Question 998425: In a 20-mile bicycle race, Matthew went twice as fast as Tim, and Sally went
twice as fast as Matthew. Sally finished 3.75 hours ahead of Tim. What was
Matthew’s average speed (in miles per hour) for the race?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula to use is rate * time = distance.
we let x = tim's rate.
distance = 20.
since matthew is twice as fast as tim, then 2x = matthew's rate.
since sally is twice as fast as matthew, then 4x = sally's rate.
we let y = sally's time.
we are given that sally takes 3.75 hours less than tim.
this means that tim take 3.75 hours more than sally to finish the race.
tim's time is therefore equal to sally's time + 3.75 hours.
we can therefore represent tim's time as y + 3.75.
since sally runs twice as fast as matthew, then matthew must take twice as long.
since sally takes y hours, matthew must take 2y hours.
you get the following table:
rate time distance
matthew 2x 2y 20
tim x y + 3.75 20
sally 4x y 20
we can use matthew's numbers or sally's numbers to solve for y in terms of x.
using sally, we get:
rate * time = distance becomes 4x * y = 20
solve for y to get y = 20/4x.
matthew would have given you the same result.
now that we have the value of y in terms of x, we can use tim's numbers to solve for y.
using tim, we get:
rate * time = distance becomes x * (y + 3.75) = 20
since y = 20/4x, the fomula becomes:
x * (20/4x + 3.75) = 20
if we remove parentheses on the left side of this equation, we get:
20x/4x + 3.75x = 20
simplify to get:
5 + 3.75x = 20
subtract 5 from both sides of this equation to get:
3.75x = 15
divide both sides of this equation by 3.75 to get:
x = 4
we now have the rate of speed for tim.
it's 4 miles per hour.
since matthew is twice as fast as tim, his rate is 8 miles per hour.
since sally is twice as fast as matthew, her rate is 16 miles per hour.
since we now have each person's rate, we can solve for each person's time because rate * time = 20.
for matthew, 8 * time = 20 gets time = 20/8 = 2.5 hours.
for tim, 4 * time = 20 gets time = 5 hours.
for sally, 16 * time = 20 gets time = 1.25 hours.
we get the following results:
rate time distance
matthew 8 2.5 20
tim 4 5 20
sally 16 1.25 20
you were asked for matthew's rate of speed.
it's 8 miles per hour.
the fastest time in a marathon is equal to 2 hours approximately (actually 2 hours 3 minutes and 2 seconds).
that's an average speed of 26 / 2 = 13 miles per hour.
since sally's average speed is 16 miles per hour, she's really moving.
what this means is that the results are a little unrealistic, but still accurate based on the facts given in the problem statement.
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