Questions: A swimming pool holds 540,000 liters of water. The pool has two drainage pipes. When the pool is completely full, the first pipe alone can empty it in 225 minutes, and the second pipe alone can empty it in 150 minutes. When both pipes are draining together, how long does it take them to empty the pool? minutes

A swimming pool holds 540,000 liters of water. The pool has two drainage pipes. When the pool is completely full, the first pipe alone can empty it in 225 minutes, and the second pipe alone can empty it in 150 minutes. When both pipes are draining together, how long does it take them to empty the pool?
minutes

Transcript text: A swimming pool holds 540,000 liters of water. The pool has two drainage pipes. When the pool is completely full, the first pipe alone can empty it in 225 minutes, and the second pipe alone can empty it in 150 minutes. When both pipes are draining together, how long does it take them to empty the pool? $\square$ minutes

Solution

Solution Steps

Step 1: Determine the Drainage Rate of Each Pipe

First, we need to calculate the drainage rate of each pipe. The first pipe can empty the pool in 225 minutes, so its drainage rate is:

\[ \text{Rate of Pipe 1} = \frac{1}{225} \text{ pools per minute} \]

Similarly, the second pipe can empty the pool in 150 minutes, so its drainage rate is:

\[ \text{Rate of Pipe 2} = \frac{1}{150} \text{ pools per minute} \]

Step 2: Calculate the Combined Drainage Rate

When both pipes are working together, their combined drainage rate is the sum of their individual rates:

\[ \text{Combined Rate} = \frac{1}{225} + \frac{1}{150} \]

To add these fractions, we need a common denominator. The least common multiple of 225 and 150 is 450. Convert each rate to have this common denominator:

\[ \frac{1}{225} = \frac{2}{450}, \quad \frac{1}{150} = \frac{3}{450} \]

Thus, the combined rate is:

\[ \text{Combined Rate} = \frac{2}{450} + \frac{3}{450} = \frac{5}{450} = \frac{1}{90} \text{ pools per minute} \]

Step 3: Calculate the Time to Empty the Pool

The combined rate of $\frac{1}{90}$ pools per minute means that together, the pipes can empty $\frac{1}{90}$ of the pool in one minute. Therefore, the time it takes to empty the entire pool is the reciprocal of this rate:

\[ \text{Time} = \frac{1}{\frac{1}{90}} = 90 \text{ minutes} \]