Solve the following problems involving rational Functions

1. 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?

2. One-sixth of a number is 3 less than one - half of the same number. What is the number?​

Let's solve these problems one by one.

### Problem 1: Pool Emptying with Two Pipes

- **Large pipe:** Can empty the pool in 6 hours. So, its rate is \( \frac{1}{6} \) of the pool per hour.
- **Small pipe:** Can empty the pool in 9 hours. So, its rate is \( \frac{1}{9} \) of the pool per hour.

When both pipes work together, their combined rate is the sum of their individual rates. So, the total rate is:

\[
\text{Combined rate} = \frac{1}{6} + \frac{1}{9}
\]

To add these, we need a common denominator:

\[
\frac{1}{6} + \frac{1}{9} = \frac{3}{18} + \frac{2}{18} = \frac{5}{18}
\]

Thus, the combined rate is \( \frac{5}{18} \) of the pool per hour. Now, to find how long it will take to empty the pool, we can use the formula:

\[
\text{Time} = \frac{1}{\text{Rate}}
\]

\[
\text{Time} = \frac{1}{\frac{5}{18}} = \frac{18}{5} = 3.6 \text{ hours}
\]

So, it will take **3.6 hours** (or 3 hours and 36 minutes) for both pipes to empty the pool.

---

### Problem 2: Solving the Equation

We are given:

\[
\frac{1}{6} \text{ of a number} = \frac{1}{2} \text{ of the same number} - 3
\]

Let \( x \) be the unknown number. Then the equation becomes:

\[
\frac{x}{6} = \frac{x}{2} - 3
\]

To eliminate the fractions, multiply every term by 6:

\[
x = 3x - 18
\]

Now, subtract \( 3x \) from both sides:

\[
x - 3x = -18
\]

\[
-2x = -18
\]

Now, divide by -2:

\[
x = 9
\]

So, the number is **9**.

---

### Summary:
1. It will take **3.6 hours** for both pipes to empty the pool together.
2. The number is **9**.

---

Would you like more details or have any questions? Here are five related questions for further practice:

1. How long will it take if the large pipe and two small pipes are used together to empty the pool?
2. If a third pipe can empty the pool in 12 hours, how long will it take if all three pipes are used together?
3. If the large pipe becomes clogged and can only work at half its rate, how long will it take to empty the pool with both pipes working?
4. What if one-fifth of a number is 4 less than one-third of the same number? What is the number?
5. If one-fourth of a number is 5 more than one-seventh of the same number, what is the number?

**Tip:** For work-related problems, always express the rates of work as fractions of the whole task per unit of time!

1. 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? 2. One-sixth of a number is 3 less than one-half of the same number. What is the number?

Learn how to solve a problem involving two pipes emptying a pool with different rates and an algebraic equation relating fractions of a number. This guide covers key concepts in rational functions, work rates, and basic algebra suitable for students in Grades 7-9.

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