A and B together can complete a piece of work in 16 days while B and C together can complete the same work in 12 days and A and C together in 24 days. Find out the number of days that A will take to compete the work, working alone.

Options

A. 36 days
B. 48 days
C. 96 days
D. 80 days

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C together and then isolate A's work.

### Step 1: Calculate the work done by A and B together.
A and B together can complete the work in 16 days. Therefore, their combined work rate is:
\[ \text{Work rate of A and B} = \frac{1}{16} \text{ (work per day)} \]

### Step 2: Calculate the work done by B and C together.
B and C together can complete the work in 12 days. Therefore, their combined work rate is:
\[ \text{Work rate of B and C} = \frac{1}{12} \text{ (work per day)} \]

### Step 3: Calculate the work done by A and C together.
A and C together can complete the work in 24 days. Therefore, their combined work rate is:
\[ \text{Work rate of A and C} = \frac{1}{24} \text{ (work per day)} \]

### Step 4: Set up the equations for total work rates.
Let the work rates of A, B, and C be \( a \), \( b \), and \( c \) respectively. We can write the following equations based on the work rates:
1. \( a + b = \frac{1}{16} \) (Equation 1)
2. \( b + c = \frac{1}{12} \) (Equation 2)
3. \( a + c = \frac{1}{24} \) (Equation 3)

### Step 5: Add all three equations.
Adding all three equations gives:
\[ (a + b) + (b + c) + (a + c) = \frac{1}{16} + \frac{1}{12} + \frac{1}{24} \]
This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{16} + \frac{1}{12} + \frac{1}{24} \]

### Step 6: Find a common denominator and simplify.
The least common multiple of 16, 12, and 24 is 48. So, we convert each fraction:
\[ \frac{1}{16} = \frac{3}{48}, \quad \frac{1}{12} = \frac{4}{48}, \quad \frac{1}{24} = \frac{2}{48} \]
Adding these gives:
\[ \frac{3}{48} + \frac{4}{48} + \frac{2}{48} = \frac{9}{48} \]
Thus, we have:
\[ 2a + 2b + 2c = \frac{9}{48} \]
Dividing by 2:
\[ a + b + c = \frac{9}{96} \]

### Step 7: Isolate A's work rate.
Now, we can find A's work rate by subtracting the work rates of B and C from the total:
1. From Equation 2, we know \( b + c = \frac{1}{12} = \frac{4}{48} = \frac{8}{96} \).
2. Thus, we can find A's work rate:
\[ a = (a + b + c) - (b + c) = \frac{9}{96} - \frac{8}{96} = \frac{1}{96} \]

### Step 8: Calculate the number of days A will take to complete the work alone.
Since A's work rate is \( \frac{1}{96} \), it means A can complete the work alone in:
\[ \text{Days taken by A} = \frac{1}{\frac{1}{96}} = 96 \text{ days} \]

### Final Answer:
A will take 96 days to complete the work alone.

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