A, B and C can complete a piece of work in 25, 30 and 50 days. Respectively. They started the work together But A and C left 2 days before the completion of the work. In how many days will the work is completed ?

Options

A. 14
B. 12
C. 18
D. 10

Explanation

To solve the problem step by step, we will determine how long it takes for A, B, and C to complete the work together and how long it takes for B to finish the remaining work after A and C leave.

### Step 1: Determine the work done by A, B, and C in one day.

- A can complete the work in 25 days. Therefore, the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{25} \text{ of the work} \]

- B can complete the work in 30 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{30} \text{ of the work} \]

- C can complete the work in 50 days. Therefore, the work done by C in one day is:
\[ \text{Work done by C in one day} = \frac{1}{50} \text{ of the work} \]

### Step 2: Calculate the total work done by A, B, and C in one day.

To find the total work done by A, B, and C together in one day, we add their individual contributions:
\[ \text{Total work done in one day} = \frac{1}{25} + \frac{1}{30} + \frac{1}{50} \]

To add these fractions, we need to find a common denominator. The LCM of 25, 30, and 50 is 150.

Now, convert each fraction:
\[ \frac{1}{25} = \frac{6}{150}, \quad \frac{1}{30} = \frac{5}{150}, \quad \frac{1}{50} = \frac{3}{150} \]

Adding these together:
\[ \text{Total work done in one day} = \frac{6 + 5 + 3}{150} = \frac{14}{150} = \frac{7}{75} \]

### Step 3: Determine how many days they worked together before A and C left.

Let \( x \) be the total number of days they worked together. Since A and C left 2 days before the work was completed, they worked together for \( x - 2 \) days.

The total work done by A, B, and C in \( x - 2 \) days is:
\[ \text{Work done} = \left(\frac{7}{75}\right)(x - 2) \]

### Step 4: Calculate the remaining work after A and C leave.

After A and C leave, only B is left to complete the work. The work done by B in one day is \( \frac{1}{30} \).

The total work done by B in 2 days is:
\[ \text{Work done by B in 2 days} = 2 \times \frac{1}{30} = \frac{2}{30} = \frac{1}{15} \]

### Step 5: Set up the equation for total work.

The total work is 1 (the whole job), so we can set up the equation:
\[ \left(\frac{7}{75}\right)(x - 2) + \frac{1}{15} = 1 \]

### Step 6: Solve the equation.

First, convert \( \frac{1}{15} \) to have a common denominator of 75:
\[ \frac{1}{15} = \frac{5}{75} \]

Now substitute back into the equation:
\[ \left(\frac{7}{75}\right)(x - 2) + \frac{5}{75} = 1 \]

Multiply through by 75 to eliminate the denominator:
\[ 7(x - 2) + 5 = 75 \]
\[ 7x - 14 + 5 = 75 \]
\[ 7x - 9 = 75 \]
\[ 7x = 84 \]
\[ x = 12 \]

### Step 7: Conclusion

The total time taken to complete the work is 12 days.