Cátia leaves school every day at the same time and returns home by bicycle. When she cycles at $20 \mathrm{~km} / \mathrm{h}$, she arrives home at 4:30 PM. If she cycles at $10 \mathrm{~km} / \mathrm{h}$, she arrives home at 5:15 PM. At what speed should she cycle to arrive home at 5:00 PM?
Let $t$ be the time she spends cycling at $20 \mathrm{~km} / \mathrm{h}$. Cycling at $10 \mathrm{~km} / \mathrm{h}$, she takes twice the time she spends cycling at $20 \mathrm{~km} / \mathrm{h}$, which is $2 t$. However, since she takes 45 minutes longer, we have:

$$
2 t-t=45 \Longrightarrow t=45 \mathrm{~min}
$$

Therefore, she leaves school daily at

$$
4: 30 \mathrm{~h}-45 \mathrm{~min}=3: 45 \mathrm{~h}
$$

and the distance to her home is

$$
45 \mathrm{~min} \times 20 \mathrm{~km} / \mathrm{h}=\frac{3}{4} \times 20=15 \mathrm{~km}
$$

To cover $15 \mathrm{~km}$ in $5: 00 \mathrm{~h}-3: 45 \mathrm{~h}=1: 15 \mathrm{~h}=\frac{5}{4} \mathrm{~h}$, she needs to maintain a speed of

$$
\frac{15 \mathrm{~km}}{\frac{5}{4} \mathrm{~h}}=12 \mathrm{~km} / \mathrm{h}
$$