Pipe X can fill a tank in 20 hours and Plpe Y can fill the tank in 35 hours. Both the pipes are opened on alternate hours. Plpe Y is opened first, then in how much time (in hours) the tank will be full?

Options

A. `(269)/(11)`
B. `(286)/(11)`
C. `(179)/(7)`
D. `(172)/(7)`

Answer

To solve the problem of how long it will take to fill the tank with pipes X and Y working alternately, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Filling Rates of Each Pipe:**
- Pipe X can fill the tank in 20 hours. Therefore, its rate of filling is:
\[ \text{Rate of Pipe X} = \frac{1 \text{ tank}}{20 \text{ hours}} = \frac{1}{20} \text{ tanks per hour} \]
- Pipe Y can fill the tank in 35 hours. Therefore, its rate of filling is:
\[ \text{Rate of Pipe Y} = \frac{1 \text{ tank}}{35 \text{ hours}} = \frac{1}{35} \text{ tanks per hour} \]

2. **Calculate the Amount Filled in Two Hours:**
- In the first hour, Pipe Y is opened, and it fills:
\[ \text{Amount filled by Y in 1 hour} = \frac{1}{35} \text{ tanks} \]
- In the second hour, Pipe X is opened, and it fills:
\[ \text{Amount filled by X in 1 hour} = \frac{1}{20} \text{ tanks} \]
- Therefore, in two hours, the total amount filled by both pipes is:
\[ \text{Total in 2 hours} = \frac{1}{35} + \frac{1}{20} \]
- To add these fractions, find a common denominator (LCM of 35 and 20 is 140):
\[ \frac{1}{35} = \frac{4}{140}, \quad \frac{1}{20} = \frac{7}{140} \]
\[ \text{Total in 2 hours} = \frac{4}{140} + \frac{7}{140} = \frac{11}{140} \text{ tanks} \]

3. **Calculate How Many Sets of Two Hours are Needed to Fill the Tank:**
- The tank has a total capacity of 1 tank (or 140/140).
- To find how many sets of 2 hours are needed to fill the tank:
\[ \text{Number of 2-hour sets} = \frac{1 \text{ tank}}{\frac{11}{140} \text{ tanks per 2 hours}} = \frac{140}{11} \approx 12.73 \text{ sets} \]
- This means it takes 12 complete sets of 2 hours (24 hours) to fill:
\[ \text{Amount filled in 24 hours} = 12 \times \frac{11}{140} = \frac{132}{140} \text{ tanks} \]

4. **Calculate Remaining Amount After 24 Hours:**
- After 24 hours, the amount filled is:
\[ 1 - \frac{132}{140} = \frac{8}{140} = \frac{2}{35} \text{ tanks} \]

5. **Fill the Remaining Amount:**
- In the 25th hour, Pipe Y is opened again, and it fills:
\[ \text{Amount filled by Y in 1 hour} = \frac{1}{35} \text{ tanks} \]
- Since the remaining amount is \(\frac{2}{35}\), Pipe Y will fill it in:
\[ \text{Time taken by Y} = \frac{\frac{2}{35}}{\frac{1}{35}} = 2 \text{ hours} \]

6. **Total Time Taken:**
- The total time taken to fill the tank is:
\[ \text{Total time} = 24 \text{ hours} + 2 \text{ hours} = 26 \text{ hours} \]

### Final Answer:
The tank will be full in **26 hours**.