Three pipes m, n and p can fill a tank separately in 4, 5 and 10 h, respectively. Find the time taken by all the three pipes to fill the tank when the pipes are opened together.

Answer

To solve the problem of how long it takes for three pipes (M, N, and P) to fill a tank together, we can follow these steps:

### Step 1: Determine the rate of each pipe
- Pipe M can fill the tank in 4 hours. Therefore, the rate of Pipe M is:
\[ \text{Rate of M} = \frac{1}{4} \text{ tank/hour} \]

- Pipe N can fill the tank in 5 hours. Therefore, the rate of Pipe N is:
\[ \text{Rate of N} = \frac{1}{5} \text{ tank/hour} \]

- Pipe P can fill the tank in 10 hours. Therefore, the rate of Pipe P is:
\[ \text{Rate of P} = \frac{1}{10} \text{ tank/hour} \]

### Step 2: Add the rates of all pipes together
To find the combined rate when all three pipes are opened together, we add their individual rates:
\[ \text{Combined Rate} = \text{Rate of M} + \text{Rate of N} + \text{Rate of P} \]
Substituting the values we calculated:
\[ \text{Combined Rate} = \frac{1}{4} + \frac{1}{5} + \frac{1}{10} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of the denominators (4, 5, and 10) is 20. We will convert each fraction to have this common denominator:
- For \(\frac{1}{4}\):
\[ \frac{1}{4} = \frac{5}{20} \]
- For \(\frac{1}{5}\):
\[ \frac{1}{5} = \frac{4}{20} \]
- For \(\frac{1}{10}\):
\[ \frac{1}{10} = \frac{2}{20} \]

### Step 4: Add the converted fractions
Now we can add these fractions:
\[ \text{Combined Rate} = \frac{5}{20} + \frac{4}{20} + \frac{2}{20} = \frac{11}{20} \text{ tank/hour} \]

### Step 5: Calculate the time to fill the tank
To find the time taken to fill the tank when all three pipes are opened together, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{11}{20}} = \frac{20}{11} \text{ hours} \]

### Step 6: Convert to mixed fraction
To express \(\frac{20}{11}\) as a mixed fraction:
- Divide 20 by 11, which gives 1 with a remainder of 9. Therefore:
\[ \frac{20}{11} = 1 \frac{9}{11} \text{ hours} \]

### Final Answer
Thus, the time taken by all three pipes to fill the tank together is:
\[ \text{Time} = 1 \frac{9}{11} \text{ hours} \]

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