5. A hiking trail leaves from point $P$. The trail consists of a flat section from point $P$ to point $Q$, followed by an uphill section from $Q$ to a viewing platform at point $R$. A hiker walked from $P$ to $Q$ to $R$ and back from $R$ to $Q$ to $P$. The hiker's speed was $4 \mathrm{~km} / \mathrm{h}$ on the flat section in both directions, $3 \mathrm{~km} / \mathrm{h}$ while walking uphill, and $6 \mathrm{~km} / \mathrm{h}$ while walking downhill. If the hiker left $P$ at 1:00 p.m., spent exactly 1 hour at $R$, and returned to $P$ at 8:00 p.m. on the same day, what was the total distance walked by the hiker?
5. Let the distance between $P$ and $Q$ be $x \mathrm{~km}$ and the distance between $Q$ and $R$ be $y \mathrm{~km}$ as shown in the diagram below.

The total distance travelled by the hiker was $(2 x+2 y) \mathrm{km}$.
The total distance walked on flat ground was $2 x \mathrm{~km}$ and this was done at a speed of $4 \mathrm{~km} / \mathrm{h}$.
Therefore, the total time the hiker spent walking on flat ground was $\frac{2 x \mathrm{~km}}{4 \mathrm{~km} / \mathrm{h}}=\frac{x}{2} \mathrm{~h}$.
The total distance walked uphill was $y \mathrm{~km}$ and this was done at a speed of $3 \mathrm{~km} / \mathrm{h}$.
Therefore, the total time spent walking uphill was $\frac{y}{3} \mathrm{~h}$.
The total distance walked downhill was $y \mathrm{~km}$ and this was done at a speed of $6 \mathrm{~km} / \mathrm{h}$.
Therefore, the total time spent walking downhill was $\frac{y}{6} \mathrm{~h}$.
Since the hiker left point $P$ at 1:00 p.m. and returned to point $P$ at 8:00 p.m., the total time for the trip, including the hour spent at the viewing platform, was 7 hours.
Of these 7 hours, $7-1=6$ of them were spent walking.
The total times spent walking on flat ground, uphill, and downhill were $\frac{x}{2} \mathrm{~h}, \frac{y}{3} \mathrm{~h}$, and $\frac{y}{6} \mathrm{~h}$. Therefore, we have
$$
\frac{x}{2}+\frac{y}{3}+\frac{y}{6}=6
$$

Combining the left side into a single fraction, we get $\frac{3 x+2 y+y}{6}=6$ or $\frac{3 x+3 y}{6}=6$.
This simplifies to $\frac{x+y}{2}=6$ or $x+y=12$, and after multiplying both sides of this equation by 2 , we get $2(x+y)^{2}=24$ or $2 x+2 y=24$.
From earlier, the total distance travelled by the hiker was $(2 x+2 y) \mathrm{km}$, so the answer is 24 $\mathrm{km}$.
ANSWER: $24 \mathrm{~km}$