It takes 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 4 hours and the pipe of smaller diameter is used for 9 hours, only half of the pool is filled. How long would each pipe take to fill the swimming pool ?

Answer

To solve the problem of how long each pipe takes to fill the swimming pool, we can follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = time taken by pipe A (larger diameter) to fill the pool (in hours).
- \( y \) = time taken by pipe B (smaller diameter) to fill the pool (in hours).

### Step 2: Set Up the First Equation
From the problem, we know that both pipes together can fill the pool in 12 hours. Therefore, the rate at which both pipes fill the pool can be expressed as:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{12} \]
This is our **first equation**.

### Step 3: Set Up the Second Equation
According to the problem, when pipe A is used for 4 hours and pipe B for 9 hours, they fill half the pool. Thus, we can express this as:
\[ \frac{4}{x} + \frac{9}{y} = \frac{1}{2} \]
This is our **second equation**.

### Step 4: Rewrite the Equations
We can rewrite the first equation:
\[ 12 \left(\frac{1}{x} + \frac{1}{y}\right) = 1 \implies 12a + 12b = 1 \quad \text{(where } a = \frac{1}{x} \text{ and } b = \frac{1}{y}\text{)} \]
And the second equation can be rewritten as:
\[ 8a + 18b = 1 \]

### Step 5: Solve the System of Equations
Now we have the system of equations:
1. \( 12a + 12b = 1 \)
2. \( 8a + 18b = 1 \)

We can multiply the first equation by 2:
\[ 24a + 24b = 2 \]

Now we can subtract the second equation from this:
\[ (24a + 24b) - (8a + 18b) = 2 - 1 \]
This simplifies to:
\[ 16a + 6b = 1 \]

### Step 6: Isolate One Variable
From \( 16a + 6b = 1 \), we can express \( a \) in terms of \( b \):
\[ 16a = 1 - 6b \implies a = \frac{1 - 6b}{16} \]

### Step 7: Substitute Back
Now substitute \( a \) back into the first equation:
\[ 12\left(\frac{1 - 6b}{16}\right) + 12b = 1 \]
This simplifies to:
\[ \frac{12 - 72b}{16} + 12b = 1 \]
Multiply through by 16 to eliminate the fraction:
\[ 12 - 72b + 192b = 16 \]
Combine like terms:
\[ 120b = 4 \implies b = \frac{1}{30} \]

### Step 8: Find \( a \)
Now substitute \( b \) back to find \( a \):
\[ a = \frac{1 - 6 \cdot \frac{1}{30}}{16} = \frac{1 - \frac{6}{30}}{16} = \frac{1 - \frac{1}{5}}{16} = \frac{\frac{4}{5}}{16} = \frac{4}{80} = \frac{1}{20} \]

### Step 9: Find \( x \) and \( y \)
Since \( a = \frac{1}{x} \) and \( b = \frac{1}{y} \):
\[ x = 20 \quad \text{and} \quad y = 30 \]

### Final Answer
- Pipe A takes **20 hours** to fill the pool.
- Pipe B takes **30 hours** to fill the pool.