
We can solve these equations to find the individual efficiencies of A, B, and C.

1. From \( A + B = 6 \) (Equation 1)
2. From \( C + A = 3 \) (Equation 2)
- Rearranging gives \( C = 3 - A \)
3. From \( B + C = 4 \) (Equation 3)
- Substitute \( C \) from Equation 2 into Equation 3:
\[ B + (3 - A) = 4 \implies B = 1 + A \]
4. Substitute \( B = 1 + A \) into Equation 1:
\[ A + (1 + A) = 6 \implies 2A + 1 = 6 \implies 2A = 5 \implies A = 2.5 \]
5. Now substitute \( A \) back to find \( B \) and \( C \):
\[ B = 1 + 2.5 = 3.5 \]
\[ C = 3 - 2.5 = 0.5 \]

### Step 5: Calculate the total efficiency of A, B, and C
Now, we have:
- Efficiency of A = 2.5 units/day
- Efficiency of B = 3.5 units/day
- Efficiency of C = 0.5 units/day

### Step 6: Calculate the total work done in 12 days
When A, B, and C work together for 12 days:
\[ \text{Total efficiency of A + B + C} = A + B + C = 2.5 + 3.5 + 0.5 = 6.5 \text{ units/day} \]
\[ \text{Work done in 12 days} = 12 \times 6.5 = 78 \text{ units} \]

### Step 7: Calculate the remaining work
\[ \text{Remaining work} = 96 - 78 = 18 \text{ units} \]

### Step 8: Calculate the time taken by C to complete the remaining work
Since C's efficiency is 0.5 units/day, the time taken by C to complete the remaining work is:
\[ \text{Time} = \frac{\text{Remaining work}}{\text{Efficiency of C}} = \frac{18}{0.5} = 36 \text{ days} \]

### Final Answer
C alone will complete the remaining work in **36 days**.

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