A can do a piece of work in 24 hours while B alone can do it in 16 hours. If A, B and C working together can finish in 8 hours, in how many hours will C alone do the work?

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Determine the work rates of A and B
- A can complete the work in 24 hours, so A's work rate is:
\[ \text{Work rate of A} = \frac{1}{24} \text{ (work per hour)} \]
- B can complete the work in 16 hours, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{16} \text{ (work per hour)} \]

### Step 2: Set up the equation for A, B, and C working together
- Let C's work rate be \( \frac{1}{X} \) (where X is the number of hours C takes to complete the work alone).
- When A, B, and C work together, their combined work rate is:
\[ \frac{1}{24} + \frac{1}{16} + \frac{1}{X} = \frac{1}{8} \]

### Step 3: Find a common denominator and simplify
- The least common multiple (LCM) of 24, 16, and 8 is 48.
- Convert each fraction to have a denominator of 48:
\[ \frac{1}{24} = \frac{2}{48}, \quad \frac{1}{16} = \frac{3}{48}, \quad \frac{1}{8} = \frac{6}{48} \]
- Substitute these into the equation:
\[ \frac{2}{48} + \frac{3}{48} + \frac{1}{X} = \frac{6}{48} \]

### Step 4: Combine the fractions
- Combine the fractions on the left side:
\[ \frac{5}{48} + \frac{1}{X} = \frac{6}{48} \]

### Step 5: Isolate \( \frac{1}{X} \)
- Rearranging gives:
\[ \frac{1}{X} = \frac{6}{48} - \frac{5}{48} \]
- Simplifying the right side:
\[ \frac{1}{X} = \frac{1}{48} \]

### Step 6: Solve for X
- Taking the reciprocal gives:
\[ X = 48 \]

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