A, B and C can do a piece of work in 16, 32 and 48 days, respectively. They all begin together. A work continously till it is finished , C leaves the work 2 days before its completion and B leaves the work 1 day before its completion . In what time is the work finished ?

Options

A. 10 days
B. 20 days
C. `9 4/11 days`
D. 30 days

Answer

To solve the problem, we will first determine the work done by A, B, and C in one day, and then calculate how long it takes for them to finish the work given the conditions stated.

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

- A can complete the work in 16 days, so the work done by A in one day is:
\[ \text{Work done by A in 1 day} = \frac{1}{16} \]

- B can complete the work in 32 days, so the work done by B in one day is:
\[ \text{Work done by B in 1 day} = \frac{1}{32} \]

- C can complete the work in 48 days, so the work done by C in one day is:
\[ \text{Work done by C in 1 day} = \frac{1}{48} \]

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

To find the total work done by A, B, and C in one day, we add their individual contributions:
\[ \text{Total work done in 1 day} = \frac{1}{16} + \frac{1}{32} + \frac{1}{48} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 16, 32, and 48 is 96.

Converting each fraction:
\[ \frac{1}{16} = \frac{6}{96}, \quad \frac{1}{32} = \frac{3}{96}, \quad \frac{1}{48} = \frac{2}{96} \]

Now, adding them:
\[ \text{Total work done in 1 day} = \frac{6}{96} + \frac{3}{96} + \frac{2}{96} = \frac{11}{96} \]

### Step 3: Let the total number of days taken to finish the work be \( x \).

Since A works continuously until the end, and C leaves 2 days before the work is finished while B leaves 1 day before, we can set up the equation based on the work done.

- A works for \( x \) days.
- B works for \( x - 1 \) days.
- C works for \( x - 2 \) days.

The total work done can be expressed as:
\[ \text{Work done by A} + \text{Work done by B} + \text{Work done by C} = 1 \]
Substituting the work done:
\[ \frac{x}{16} + \frac{x - 1}{32} + \frac{x - 2}{48} = 1 \]

### Step 4: Solve the equation.

To solve this equation, we will first find a common denominator, which is 96:
\[ \frac{6x}{96} + \frac{3(x - 1)}{96} + \frac{2(x - 2)}{96} = 1 \]

Combining the fractions:
\[ \frac{6x + 3(x - 1) + 2(x - 2)}{96} = 1 \]

Expanding:
\[ 6x + 3x - 3 + 2x - 4 = 96 \]
\[ 11x - 7 = 96 \]
\[ 11x = 103 \]
\[ x = \frac{103}{11} \approx 9.36 \]

### Step 5: Calculate the time taken to finish the work.

Since \( x \) represents the total days taken to finish the work, we round it to the nearest whole number. Thus, the work is finished in approximately 9 days.

### Final Answer:
The work is finished in approximately **9 days**.

---