A flask has two holes. The `1^("st")` hole alone makes the flask empty in 9 minutes and `2^("nd")` hole alone makes the flask empty in 16 minutes. If water leaks out at a constant rate, how long in minutes does it take both the holes empty the flask?

Options

A. `19/25`
B. `4(19)/25`
C. `5(20)/25`
D. `5(19)/25`

Answer

To solve the problem of how long it takes for both holes to empty the flask together, we can follow these steps:

### Step 1: Determine the rate of each hole
- The first hole empties the flask in 9 minutes. Therefore, its rate of emptying is:
\[ \text{Rate of 1st hole} = \frac{1 \text{ flask}}{9 \text{ minutes}} = \frac{1}{9} \text{ flasks per minute} \]

- The second hole empties the flask in 16 minutes. Therefore, its rate of emptying is:
\[ \text{Rate of 2nd hole} = \frac{1 \text{ flask}}{16 \text{ minutes}} = \frac{1}{16} \text{ flasks per minute} \]

### Step 2: Combine the rates of both holes
- When both holes are open, their rates add up:
\[ \text{Combined rate} = \frac{1}{9} + \frac{1}{16} \]

### Step 3: Find a common denominator
- The least common multiple (LCM) of 9 and 16 is 144. We convert each rate to have this common denominator:
\[ \frac{1}{9} = \frac{16}{144} \quad \text{(since } 144 \div 9 = 16\text{)} \]
\[ \frac{1}{16} = \frac{9}{144} \quad \text{(since } 144 \div 16 = 9\text{)} \]

### Step 4: Add the rates
- Now we can add the two rates:
\[ \text{Combined rate} = \frac{16}{144} + \frac{9}{144} = \frac{25}{144} \text{ flasks per minute} \]

### Step 5: Find the time taken to empty the flask
- Let \( t \) be the time taken to empty the flask when both holes are open. The relationship between rate and time is given by:
\[ \text{Rate} = \frac{1 \text{ flask}}{t \text{ minutes}} \]
- Therefore, we can set up the equation:
\[ \frac{1}{t} = \frac{25}{144} \]
- Solving for \( t \):
\[ t = \frac{144}{25} \text{ minutes} \]

### Step 6: Convert to a mixed number
- To convert \( \frac{144}{25} \) into a mixed number:
\[ 144 \div 25 = 5 \quad \text{(with a remainder of 19)} \]
Thus,
\[ t = 5 \frac{19}{25} \text{ minutes} \]

### Final Answer
- Therefore, it takes both holes \( 5 \frac{19}{25} \) minutes to empty the flask together.

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