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Question 86663: Mike, Joe, and Bill are painting a fence. The painting can be finished if MIke and JOe work together for 4 horus and Bill works alone for 2 hours, or if Mike and Joe work together for 2 horus and Bill works alone for 5 hours, or if MIke works alone for 6 hours, Joe works alone for 2 hours, and Billw orks alone for 1 hour. how much time would it take each man to paint the entire fence by himself?
Found 2 solutions by ankor@dixie-net.com, Edwin McCravy:
Answer by ankor@dixie-net.com(15647)   (Show Source):
You can put this solution on YOUR website!: Mike, Joe, and Bill are painting a fence.
:
Let M = hrs required if Mike works alone
Let J = hrs required if Joe works alone
Let B = hrs required if Bill works alone
Let completed job = 1
:
The painting can be finished if Mike and Joe work together for 4 hrs and Bill works alone for 2 hours,
 +  +  = 1
:
or if Mike and Joe work together for 2 hrs and Bill works alone for 5 hours,
 +  +  = 1
:
or if MIke works alone for 6 hours, Joe works alone for 2 hours, and Bill works alone for 1 hour.
 + +  = 1
:
how much time would it take each man to paint the entire fence by himself?
:
Multiply the 2nd equation by 2 and subtract the 1st equation:
+ +  = 2
+ + = 1
---------------------------Subtracting eliminates M and J
 +  +  = 1
= 1
B = 8 hrs, Bill alone
:
Substitute 8 for B in 2nd and 3rd equations:
+ +  = 1: subtract (5/8) from both sides resulting in: + = 
+ +  = 1: subtract (1/8) from both sides resulting in: + = 
:
Using the two resulting equations
+ =
+ =
------------------------Subtracting eliminates J
+ = 
= : cross multiply
4M = 4*8
4M = 32
M = 32/4
M = 8 hrs, Mike alone
:
Find J using the 1st equation;
+ + = 1
+ +  = 1
+  = 1
= 1 -
= ; cross multiply
2J = 4 * 8
2J = 32
J = 16 hrs, Joe alone
:
Check our solutions M=8, J=16, B=8, in the 2nd equation:
+  + =
+ + = 1
Answer by Edwin McCravy(8904)  (Show Source):
You can put this solution on YOUR website!Mike, Joe, and Bill are painting a fence. The painting can be finished if MIke and Joe work together for 4 hours and Bill works alone for 2 hours, or if Mike and Joe work together for 2 hours and Bill works alone for 5 hours, or if MIke works alone for 6 hours, Joe works alone for 2 hours, and Bill works alone for 1 hour. how much time would it take each man to paint the entire fence by himself?
Let m = the number of hours Mike requires to paint a fence.
Let j = the number of hours Joe requires to paint a fence.
Let b = the number of hours Bill requires to paint a fence
Make this chart, flling in all the times
Number of jobs
or
Fraction of job Rate in Time in
done jobs/hr hours
Mike alone for m hrs m
Mike alone for 6 hrs 6
Joe alone for j hrs j
Joe alone for 2 hrs 2
Bill alone for b hrs b
Bill alone for 1 hr 1
Bill alone for 2 hrs 2
Bill alone for 5 hrs 5
Mike and Joe for 2 hrs 2
Mike and Joe for 4 hrs 4
Since Mike requires m hours to paint 1 fence, fill in 1 fence for
number of jobs job done or fraction of jobs done when he works
m hours.
Since Joe requires j hours to paint 1 fence, fill in 1 fence for
number of jobs job done or fraction of jobs done when he works
j hours.
Since Bill requires b hours to paint 1 fence, fill in 1 fence for
number of jobs job done or fraction of jobs done when he works
b hours.
Number of jobs
or
Fraction of job Rate in Time in
done jobs/hr hours
Mike alone for m hrs 1 m
Mike alone for 6 hrs 6
Joe alone for j hrs 1 j
Joe alone for 2 hrs 2
Bill alone for b hrs 1 b
Bill alone for 1 hr 1
Bill alone for 2 hrs 2
Bill alone for 5 hrs 5
Mike and Joe for 2 hrs 2
Mike and Joe for 4 hrs 4
Now fill in Mike's Joe's and Bill's "working alone" rates
by using
Number of jobs or fraction of job done
Rate = -------------------------------------------
Time
Number of jobs
or
Fraction of job Rate in Time in
done jobs/hr hours
Mike alone for m hrs 1 1/m m
Mike alone for 6 hrs 6
Joe alone for j hrs 1 1/j j
Joe alone for 2 hrs 2
Bill alone for b hrs 1 1/b b
Bill alone for 1 hr 1
Bill alone for 2 hrs 2
Bill alone for 5 hrs 5
Mike and Joe for 2 hrs 2
Mike and Joe for 4 hrs 4
Now fill in those same rates for all cases of Mike working alone,
for all cases of Joe working alone, and for all cases of Bill
working alone.
Number of jobs
or
Fraction of job Rate in Time in
done jobs/hr hours
Mike alone for m hrs 1 1/m m
Mike alone for 6 hrs 1/m 6
Joe alone for j hrs 1 1/j j
Joe alone for 2 hrs 1/j 2
Bill alone for b hrs 1 1/b b
Bill alone for 1 hr 1/b 1
Bill alone for 2 hrs 1/b 2
Bill alone for 5 hrs 1/b 5
Mike and Joe for 2 hrs 2
Mike and Joe for 4 hrs 4
Now use
Number of jobs or fraction of job done = Rate × Time
to fill in the number of jobs or fraction of a job
which Mike does alone for 6 hrs, Joe does alone for 2
hrs, and Bill does alone for 2 and for 5 hrs.
Number of jobs
or
Fraction of job Rate in Time in
done jobs/hr hours
Mike alone for m hrs 1 1/m m
Mike alone for 6 hrs 6/m 1/m 6
Joe alone for j hrs 1 1/j j
Joe alone for 2 hrs 2/j 1/j 2
Bill alone for b hrs 1 1/b b
Bill alone for 1 hr 1/b 1/b 1
Bill alone for 2 hrs 2/b 1/b 2
Bill alone for 5 hrs 5/b 1/b 5
Mike and Joe for 2 hrs 2
Mike and Joe for 4 hrs 4
Now when Mike and Joe work together, their
combined rate is the sum of their individual
rates, so fill in 1/m + 1/j for their
combined rate in the last two slots:
Number of jobs
or
Fraction of job Rate in Time in
done jobs/hr hours
Mike alone for m hrs 1 1/m m
Mike alone for 6 hrs 6/m 1/m 6
Joe alone for j hrs 1 1/j j
Joe alone for 2 hrs 2/j 1/j 2
Bill alone for b hrs 1 1/b b
Bill alone for 1 hr 1/b 1/b 1
Bill alone for 2 hrs 2/b 1/b 2
Bill alone for 5 hrs 5/b 1/b 5
Mike and Joe for 2 hrs 1/m + 1/j 2
Mike and Joe for 4 hrs 1/m + 1/j 4
Now use:
Number of jobs or fraction of job done = Rate × Time
to fill in the number of jobs or fraction of a job
which Mike and Joe do in both 2 hours and 4 hours:
Number of jobs
or
Fraction of job Rate in Time in
done jobs/hr hours
Mike alone for m hrs 1 1/m m
Mike alone for 6 hrs 6/m 1/m 6
Joe alone for j hrs 1 1/j j
Joe alone for 2 hrs 2/j 1/j 2
Bill alone for b hrs 1 1/b b
Bill alone for 1 hr 1/b 1/b 1
Bill alone for 2 hrs 2/b 1/b 2
Bill alone for 5 hrs 5/b 1/b 5
Mike and Joe for 2 hrs 2(1/m + 1/n) 1/m + 1/n 2
Mike and Joe for 4 hrs 4(1/m + 1/n) 1/m + 1/n 4
Now that we have the chart filled in completely, we can start
getting the equations:
>>...The painting can be finished if MIke and Joe work together
for 4 hours and Bill works alone for 2 hours...<<
4(1/m + 1/j) + 2/b = 1 fence painted
That simplifies to
4/m + 4/j + 2/b = 1
>>...The painting can be finished...if Mike and Joe work together
for 2 hours and Bill works alone for 5 hours...<<
2(1/m + 1/j) + 5/b = 1 fence painted
That simplifies to
2/m + 2/j + 5/b = 1
>>...The painting can be finished...if MIke works alone for 6 hours,
Joe works alone for 2 hours, and Bill works alone for 1 hour.
6/m + 2/j + 1/b = 1 fence painted
So you have this system of equations:
4/m + 4/j + 2/b = 1
2/m + 2/j + 5/b = 1
6/m + 2/j + 1/b = 1
Don't clear of fractions to solve that. Instead, let
x = 1/m, y = 1/j and z = 2/b
Then the system becomes:
4x + 4y + 2z = 1
2x + 2y + 5z = 1
6x + 2y + z = 1
Solve that and get x = 1/8, y = 1/16, 1/8
Now since x = 1/m, y = 1/j and z = 1/b,
m = 1/x, j = 1/y, b = 1/z
so
m = 1/(1/8) = 8
j = 1/(1/16) = 16
b = 1/(1/8) = 8
So
working alone, it would take Mike 8 hours to paint a fence,
working alone, it would take Joe 16 hours to paint a fence, and
working alone, it would take Bill 8 hours to paint a fence.
Edwin
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