, 2014

As Nadia walks from $N$ to $G$, suppose that she walks $x$ km uphill and $y$ km downhill. We are told that she walks 2.5 km on flat ground.

This means that when she walks from $G$ to $N$, she will walk $x$ km downhill, $y$ km uphill, and again 2.5 km on flat ground. This is because downhill portions become uphill portions on the return trip, while uphill portions become downhill portions on the return trip.

We are told that Nadia walks at 5 km/h on flat ground, 4 km/h uphill, and 6 km/h downhill.

Since $\text{speed = distance time}$, then $\text{distance = speed time}$ and $\text{time = distance speed}$.

Thus, on her trip from $N$ to $G$, her time walking uphill is $\dfrac{x}{4}$ hours, her time walking downhill is $\dfrac{y}{6}$ hours, and her time walking on flat ground is $\dfrac{2.5}{5}$ hours.

Since it takes her 1 hour and 36 minutes (which is 96 minutes or $\dfrac{96}{60}$ hours), then $$\dfrac{x}{4}+\dfrac{y}{6} + \dfrac{2.5}{5} = \dfrac{96}{60}$$ A similar analysis of the return trip gives $$\dfrac{x}{6}+\dfrac{y}{4} + \dfrac{2.5}{5} = \dfrac{99}{60}$$ We are asked for the total distance from $N$ to $G$, which equals $x+y+2.5$ km. Therefore, we need to determine $x+y$.

We add the two equations above and simplify to obtain $$\begin{aligned} \dfrac{x}{4}+\dfrac{x}{6} + \dfrac{y}{6}+\dfrac{y}{4} + 1 &= \dfrac{195}{60}\\ x\left(\dfrac{1}{4}+\dfrac{1}{6}\right) + y\left(\dfrac{1}{4}+\dfrac{1}{6}\right) & = \dfrac{135}{60} \\ \dfrac{5}{12}x + \dfrac{5}{12}y & = \dfrac{9}{4} \\ x + y & = \dfrac{12}{5}\left(\dfrac{9}{4}\right)\end{aligned}$$ Thus, $x+y = \dfrac{108}{20} = \dfrac{27}{5} = 5.4$ km.

Finally, the distance from $N$ to $G$ is $5.4 + 2.5 = 7.9$ km.