A glass containing water has two holes. The `1^(st)` hole alone empties the gas in 9 minutes and `2^(nd)` hole alone empties the glass in 3 minutes. If water leaks out at a constant rate, how many minutes will it take if both the holes together empty the glass?

Options

A. `(1)/(4)`
B. `2 (2)/(4)`
C. `3 (1)/(4) `
D. ` 2(1)/(4)`

Answer

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

### Step 1: Determine the individual efficiencies of each hole.
- The first hole empties the glass in 9 minutes. Therefore, its efficiency is:
\[ \text{Efficiency of 1st hole} = \frac{1 \text{ glass}}{9 \text{ minutes}} = \frac{1}{9} \text{ glasses per minute} \]

- The second hole empties the glass in 3 minutes. Therefore, its efficiency is:
\[ \text{Efficiency of 2nd hole} = \frac{1 \text{ glass}}{3 \text{ minutes}} = \frac{1}{3} \text{ glasses per minute} \]

### Step 2: Calculate the combined efficiency of both holes.
- To find the total efficiency when both holes are working together, we add their individual efficiencies:
\[ \text{Total Efficiency} = \text{Efficiency of 1st hole} + \text{Efficiency of 2nd hole} = \frac{1}{9} + \frac{1}{3} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 9 and 3 is 9.
\[ \frac{1}{3} = \frac{3}{9} \]
- Now, we can add:
\[ \text{Total Efficiency} = \frac{1}{9} + \frac{3}{9} = \frac{4}{9} \text{ glasses per minute} \]

### Step 3: Calculate the time taken to empty the glass using the combined efficiency.
- The total work (which is emptying one glass) can be represented as 1 glass. The time taken to empty the glass when both holes are working together is given by:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Total Efficiency}} = \frac{1 \text{ glass}}{\frac{4}{9} \text{ glasses per minute}} = 1 \times \frac{9}{4} = \frac{9}{4} \text{ minutes} \]
- Converting \(\frac{9}{4}\) minutes into minutes and seconds, we get:
\[ \frac{9}{4} = 2.25 \text{ minutes} = 2 \text{ minutes and } 15 \text{ seconds} \]

### Final Answer:
It will take \(2.25\) minutes (or \(2\) minutes and \(15\) seconds) for both holes together to empty the glass.

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