Question

Alice and Bill live at opposite ends of the same street. They leave their houses at the same time and each walk,

at constant speed, from their house to the other house and back. The first time they meet, they are 400 yards
from Alice's house, and the second time they meet, they are 300 yards from Bill's house. Both times they aretraveling in opposite directions. What is the distance, in yards, between the two houses?

Answer 1

By setting up a ratio based on the two meeting points between Alice and Bill, we can establish that the distance between their houses is 1600 yards.

Step-by-step explanation:

To solve the problem of finding the distance between Alice and Bill's houses, we can set up a relative motion problem. Since we know the two points where Alice and Bill meet and they travel at constant speeds, we can use ratios to determine the total distance between the two houses.

Let's denote the distance between Alice and Bill's houses as d yards. Alice and Bill meet for the first time when Alice has traveled 400 yards and the second time when Bill has traveled d - 300 yards. At their first meeting, since both have walked the same amount of time, the ratio of their distances will be the same when they meet for the second time.

At the first meeting:
Alice's Distance: 400 yards
Bill's Distance: d - 400 yards

At the second meeting, after they have both turned around and met again:
Alice's Distance: d - (d - 300) yards = 300 yards
Bill's Distance: 2d - (400 + d - 300) yards = 2d - d + 100 yards = d + 100 yards

Setting up the ratios:
400 / (d - 400) = 300 / (d + 100)

Now we can solve for d by cross-multiplying and simplifying:

400(d + 100) = 300(d - 400)
400d + 40,000 = 300d - 120,000
400d - 300d = 120,000 + 40,000
100d = 160,000
d = 160,000 / 100
d = 1600 yards

So, the distance between Alice and Bill's houses is 1600 yards.