A, B and C can complete a piece of work in 10 days, 20 days and 25 days, respectively. If they tak 40 days to complete a piece of work, then in how many days can C alone complete the work ?

Options

A. 65 days
B. 76 days
C. 95 days
D. 190 days

Explanation

To solve the problem step by step, let's break it down:

### Step 1: Determine the work rates of A, B, and C
- A can complete the work in 10 days, so A's work rate is \( \frac{1}{10} \) of the work per day.
- B can complete the work in 20 days, so B's work rate is \( \frac{1}{20} \) of the work per day.
- C can complete the work in 25 days, so C's work rate is \( \frac{1}{25} \) of the work per day.

### Step 2: Calculate the combined work rate of A, B, and C
The combined work rate of A, B, and C is:
\[ \text{Combined work rate} = \frac{1}{10} + \frac{1}{20} + \frac{1}{25} \]

### Step 3: Find the LCM of the denominators
The denominators are 10, 20, and 25. The least common multiple (LCM) of these numbers is 100.

### Step 4: Convert each fraction to have a common denominator
- \( \frac{1}{10} = \frac{10}{100} \)
- \( \frac{1}{20} = \frac{5}{100} \)
- \( \frac{1}{25} = \frac{4}{100} \)

### Step 5: Add the fractions
\[ \text{Combined work rate} = \frac{10}{100} + \frac{5}{100} + \frac{4}{100} = \frac{19}{100} \]

### Step 6: Calculate the time taken by A, B, and C together to complete the work
If A, B, and C work together, they can complete the work in:
\[ \text{Time} = \frac{1}{\text{Combined work rate}} = \frac{100}{19} \text{ days} \]

### Step 7: Set up the equation based on the problem statement
According to the problem, A, B, and C together take 40 days to complete the work. We can set up the equation:
\[ \frac{100}{19} = 40 \]

### Step 8: Calculate the work done by C alone
Let \( x \) be the number of days C takes to complete the work alone. We know that:
\[ \frac{100}{19} \text{ (time taken by A, B, and C)} = 40 \text{ (given time)} \]
Now, we can find \( x \) using the relationship:
\[ \frac{100}{19} \cdot x = 25 \cdot 40 \]

### Step 9: Solve for \( x \)
Cross-multiplying gives:
\[ 100x = 19 \cdot 25 \cdot 40 \]
Calculating the right side:
\[ 100x = 19000 \]
Now, divide both sides by 100:
\[ x = 190 \]

### Step 10: Find the number of days C alone can complete the work
Since we need to find how many days C alone can complete the work:
\[ x = 190 \text{ days} \]

### Final Answer
C can complete the work alone in **190 days**.