(A + B) can do a piece of work in 12 days. (B + C) can do same work in 16 days. A work for 5 days, B works for 7 days and C completed the remaining work in 13 days. In how many days will A, B and C do this work alone ?

Answer

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

### Step 1: Determine the total work
Given:
- A + B can complete the work in 12 days.
- B + C can complete the work in 16 days.

To find the total work, we can take the least common multiple (LCM) of the two durations:
- LCM of 12 and 16 is 48.

Thus, the total work is 48 units.

### Step 2: Calculate the efficiencies of A + B and B + C
- The efficiency of A + B:
\[ \text{Efficiency of } (A + B) = \frac{\text{Total Work}}{\text{Days}} = \frac{48}{12} = 4 \text{ units/day} \]

- The efficiency of B + C:
\[ \text{Efficiency of } (B + C) = \frac{\text{Total Work}}{\text{Days}} = \frac{48}{16} = 3 \text{ units/day} \]

### Step 3: Set up the equations based on the work done
Let:
- Efficiency of A = a
- Efficiency of B = b
- Efficiency of C = c

From the information:
1. \( a + b = 4 \) (from A + B)
2. \( b + c = 3 \) (from B + C)

### Step 4: Calculate the work done by A, B, and C
- A works for 5 days:
\[ \text{Work done by A} = 5a \]

- B works for 7 days:
\[ \text{Work done by B} = 7b \]

- C completes the remaining work in 13 days:
\[ \text{Work done by C} = 13c \]

### Step 5: Set up the equation for total work
The total work done by A, B, and C should equal the total work:
\[ 5a + 7b + 13c = 48 \]

### Step 6: Substitute the values of a and b
From the equations:
1. \( a + b = 4 \) → \( a = 4 - b \)
2. Substitute \( a \) in the total work equation:
\[ 5(4 - b) + 7b + 13c = 48 \]
Simplifying this gives:
\[ 20 - 5b + 7b + 13c = 48 \]
\[ 2b + 13c = 28 \]

### Step 7: Substitute \( c \) from \( b + c = 3 \)
From \( b + c = 3 \):
\[ c = 3 - b \]
Substituting \( c \) in the equation \( 2b + 13c = 28 \):
\[ 2b + 13(3 - b) = 28 \]
Expanding this gives:
\[ 2b + 39 - 13b = 28 \]
Combining like terms:
\[ -11b + 39 = 28 \]
\[ -11b = 28 - 39 \]
\[ -11b = -11 \]
\[ b = 1 \]

### Step 8: Find a and c
Using \( b = 1 \):
- From \( a + b = 4 \):
\[ a = 4 - 1 = 3 \]

- From \( b + c = 3 \):
\[ c = 3 - 1 = 2 \]

### Step 9: Calculate the time taken by A, B, and C to complete the work alone
- Time taken by A:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency}} = \frac{48}{3} = 16 \text{ days} \]

- Time taken by B:
\[ \text{Time} = \frac{48}{1} = 48 \text{ days} \]

- Time taken by C:
\[ \text{Time} = \frac{48}{2} = 24 \text{ days} \]

### Final Answer
- A can complete the work alone in 16 days.
- B can complete the work alone in 48 days.
- C can complete the work alone in 24 days.