A can do a piece of work in 24 days while B alone can do it in 16 days. But with the help of C, they finish the work in 8 days. C alone can do the work in….

Options

A. 32 days
B. 36 days
C. 40 days
D. 48 days

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C, and then find out how long C alone would take to complete the work.

### Step 1: Determine the work done by A and B
- A can complete the work in 24 days.
- B can complete the work in 16 days.

To find the amount of work done, we can use the concept of efficiency, which is defined as the amount of work done in one day.

**Efficiency of A**:
\[ \text{Efficiency of A} = \frac{1 \text{ work}}{24 \text{ days}} = \frac{1}{24} \text{ work per day} \]

**Efficiency of B**:
\[ \text{Efficiency of B} = \frac{1 \text{ work}}{16 \text{ days}} = \frac{1}{16} \text{ work per day} \]

### Step 2: Calculate the combined efficiency of A and B
To find the combined efficiency of A and B, we add their efficiencies:
\[ \text{Combined Efficiency of A and B} = \frac{1}{24} + \frac{1}{16} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 24 and 16 is 48.

Converting the fractions:
\[ \frac{1}{24} = \frac{2}{48}, \quad \frac{1}{16} = \frac{3}{48} \]
Now, adding them:
\[ \text{Combined Efficiency of A and B} = \frac{2}{48} + \frac{3}{48} = \frac{5}{48} \text{ work per day} \]

### Step 3: Calculate the combined efficiency of A, B, and C
We know that A, B, and C together complete the work in 8 days. Thus, their combined efficiency is:
\[ \text{Combined Efficiency of A, B, and C} = \frac{1 \text{ work}}{8 \text{ days}} = \frac{1}{8} \text{ work per day} \]

### Step 4: Find the efficiency of C
Now we can find the efficiency of C by subtracting the combined efficiency of A and B from the combined efficiency of A, B, and C:
\[ \text{Efficiency of C} = \text{Combined Efficiency of A, B, and C} - \text{Combined Efficiency of A and B} \]
\[ \text{Efficiency of C} = \frac{1}{8} - \frac{5}{48} \]

To perform this subtraction, we need a common denominator. The LCM of 8 and 48 is 48:
\[ \frac{1}{8} = \frac{6}{48} \]
Now subtract:
\[ \text{Efficiency of C} = \frac{6}{48} - \frac{5}{48} = \frac{1}{48} \text{ work per day} \]

### Step 5: Calculate the time taken by C to complete the work alone
To find out how long C will take to complete the work alone, we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of C}} \]
The total work is 1 (as we defined it), and the efficiency of C is \(\frac{1}{48}\):
\[ \text{Time taken by C} = \frac{1}{\frac{1}{48}} = 48 \text{ days} \]

### Final Answer
C alone can complete the work in **48 days**.

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