Question 407455
If it takes workers A and B 2 days to complete a job.
 And it takes workers B and C 3 days to complete the same job.
 And it takes workers A and C 4 days to complete this job how long does it take
 each worker to do this job alone.
:
Let a, b, c, represent the time required by each work alone
:
Let the completed job = 1
:
Write a shared work equation for each statement:
:
"If it takes workers A and B 2 days to complete a job."
{{{2/a}}} + {{{2/b}}} = 1
:
"If it takes workers B and C 3 days to complete the same job."
{{{3/b}}} + {{{3/c}}} = 1
:
"it takes workers A and C 4 days to complete this job"
{{{4/a}}} + {{{4/c}}} = 1
:
Add all three equations to give us another equation to work with
{{{2/a}}} + {{{2/b}}} + {{{0/c}}} = 1
{{{0/a}}} + {{{3/b}}} + {{{3/c}}} = 1
{{{4/a}}} + {{{0/b}}} + {{{4/c}}} = 1
---------------------------------------
{{{6/a}}} + {{{5/b}}} + {{{7/c}}} = 3
:
Multiply the 1st equation by 3, subtract the above equation from it
{{{6/a}}} + {{{6/b}}} + {{{0/c}}} = 3
{{{6/a}}} + {{{5/b}}} + {{{7/c}}} = 3
---------------------------------------subtraction eliminates a
{{{0/a}}} + {{{1/b}}} - {{{7/c}}} = 0
:
{{{1/b}}} = {{{7/c}}}
Cross multiply
c = 7b
:
In equation: {{{3/b}}} + {{{3/c}}} = 1, replace c with 7b
{{{3/b}}} + {{{3/7b}}} = 1
Multiply by 7b, results
7(3) + 3 = 7b
21 + 3 = 7b
24 = 7b
b = {{{24/7}}}
b = 3.43 hrs, B alone
:
Find c
c = 7b
c = 7(3.43)
c = 24 hrs, C alone
:
Find a using, {{{4/a}}} + {{{4/c}}} = 1; replace c with 24
{{{4/a}}} + {{{4/24}}} = 1
multiply by 24a
24(4) + 4a = 24a
96 = 24a - 4a
96 = 20a
a = {{{96/20}}}
a = 4.8 hrs, A alone
:
:
Check this in the 1st equation
{{{2/4.8}}} + {{{2/3.43}}} = 1 
.42 + .58 = 1 confirms our solutions of:
:
A = 4.8 hrs, B = 3.43 hrs, C = 24 hrs (almost useless)