If A, B and C together do a job in 4 days, A and C together do the job in 4.5 days and B and C together do the job in 12 days then in how many days can C alone do the job?

Options

A. 36
B. 6
C. 18
D. 12

Answer

To solve the problem step by step, let's denote the work done by A, B, and C in terms of their efficiencies.

### Step 1: Determine the work done by A, B, and C together
Let the total work be represented as 1 unit of work. If A, B, and C together can complete the job in 4 days, their combined efficiency is:
\[ \text{Efficiency of A, B, and C} = \frac{1 \text{ unit of work}}{4 \text{ days}} = \frac{1}{4} \text{ units/day} \]

### Step 2: Determine the work done by A and C together
If A and C together can complete the job in 4.5 days, their combined efficiency is:
\[ \text{Efficiency of A and C} = \frac{1 \text{ unit of work}}{4.5 \text{ days}} = \frac{1}{4.5} \text{ units/day} = \frac{2}{9} \text{ units/day} \]

### Step 3: Determine the work done by B and C together
If B and C together can complete the job in 12 days, their combined efficiency is:
\[ \text{Efficiency of B and C} = \frac{1 \text{ unit of work}}{12 \text{ days}} = \frac{1}{12} \text{ units/day} \]

### Step 4: Set up equations for individual efficiencies
Let the efficiencies of A, B, and C be represented as \( a \), \( b \), and \( c \) respectively. We can set up the following equations based on the efficiencies calculated:
1. \( a + b + c = \frac{1}{4} \) (from A, B, and C)
2. \( a + c = \frac{2}{9} \) (from A and C)
3. \( b + c = \frac{1}{12} \) (from B and C)

### Step 5: Solve the equations
From equation (2), we can express \( a \) in terms of \( c \):
\[ a = \frac{2}{9} - c \]

From equation (3), we can express \( b \) in terms of \( c \):
\[ b = \frac{1}{12} - c \]

Now substitute \( a \) and \( b \) into equation (1):
\[ \left(\frac{2}{9} - c\right) + \left(\frac{1}{12} - c\right) + c = \frac{1}{4} \]

Combine like terms:
\[ \frac{2}{9} + \frac{1}{12} - c = \frac{1}{4} \]

### Step 6: Find a common denominator and simplify
The common denominator of 9, 12, and 4 is 36. Rewrite each fraction:
\[ \frac{2}{9} = \frac{8}{36}, \quad \frac{1}{12} = \frac{3}{36}, \quad \frac{1}{4} = \frac{9}{36} \]

Now substitute:
\[ \frac{8}{36} + \frac{3}{36} - c = \frac{9}{36} \]
\[ \frac{11}{36} - c = \frac{9}{36} \]
\[ c = \frac{11}{36} - \frac{9}{36} = \frac{2}{36} = \frac{1}{18} \]

### Step 7: Calculate the time taken by C alone
Since \( c = \frac{1}{18} \), this means C can complete the job alone in:
\[ \text{Time taken by C} = \frac{1 \text{ unit of work}}{c} = \frac{1}{\frac{1}{18}} = 18 \text{ days} \]

Thus, C alone can do the job in **18 days**.