Question 499228
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 does it take each man working alone to complete the painting.
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Write an equation for each scenario, let the completed job = 1
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The painting can be finished if Mike and Joe work together for 4 hours and Bill works alone for 2 hours,"
{{{4/m}}} + {{{4/j}}} + {{{2/b}}} = 1
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"if Mike and Joe work together for 2 hours and Bill works alone for 5 hours,"
{{{2/m}}} + {{{2/j}}} + {{{5/b}}} = 1
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"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
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We are going to use elimination here multiply the 2nd eq by 2, subtract the 1st equation
{{{4/m}}} + {{{4/j}}} + {{{10/b}}} = 2
{{{4/m}}} + {{{4/j}}} + {{{2/b}}} = 1
---------------------------------------subtraction eliminates m and j, find b
{{{8/b}}} = 1
b = 8 hrs for Bill alone
:
In the 2nd equation replace b with 8
{{{2/m}}} + {{{2/j}}} + {{{5/8}}} = 1
{{{2/m}}} + {{{2/j}}} = 1 - {{{5/8}}}
{{{2/m}}} + {{{2/j}}} = {{{3/8}}}
Do the same with the 3rd equation
{{{6/m}}} + {{{2/j}}} + {{{1/8}}} = 1 
{{{6/m}}} + {{{2/j}}} = {{{7/8}}} 
use these two equation for elimination
{{{6/m}}} + {{{2/j}}} + {{{7/8}}}
{{{2/m}}} + {{{2/j}}} + {{{3/8}}}
----------------------------------Subtraction eliminates j, find m
{{{4/m}}} = {{{4/8}}}
m = 8 hrs for Mike alone
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find j using eq:{{{2/m}}} + {{{2/j}}} = {{{3/8}}}, replace m with 8
{{{2/8}}} + {{{2/j}}} = {{{3/8}}}
{{{2/j}}} = {{{3/8}}} - {{{2/8}}}
{{{2/j}}} = {{{1/8}}}
Cross multiply
j = 2*8
j = 16 hrs for Joe alone
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to do the job alone: m = 8 hrs, j = 16 hrs, b = 8hrs
:
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Confirm this in the 1st equation
{{{4/8}}} + {{{4/16}}} + {{{2/8}}} = 1
which is
{{{4/8}}} + {{{2/8}}} + {{{2/8}}} = 1