Question: Celeste rides her bike to work, averaging $12$ miles per hour. On her way home she averages $8$ miles per hour. If the total trip takes one hour, how far does she live from her work?
1. **State the problem:**
Celeste rides to work at $12$ mph and returns at $8$ mph. The total time for the round trip is $1$ hour. We need to find the distance from her home to work.
2. **Define variables:**
Let $d$ be the distance from home to work (in miles).
3. **Write the time expressions:**
Time to go to work = $\frac{d}{12}$ hours.
Time to return home = $\frac{d}{8}$ hours.
4. **Use the total time equation:**
$$\frac{d}{12} + \frac{d}{8} = 1$$
5. **Find a common denominator and combine:**
The common denominator of $12$ and $8$ is $24$.
$$\frac{2d}{24} + \frac{3d}{24} = 1$$
$$\frac{2d + 3d}{24} = 1$$
$$\frac{5d}{24} = 1$$
6. **Solve for $d$:**
Multiply both sides by $24$:
$$\cancel{\frac{5d}{\cancel{24}}} \times 24 = 1 \times 24$$
$$5d = 24$$
Divide both sides by $5$:
$$\frac{\cancel{5}d}{\cancel{5}} = \frac{24}{5}$$
$$d = \frac{24}{5} = 4.8$$
7. **Interpretation:**
Celeste lives $4.8$ miles from her work.
8. **Fill the Rate column:**
$a = 12$ (to work), $b = 8$ (to home).