It takes 8,12 and 16 days for X, Y and Z respectively to complete a work. How many days will it take to complete the work if X works on it for 2 days, and then Y works on it for until 25% of the work is left for Z to do, and then Z complete the work?

Options

A. 10 days
B. 12 days
C. 8 days
D. 14 days

Answer

To solve the problem step-by-step, we will break it down into manageable parts.

### Step 1: Determine the work done by X, Y, and Z in one day.
- X completes the work in 8 days, so the work done by X in one day is:
\[ \text{Work done by X in 1 day} = \frac{1}{8} \]

- Y completes the work in 12 days, so the work done by Y in one day is:
\[ \text{Work done by Y in 1 day} = \frac{1}{12} \]

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

### Step 2: Calculate the work done by X in 2 days.
- In 2 days, X will complete:
\[ \text{Work done by X in 2 days} = 2 \times \frac{1}{8} = \frac{2}{8} = \frac{1}{4} \]

### Step 3: Calculate the remaining work after X works for 2 days.
- The total work is considered as 1 (or 100%). After X completes \(\frac{1}{4}\) of the work, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 4: Determine how much work Y needs to do before leaving 25% of the work for Z.
- 25% of the total work is:
\[ 25\% = \frac{25}{100} = \frac{1}{4} \]
- Therefore, Y will work until \(\frac{1}{4}\) of the work is left, which means Y will complete:
\[ \text{Work done by Y} = \frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \]

### Step 5: Calculate how many days Y takes to complete \(\frac{1}{2}\) of the work.
- Since Y does \(\frac{1}{12}\) of the work in one day, the number of days Y needs to complete \(\frac{1}{2}\) of the work is:
\[ \text{Days taken by Y} = \frac{\frac{1}{2}}{\frac{1}{12}} = \frac{1}{2} \times \frac{12}{1} = 6 \text{ days} \]

### Step 6: Calculate the work remaining for Z.
- After Y completes \(\frac{1}{2}\) of the work, the remaining work for Z is:
\[ \text{Remaining work for Z} = \frac{1}{4} \text{ (as calculated earlier)} \]

### Step 7: Calculate how many days Z takes to complete the remaining work.
- Since Z completes \(\frac{1}{16}\) of the work in one day, the number of days Z needs to complete \(\frac{1}{4}\) of the work is:
\[ \text{Days taken by Z} = \frac{\frac{1}{4}}{\frac{1}{16}} = \frac{1}{4} \times \frac{16}{1} = 4 \text{ days} \]

### Step 8: Calculate the total time taken to complete the work.
- The total time taken is the sum of the days worked by X, Y, and Z:
\[ \text{Total days} = 2 \text{ (days by X)} + 6 \text{ (days by Y)} + 4 \text{ (days by Z)} = 12 \text{ days} \]

### Final Answer:
The total number of days required to complete the work is **12 days**.

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