# Distance and speed of two people walking 100 miles

This is for GRE math prep. Can you explain why the answer to this is 54? Five hours after Sasha began walking from A to B, a distance of 100 miles, Mario started walking along the same road from B to A. If Sasha's walking rate was 2 miles per hour and Mario's was 3 miles per hour, how many miles had Mario walked when they met?

[ ] 42
[ ] 46
[X] 54
[ ] 58
[ ] 64


I know the formula $$Rate=\frac{Distance}{Time}$$ but not sure how to use it here to solve this. Do you have a general rule for helping solve these mind-twisting word puzzles? Thank you.

• Sasha has already reduced the distance between Mario and him by 10 miles to just 90 miles. Every hour that the two walk, the distance between them is reduced by another (3+2) miles. In how many more hours will they meet up with each other? In that many hours, how far has Mario walked? (P.S. Your title doesn't seem to describe this problem. Aren't they on foot?) – colormegone Apr 19 '14 at 2:31
• Thanks, @RecklessReckoner. The title was from another question I was going to post, but ended up posting this more difficult one. Updated title for this one :) – Alex Apr 19 '14 at 2:44

Let $t$ be the number of hours that Mario walks before they meet. Then Sasha has walked $5+t$ hours, and $$2(5+t)+3t=100.$$

Or else, without "algebra": Sasha has walked $10$ miles before Mario sets out, so at that time they are $90$ miles apart. Then distance between them shrinks $5$ miles per hour, so it takes $18$ hours to shrink to $0$.

• Thanks, @André Nicolas. Been away from this stuff a looong time! – Alex Apr 19 '14 at 2:38
• You are welcome. This kind of test requires practice, the questions change (a little) but the themes do not. – André Nicolas Apr 19 '14 at 2:40

Let M be mario's distance and let S be Sasha's distance. Then $S = 2t$ and $M = 100 - 3(t-5)$. Equate them and you have $2t = 100 - 3(t-5)$ simplifying gives $5t = 113$ thus $t = 23$. Thus mario has walked 3*(23-5) = 54 miles.

• Please use LaTeX thoroughout your response. Thums down! – Sandeep Silwal May 20 '14 at 3:44