SOLUTION: Sara can walk the length of the sidewalk in 8 minutes if it is not moving. If she stands on the sidewalk as it moves, she can travel the length in 6 minutes. If Sara walks on the s

Question 157341: Sara can walk the length of the sidewalk in 8 minutes if it is not moving. If she stands on the sidewalk as it moves, she can travel the length in 6 minutes. If Sara walks on the sidewalk as it moves,how many minutes will it takes her to travel the same distance? Assume she always at the same rate, and express the answer as a decimal.
Could you please explain this , please?
Found 2 solutions by josmiceli, gonzo:
Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!
The solution is to add two rates:
(1) Sarah's rate of walking
(2) The sidewalk's rate of moving
----------------------------------
Added together, these rates equal
the rate of Sarah walking on the
moving sidewalk.
Call the length in the problem
Sarah's rate of walking = length/min
Sidewalk's rate of moving = length/min
Added together:
Let = number of minutes for Sarah to
travel length when sidewalk is moving

Notice I can divide both sides of this equation
by , resulting in:

multiply both sides by

min

Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
sara moves x feet in 8 minutes at the rate of r1.
the sidewalk moves x feet in 6 minutes at the rate of r2
if sara and the sidewalk move together, then sara moves x feet at the combined rate of (r1+r2)
distance = rate * time.
x = 8*r1
x = 6*r2
x = y*(r1+r2)
so 8*r1 = 6*r2 = y*(r1+r2)
now if x = 8*r1, then r1 = x/8.
similarly, if x = 6*r2, then r2 = x/6.
to solve the equation x = y*(r1+r2), we can substitute x/8 for r1 and x/6 for r2 since they are equivalent. this allows us to solve for 1 unknown rather than 3.
the equation then becomes
we multiply both sides of the equation to remove the fractions using a multiplier of 24 because both 8 and 6 divide into 24 without a remainder.
the equation then becomes
which then becomes
which then becomes
which then becomes
which finally becomes because the x cancels out.
this equals the time it takes to travel x feet at y feet per minute.
y = 24/7.
expressed as a decimal, then y = 3.428571429 minutes which is approximately 3 minutes and 26 seconds.
to prove the answer is correct, r1 and r2 should equal (r1+r2).
we don't know the distance, nor do we know the actual rate, but we can make an assumption.
say the distance is anything. we can say the distance is anything because we're dealing with ratios.
let's use 1000 feet.
so r1 must be 1000 / 8 = 125 feet per minute.
likewise r2 must be 1000 / 6 = 166.6666666.... feet per minute.
if r3 = r1 + r2, then r3 must be 125 + 166.6666666... feet per minute.
that makes r3 = 291.66666..... feet per minute.
since rate * time = distance, then time = distance / rate, so ...........
1000 feet / 291.666..... feet per minute = 3.428571429 minutes to cover the distance which is exactly the number of minutes we calculated previously.

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