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Question 150999: Mike, Joe, and Bill are painting a fence. The painting can be finished if Mike and Joe each work for 4 hours and Bill works for 2 hours; or if Mike and Joe each work for 2 hours and Bill works for 5 hours; or if Mike works for 6 hours, Joe works for 2 hours, and Bill works for 1 hour. How much time does it take for each man working alone to complete the painting?
Answer by ptaylor(2198)   (Show Source):
You can put this solution on YOUR website! Let x=Mike's rate of work (in fences (or fractions thereof) painted per hour)
Let y=Joe's rate of work
Let z=Bill's rate of work
Now we know the following:
Mikes rate of work times the number of hours worked Plus Joe's rate of work times the number of hours worked plus Bill's rate of work times the number of hours worked equals 1 fence painted, so our equations to solve are:
4x+4y+2z=1--------------------------eq1
2x+2y+5z=1----------------------------eq2
6x+2y+z=1---------------------------eq3
Multiply eq2 by 2 and subtract it form eq1
4x+4y+2z=1---eq1
4x+4y+10z=2---eq2a
and we get:
-8z=-1
z=1/8----Bill's rate of work is 1/8 fence per hour so it follows that in 8 hours he can paint the fence alone
Next, subtract eq2 from eq3
6x+2y+z=1----------------eq3
2x+2y+5z=1--------------eq2
And we get:
4x-4z=0 or
x=z=1/8-------------Mike's rate of work is also 1/8 fence per hour so in 8 hours Mike can also paint the fence alone
Now we substitute x=1/8 and z=1/8 into eq1
4*(1/8)+4y+2*(1/8)=1 simplify
1/2+4y+1/4=1 further simplify
4y + 3/4 =1 subtract 3/4 from each side
4y=1/4 divide each side by 4
y=1/16-------------------Joe's rate of work is 1/16 fence per hour so in 16 hours Joe can paint the fence alone
CK
4*(1/8)+4*(1/16)+2*(1/8)=1
1/2+1/4+ 1/4=1
1=1
same should work for the other two equations
Hope this helps----ptaylor
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