Question: Solving a Work Problem
It takes $12$ minutes to fill an entire bathtub using both the cold and hot water. If just the cold water is used, it takes $18$ minutes to fill the bathtub. How long would it take to fill the bathtub if just the hot water were used?
The equation [blank] can be used to solve for the rate, $r$, for the hot water alone to fill the bathtub.
The hot water can fill [blank] of the tub in $1$ minute.
It would take [blank] minutes for the hot water to fill the bathtub.
1. **State the problem:** We want to find how long it takes for the hot water alone to fill the bathtub.
2. **Define variables:** Let $r$ be the rate at which the hot water fills the tub (in tubs per minute).
3. **Known rates:**
- Cold water fills the tub in $18$ minutes, so its rate is $\frac{1}{18}$ tub per minute.
- Both cold and hot water together fill the tub in $12$ minutes, so their combined rate is $\frac{1}{12}$ tub per minute.
4. **Set up the equation:** The combined rate is the sum of the individual rates:
$$\frac{1}{18} + r = \frac{1}{12}$$
5. **Solve for $r$:**
$$r = \frac{1}{12} - \frac{1}{18}$$
6. **Find common denominator and subtract:**
$$r = \frac{3}{36} - \frac{2}{36} = \frac{1}{36}$$
7. **Interpretation:** The hot water fills $\frac{1}{36}$ of the tub in $1$ minute.
8. **Find time for hot water alone:**
If hot water fills $\frac{1}{36}$ of the tub per minute, it takes $36$ minutes to fill the whole tub.
**Final answers:**
- The equation to solve for $r$ is $$\frac{1}{18} + r = \frac{1}{12}$$
- The hot water fills $$\frac{1}{36}$$ of the tub in $1$ minute.
- It would take $36$ minutes for the hot water to fill the bathtub alone.