A and B complete a piece of work in 24 days, B and C do the same work in 36 days, and A, B and C together finish it in 18 days. In how many days will :
C alone,

Answer

To find out how many days C alone will take to complete the work, we can follow these steps:

### Step 1: Determine the efficiencies of A + B, B + C, and A + B + C
- A and B can complete the work in 24 days. Therefore, their combined efficiency is:
\[ \text{Efficiency of A + B} = \frac{1}{24} \]

- B and C can complete the work in 36 days. Therefore, their combined efficiency is:
\[ \text{Efficiency of B + C} = \frac{1}{36} \]

- A, B, and C together can complete the work in 18 days. Therefore, their combined efficiency is:
\[ \text{Efficiency of A + B + C} = \frac{1}{18} \]

### Step 2: Calculate the efficiency of C
To find the efficiency of C, we can use the formula:
\[ \text{Efficiency of C} = \text{Efficiency of A + B + C} - \text{Efficiency of A + B} \]

Substituting the values we found:
\[ \text{Efficiency of C} = \frac{1}{18} - \frac{1}{24} \]

### Step 3: Find a common denominator and perform the subtraction
The least common multiple (LCM) of 18 and 24 is 72. We can rewrite the fractions:
\[ \frac{1}{18} = \frac{4}{72} \quad \text{and} \quad \frac{1}{24} = \frac{3}{72} \]

Now, substituting these values back into the equation:
\[ \text{Efficiency of C} = \frac{4}{72} - \frac{3}{72} = \frac{1}{72} \]

### Step 4: Calculate the time taken by C to complete the work alone
The time taken by C to complete the work alone is the inverse of the efficiency of C:
\[ \text{Time taken by C} = \frac{1}{\text{Efficiency of C}} = \frac{1}{\frac{1}{72}} = 72 \text{ days} \]

### Final Answer
C alone will take **72 days** to complete the work.

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