Al and Bill live at opposite ends of the same street. Al had to deliver a parcel to Bill’s home, Bill one at Al’s home. They started at the same moment, each walked at constant speed and returned home immediately after leaving the parcel at its destination. They met the first time at a distance of a yards from Al’s home and the second time at a distance of b yards from Bill’s home.
Assume that Al and Bill walk less than twice as fast as each other.
a. How long is the street?
b. If a = 300 and b = 400, who walks faster?
see level 2 prob six on page 15 on this document: http://math.fau.edu/MathematicsCompetition/spring2000/spring2000solution.pdf
Now that I have the answer to those parts... how would I change the four equations we use to describe the problem if Al walks more than twice as fast as Bill?
To solve this problem, we need to set up a system of equations and use the given information to solve for the unknowns.
Let's denote the length of the street as "L" and the speed of Al as "v" and the speed of Bill as "2v" (since it is stated that they walk less than twice as fast as each other).
a. To find the length of the street, we can use the fact that they meet the first time at a distance of "a" yards from Al's home and the second time at a distance of "b" yards from Bill's home.
When they meet for the first time, Al has walked "a" yards and Bill has walked (L - a) yards. Since they walk at constant speeds, we can set up the equation:
a / v = (L - a) / (2v)
Simplifying this equation, we get:
2a = L - a
3a = L
Similarly, when they meet for the second time, Al has walked (L - b) yards and Bill has walked "b" yards. We can set up the equation:
(L - b) / v = b / (2v)
Simplifying this equation, we get:
2(L - b) = b
2L - 2b = b
2L = 3b
b. If a = 300 and b = 400, we can use the equations we derived to find the lengths of the street (L) and compare the speeds of Al and Bill.
From the equation 3a = L, substituting a = 300, we can find L:
3(300) = L
900 = L
From the equation 2L = 3b, substituting b = 400, we can find L:
2L = 3(400)
2L = 1200
L = 600
Now, we can compare the speeds of Al and Bill:
Al's speed, v = Al's distance / time = L / (2a/v)
Bill's speed, 2v = Bill's distance / time = L / (2b/(2v))
Since L is the same in both equations, we can simplify and compare the ratios:
v / (2a) = 1
v / (2b) = 1/2
From these ratios, we can conclude that Al walks faster, as v is greater than 2b.
In summary:
a. The length of the street is 900 yards.
b. Al walks faster.