Rate and Fractions Problem

by Ryan
(Bethel)

Fred washes the car in 8 minutes. Al washes the same car in 6 minutes how long will it take both of them to wash the car together?

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Sep 07, 2015
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Multiple Solutions Along Curve
by: Anonymous

Fred washes the car in 8 minutes. Al washes the same car in 6 minutes how long will it take both of them to wash the car together?

x= Percentage of car Al Washes
1-x= Percentage of car Fred Washes

x is in [0,1]

It takes Al 6*x min to wash x percent of the car
It takes Fred 8*(1-x) min to wash (1-x) percent of the car

Time = 6*(x^2)+8(1-x)^2

Any solution is possible with x between 0 and 1.

min solution:
take derivative
dTime/dx = 12x-16(1-x)
dTime/dx = 28x-16

28x-16=0
x= 16/28 = 4/7

so Al does 4/7 of the work and Fred does 3/7.
substitute back into Time equation

6*((4/7)^2)+8(3/7)^2 = 24/7

May 14, 2014
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No answer is right
by: Anonymous

Hi, i watch this same problem in the tv show "Boy meets world" and at the end they said that there is not solution, i disagreed but giving a second thought i think that they are right....is a logical problem, you can only use the method that you use, as long as you assume that the situation is "ideal" i mean that they will start to watch the car at the exact same time with the exact same technique...but is assume to much...they can work together and say "i will start the first half and then you will take over" or infinity of variables...the problem should say "how long they will take if they start at the same time" but again...what if they have the exact same technique? what if they both start in the front and at the left? then they cannot start at the same time....you have to assume that they can start watching at any point...what people forget very often is that you have to assume that the problem is talking about "ideal situations" what if they never have watched a car together? hahaha they should take at least an extra minute in get organized....Greetings!!

Aug 09, 2011
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Rate and Fractions Problem
by: Karin

Hi Ryan,
Your problem states:

Fred washes the car in 8 minutes. Al washes the same car in 6 minutes how long will it take both of them to wash the car together?

You must think of this problem in terms of rates. If Fred washes the car in 8 minutes, then that means that he washes the car at a rate of 1/8 (1/8 of the car per minute). Al washes at a rate of 1/6(1/6 of the car per minute).

If they work together, then add their two rates.
(1/6+1/8 = 7/24)

Their rate together is 7/24. In order to wash 1 car, think of the algebraic equation:

(7/24)(x) = 1 (rate x time)

x = 24/7

Therefore, it would take 3 and 3/7 minutes to wash the car together. (24/7 = 3 and 3/7)

I hope this helps,
Karin

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