Distance and speed of two people walking 100 miles?

Question

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 R=DT 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.

Answer 1

Actually, the formula is D(istance)=R(ate) * T(ime). First, find out what the problem is asking and then solve the equation to give that type of answer. The answer is looking for distance and we already have a formula for distance. If we wanted Rate, we would divide by Time giving R = D/T. For Time, T=D/R.

Second, we know that when they meet, since they are traveling in opposite directions, that they must have traveled the total distance. So Sasha'a distance + Mario's distance = Total distance. Since we know that D=R*T, we have the start of our equations: (R of Sasha * T of Sasha) + (R Mario * T Mario) = D total.

During the time that they are both walking, they are walking for time X. Since Sasha started first, we have Sasha's time as X + 5.

Now lets put in all of our known numbers: (2*(x+5)) + (3 * x) = 100. Solving gives us a value of 18 for x. So, Mario walked for 18 hours, and Sasha walked for 23 hours (18+5).

Now using the original formula of D=R*T, we have Mario's distance = 3*18 = 54. For fun, Sasha's distance would be 2 * 23 or 46. To check, we can see that 54+46 does equal 100.

Answer 2

If Mario began walking 5 hours after Sasha started and Sasha walked at a rate of 2 miles per hour, this implies she walked 5*2 10 miles before Mario began. This also means that there were now 90 miles separating them, since she'd already covered 10 out of the original 100.

So:

If she walks at 2/hour and he walks at 3/hour, this means they cover a combined 5 miles of distance per hour. Since they are 90 miles apart, this implies that it'll take a combined 18 hours for them to meet.

18 hours at 3 miles per hour means Mario walked a total of 54 miles.