Question 86663
: 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,
{{{4/M}}} + {{{4/J}}} + {{{2/B}}} = 1
: 
or if Mike and Joe work together for 2 hrs and Bill works alone for 5 hours,
{{{2/M}}} + {{{2/J}}} + {{{5/B}}} = 1 
:
or 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
:
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:
{{{4/M}}} + {{{4/J}}} + {{{10/B}}} = 2
{{{4/M}}} + {{{4/J}}} + {{{2/B}}} = 1
---------------------------Subtracting eliminates M and J
{{{0/M}}} + {{{0/J}}} + {{{8/B}}} = 1
{{{8/B}}} = 1
B = 8 hrs, Bill alone
:
Substitute 8 for B in 2nd and 3rd equations:
{{{2/M}}} + {{{2/J}}} + {{{5/8}}} = 1: subtract (5/8) from both sides resulting in:{{{2/M}}} + {{{2/J}}} = {{{3/8}}}
{{{6/M}}} + {{{2/J}}} + {{{1/8}}} = 1: subtract (1/8) from both sides resulting in: {{{6/M}}} + {{{2/J}}} = {{{7/8}}}
:
Using the two resulting equations
{{{6/M}}} + {{{2/J}}} = {{{7/8}}}
{{{2/M}}} + {{{2/J}}} = {{{3/8}}}
------------------------Subtracting eliminates J
{{{4/M}}} + {{{0/J}}} = {{{4/8}}}
{{{4/M}}} = {{{4/8}}}: cross multiply
4M = 4*8
4M = 32
M = 32/4
M = 8 hrs, Mike alone
:
Find J using the 1st equation;
{{{4/M}}} + {{{4/J}}} + {{{2/B}}} = 1
{{{4/8}}} + {{{4/J}}} + {{{2/8}}} = 1
{{{4/J}}} + {{{6/8}}} = 1
{{{4/J}}} = 1 - {{{6/8}}}
{{{4/J}}} = {{{2/8}}}; 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:
{{{2/8}}} + {{{2/16}}} + {{{5/8}}} =
{{{2/8}}} + {{{1/8}}} + {{{5/B}}} = 1