A can do a piece of work in 10 days, B in 18 days and A, B and C together in 4 days. In what time would C do it alone ?

Answer

To find out how long C would take to complete the work alone, we can follow these steps:

### Step 1: Determine the work done by A, B, and C in one day
- A can complete the work in 10 days, so A's work in one day is:
\[ \text{Work done by A in 1 day} = \frac{1}{10} \]
- B can complete the work in 18 days, so B's work in one day is:
\[ \text{Work done by B in 1 day} = \frac{1}{18} \]
- A, B, and C together can complete the work in 4 days, so their combined work in one day is:
\[ \text{Work done by A, B, and C in 1 day} = \frac{1}{4} \]

### Step 2: Set up the equation for C's work
- To find the work done by C alone in one day, we can use the formula:
\[ \text{Work done by C in 1 day} = \text{Work done by A, B, and C} - \text{Work done by A} - \text{Work done by B} \]
- Substituting the values we found:
\[ \text{Work done by C in 1 day} = \frac{1}{4} - \frac{1}{10} - \frac{1}{18} \]

### Step 3: Find a common denominator
- The least common multiple (LCM) of 4, 10, and 18 is 180. Now we convert each fraction:
\[ \frac{1}{4} = \frac{45}{180}, \quad \frac{1}{10} = \frac{18}{180}, \quad \frac{1}{18} = \frac{10}{180} \]

### Step 4: Perform the subtraction
- Now substitute these values into the equation:
\[ \text{Work done by C in 1 day} = \frac{45}{180} - \frac{18}{180} - \frac{10}{180} \]
- Combine the fractions:
\[ \text{Work done by C in 1 day} = \frac{45 - 18 - 10}{180} = \frac{17}{180} \]

### Step 5: 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 work done by C in one day:
\[ \text{Time taken by C} = \frac{1}{\text{Work done by C in 1 day}} = \frac{180}{17} \]
- Converting this to a mixed fraction:
\[ \frac{180}{17} = 10 \frac{10}{17} \text{ days} \]

### Final Answer
C would take \(10 \frac{10}{17}\) days to complete the work alone.

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