a large pipe can empty a pool in 6 hours, and  a small pipe can empty the same pool in 9 hours. how long will it take for both pipes to empty the pool?

To solve this problem, we need to determine the rate at which both pipes empty the pool together.

### Step 1: Find the individual rates of the pipes.
- The large pipe can empty the pool in 6 hours. Therefore, the rate of the large pipe is \( \frac{1}{6} \) of the pool per hour.
- The small pipe can empty the pool in 9 hours. Therefore, the rate of the small pipe is \( \frac{1}{9} \) of the pool per hour.

### Step 2: Add the rates of the two pipes.
The combined rate when both pipes are working together is the sum of their individual rates:
\[
\text{Combined Rate} = \frac{1}{6} + \frac{1}{9}
\]
To add these fractions, find the least common denominator (LCD), which is 18:
\[
\frac{1}{6} = \frac{3}{18}, \quad \frac{1}{9} = \frac{2}{18}
\]
So, the combined rate is:
\[
\text{Combined Rate} = \frac{3}{18} + \frac{2}{18} = \frac{5}{18}
\]
This means that together, the pipes empty \( \frac{5}{18} \) of the pool per hour.

### Step 3: Find the time taken to empty the pool.
The time taken to empty the entire pool when both pipes are working together is the reciprocal of their combined rate:
\[
\text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{5}{18}} = \frac{18}{5} \text{ hours}
\]
Therefore, it will take \( \frac{18}{5} \) hours, or 3.6 hours, for both pipes to empty the pool together.

Would you like further details or have any other questions?

Here are 5 related questions for further exploration:
1. If the large pipe could empty the pool in 4 hours, how long would it take for both pipes to empty the pool?
2. How long would it take for only half of the pool to be emptied if both pipes are working together?
3. If a third pipe that empties the pool in 12 hours is added, how long will it take to empty the pool with all three pipes working together?
4. What fraction of the pool would be emptied in 1 hour by both pipes together?
5. If the small pipe gets partially blocked and its time to empty the pool doubles, how long would it take for both pipes to empty the pool?

**Tip:** When solving rate problems, always express rates in terms of a single unit (e.g., per hour) and add them directly to find the combined rate.

A large pipe can empty a pool in 6 hours, and a small pipe can empty the same pool in 9 hours. How long will it take for both pipes to empty the pool?

Learn how to solve a rate problem involving two pipes working together to empty a pool. This example covers the calculation of rates, the addition of fractions, and finding the combined time to empty the pool. Suitable for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation