Two people (A and B) want to meet each other to have lunch, and plan to meet at B's house party. A starts out for B's house, but when A is a mile away from B's house, B realizes that his house is a mess and A's isn't. So B takes off to meet A.

Once they meet, they talk for a while, and decide to meet at A's house instead. A goes back to her house to wait for B, and B goes to his house to pick up his food, then goes to A's house. They arrive at A's house at exactly the same time. If B walks twice as fast as A, how far apart do the two people live?

Okay, I started out by finding where A and B met for the first time. We know since they meet between n=0 and n=1 where n is the distance from B's house (since B starts out at 0 and A starts at 1), the distance traveled by A and B (henceforth repesented by a and b, respectively)is equal to one. We know in t Time, B can travel 2 times as much distance as A, so therefore b/t=2a/t or b=2a. Since they travel 1 mile in total, b+a=1, and substituting 2a for b, 3a=1 so a=1/3 and b=2(1/3)=2/3.

Where d is the total distance to travel between A's house and B's house, B will travel 2/3+d mi to get to A's house, as B starts out at 2/3 mi away from his house and therefore travels 2/3 mi to get to his house then d distance to get to A's house. A, on the other hand, will travel d-2/3 mi to get to his house, as A is 2/3 mi away from B's house.

So, assume it takes them x time to get to A's house. We know (d+2/3)/x is equal to b and (d-2/3)/x is equal to a. Since we know b=2a, substituting the values for a and b we get (d+2/3)/x=(2d-4/3)/x. Multiplying both sides by x we get d+2/3=2d-4/3. Subtract d from both sides and we get 2/3=d-4/3. Add 4/3 to both sides and the result is 6/3=d. Since 6/3 is equal to 2, the total distance traveled between the two houses is equal to 2 miles.

This result is arrived at by assuming A and B have to travel in a straight line. If, however, they do not travel in a straight line, without further data it is impossible to determine the absolute distance between the two houses. The best we can do is essentially "taxi distance".

So, in case you'd rather not read all that, my answer is 2 if they travel in a straight line and no solution if they do not.

Posted by Sniper59 on 2003-10-31 20:25:45