A and B can do a piece of work in 8 days. B and C can do it in 24 days, while C and A can do it in 8`4/7` days. In how many days can C do it alone?

Options

A. 60
B. 40
C. 50
D. 10

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C based on the information given.

### Step 1: Determine the work done by A and B together
A and B can complete the work in 8 days. Therefore, the fraction of work done by A and B in one day is:
\[ \text{Work done by A and B in one day} = \frac{1}{8} \]

### Step 2: Determine the work done by B and C together
B and C can complete the work in 24 days. Therefore, the fraction of work done by B and C in one day is:
\[ \text{Work done by B and C in one day} = \frac{1}{24} \]

### Step 3: Determine the work done by C and A together
C and A can complete the work in \(8 \frac{4}{7}\) days. First, convert \(8 \frac{4}{7}\) to an improper fraction:
\[ 8 \frac{4}{7} = \frac{60}{7} \text{ days} \]
Thus, the fraction of work done by C and A in one day is:
\[ \text{Work done by C and A in one day} = \frac{1}{\frac{60}{7}} = \frac{7}{60} \]

### Step 4: Set up equations for the work done
Let the work done by A in one day be \(a\), by B be \(b\), and by C be \(c\). We can write the following equations based on the work done:
1. \(a + b = \frac{1}{8}\) (Equation 1)
2. \(b + c = \frac{1}{24}\) (Equation 2)
3. \(c + a = \frac{7}{60}\) (Equation 3)

### Step 5: Add the equations
Adding all three equations:
\[ (a + b) + (b + c) + (c + a) = \frac{1}{8} + \frac{1}{24} + \frac{7}{60} \]
This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{8} + \frac{1}{24} + \frac{7}{60} \]
Now, we need to find a common denominator to add the fractions on the right side. The least common multiple of 8, 24, and 60 is 120.

Converting each fraction:
- \(\frac{1}{8} = \frac{15}{120}\)
- \(\frac{1}{24} = \frac{5}{120}\)
- \(\frac{7}{60} = \frac{14}{120}\)

Now, adding these fractions:
\[ \frac{15}{120} + \frac{5}{120} + \frac{14}{120} = \frac{34}{120} \]

### Step 6: Simplify the equation
So we have:
\[ 2(a + b + c) = \frac{34}{120} \]
Dividing both sides by 2:
\[ a + b + c = \frac{17}{120} \]

### Step 7: Find the value of C's work
From Equation 1, we know:
\[ a + b = \frac{1}{8} = \frac{15}{120} \]
Now substituting into the equation:
\[ c = (a + b + c) - (a + b) = \frac{17}{120} - \frac{15}{120} = \frac{2}{120} = \frac{1}{60} \]

### Step 8: Calculate the time taken by C to complete the work alone
If C does \(\frac{1}{60}\) of the work in one day, then the number of days C will take to complete the entire work is the reciprocal:
\[ \text{Days taken by C} = \frac{1}{\frac{1}{60}} = 60 \text{ days} \]

### Final Answer
C can complete the work alone in **60 days**.