Pipe C can fill a tank in 12 hours and pipe D can fill the same tank in 40 hours. In how many hours both pipe C and D together can fill the same tank?

Options

A. A) `(60)/7` hours
B. B) `60/11` hours
C. C) `(120)/(13)` hours
D. D) `(120)/(11)` hours

Answer

To solve the problem of how long it will take for pipes C and D to fill a tank together, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Filling Rates of Each Pipe:**
- Pipe C can fill the tank in 12 hours.
- Pipe D can fill the tank in 40 hours.

2. **Calculate the Work Done by Each Pipe in One Hour:**
- The work done by Pipe C in one hour is:
\[ \text{Work by C} = \frac{1}{12} \text{ of the tank per hour} \]
- The work done by Pipe D in one hour is:
\[ \text{Work by D} = \frac{1}{40} \text{ of the tank per hour} \]

3. **Combine the Work Done by Both Pipes:**
- To find the total work done by both pipes in one hour, we add the work done by each pipe:
\[ \text{Total Work per hour} = \frac{1}{12} + \frac{1}{40} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 40 is 120.
- Convert each fraction:
\[ \frac{1}{12} = \frac{10}{120}, \quad \frac{1}{40} = \frac{3}{120} \]
- Now add them:
\[ \text{Total Work per hour} = \frac{10}{120} + \frac{3}{120} = \frac{13}{120} \]

4. **Calculate the Time Taken to Fill the Tank:**
- Since both pipes together can fill \(\frac{13}{120}\) of the tank in one hour, we need to find out how many hours it will take to fill the entire tank (1 tank):
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{13}{120} \text{ tank/hour}} = \frac{120}{13} \text{ hours} \]

5. **Final Answer:**
- Therefore, the time taken for both pipes C and D to fill the tank together is:
\[ \frac{120}{13} \text{ hours} \approx 9.23 \text{ hours} \]