Tony and Maria are training for a race by running all the way up and down a $700 \mathrm{~m}$ long ski slope. They each run up the slope at different constant speeds. Coming down the slope, each runs at double his or her uphill speed. Maria reaches the top first, and immediately starts running back down, meeting Tony $70 \mathrm{~m}$ from the top. When Maria reaches the bottom, how far behind is Tony?
(A) $140 \mathrm{~m}$
(B) $250 \mathrm{~m}$
(C) $280 \mathrm{~m}$
(D) $300 \mathrm{~m}$
(E) $320 \mathrm{~m}$
Tony and Maria are training for a race by running all the way up and down a $700 \mathrm{~m}$ long ski slope. They each run up the slope at different constant speeds. Coming down the slope, each runs at double his or her uphill speed. Maria reaches the top first, and immediately starts running back down, meeting Tony $70 \mathrm{~m}$ from the top. When Maria reaches the bottom, how far behind is Tony?
(A) $140 \mathrm{~m}$
(B) $250 \mathrm{~m}$
(C) $280 \mathrm{~m}$
(D) $300 \mathrm{~m}$
(E) $320 \mathrm{~m}$

## Solution

When Tony and Maria meet for the first time, Tony has run $700-70=630 \mathrm{~m}$.

At this point, Maria has run $700 \mathrm{~m}$ up the hill and then $70 \mathrm{~m}$ at double the speed back down the hill. This takes her the same amount of time as if she had run $700 \mathrm{~m}$ up the hill and then $35 \mathrm{~m}$ more at the same speed.

In effect, she has run $735 \mathrm{~m}$, while Tony has run $630 \mathrm{~m}$. So the ratio of their speeds is $\frac{735}{630}=\frac{7(105)}{6(105)}=\frac{7}{6}$.

This means that for every 6 metres that Tony covers, Maria will cover 7 metres.

If we think of both runners as running at constant speeds, Maria runs $700+\frac{1}{2}(700)=1050 \mathrm{~m}$ at a constant speed over the course of the race. In effect, we are saying that she would run the equivalent of $1050 \mathrm{~m}$ at the same constant speed at which she ran up the hill.

She runs $\frac{7}{6}$ as fast as Tony. In the time that Maria runs $1050 \mathrm{~m}$, Tony runs $\frac{6}{7} \times 1050 \mathrm{~m}=900 \mathrm{~m}$ at a constant speed.

So in this new way of looking at the race, Tony is $150 \mathrm{~m}$ behind Maria.

But this $150 \mathrm{~m}$ is in the new way of looking at things, so Tony is actually $2 \times 150 \mathrm{~m}=300 \mathrm{~m}$ behind Maria.

ANswer: (D)