Question: A swimming pool can be filled in $16$ hours if water enters through a pipe alone, or in $23$ hours if water enters through a hose alone. If water is entering through both the pipe and the hose, how long will it take to fill the pool? It will take [ ] hours to fill the pool if water is entering through both the pipe and the hose. (Simplify your answer. Type an integer, fraction, or mixed numeral.)
1. **State the problem:** We need to find the time it takes to fill the pool when both the pipe and the hose are working together.
2. **Identify rates:** The pipe fills the pool in $16$ hours, so its rate is $\frac{1}{16}$ pool per hour.
The hose fills the pool in $23$ hours, so its rate is $\frac{1}{23}$ pool per hour.
3. **Combined rate:** When both work together, their rates add:
$$\text{Combined rate} = \frac{1}{16} + \frac{1}{23}$$
4. **Find common denominator and add:**
$$\frac{1}{16} + \frac{1}{23} = \frac{23}{16 \times 23} + \frac{16}{16 \times 23} = \frac{23 + 16}{368} = \frac{39}{368}$$
5. **Calculate time to fill pool together:**
Time $= \frac{1}{\text{Combined rate}} = \frac{1}{\frac{39}{368}} = \frac{368}{39}$ hours.
6. **Simplify fraction:**
$\frac{368}{39}$ cannot be simplified further since $368 = 8 \times 46$ and $39 = 3 \times 13$, no common factors.
7. **Convert to mixed number:**
$\frac{368}{39} = 9 \frac{17}{39}$ hours.
**Final answer:**
It will take $9 \frac{17}{39}$ hours to fill the pool if water is entering through both the pipe and the hose.