# Rates Question (Speed/Distance/Time)

This is straight out of a maths competition. One of the few questions I can't manage to get a grip on.

Luisa cycled to and from a destination using the same route. The route included some flat roads and hills.

On the flat road she averaged 40km/h, uphill she averaged 30km/h and downhill she averaged 60km/h.

The whole journey took six hours.

What was the total distance travelled by Luisa, in km?

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One extreme case is where there are no uphill or downhill sections of road, only flat. In this case the total distance is $240$ km, which suggests that the answer to the question is $240$ km. – littleO Aug 10 '14 at 8:44
That's genius! I would accept your answer if it wasn't a comment. – Orange Peel Aug 10 '14 at 8:56

The key part of the question is that Luisa cycles to and from the destination using the same route.

Let $x$ be the total flat distance, $y$ the total uphill distance and $z$ the total downhill distance from her start point to the destination. The objective is to find $2(x+y+z)$.

Thus the time from start to destination is $$\frac{x}{40}+\frac{y}{30}+\frac{z}{60}$$ On the way back, the flat distance stays the same ($x$), but the uphill distance is now $z$ and the downhill distance is $y$. Thus the time from destination to start is $$\frac{x}{40}+\frac{y}{60}+\frac{z}{30}$$ The total time taken is $6$ hours, so we have an equation $$\frac{x}{40}+\frac{y}{30}+\frac{z}{60}+\frac{x}{40}+\frac{y}{60}+\frac{z}{30}=6\\\Rightarrow x+y+z=120$$ Therefore the total distance travelled is $240$ km.

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She travelled 240 km.

Let the times spent on flat, uphill and downhill be $t_f$, $t_u$ and $t_d$ (in hours). As she cycled to and from a destination using the same route, the total downhill and uphill distances are equal: $30t_u=60t_d$. Also, $t_f+t_u+t_d=6$.

Solving these two equations gives $t_u/2=t_d=2-t_f/3$. Thus the total distance covered was $$40t_f+30t_u+60t_d = 40t_f+30\times2(2-t_f/3)+60(2-t_f/3) = 240.$$ The result is independent of the value of $t_f$. All we know about it is $0\leq t_f\leq\frac32$, since all three times are positive.

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