Question 208065
If A,B and C work together on a job, it will take 1 and 1/3 hours. If only A and B work, it would take 1 and 5/7 hours, but if B and C work, it would take 2 and 2/5 hours. How long it would take each man, working alone to complete the job?
:
Change 1 & 1/3 hrs to {{{4/3}}}
Let the completed job = 1
:
Simplify the shared work equations
{{{(4/3)/A}}} + {{{(4/3)/B}}} + {{{(4/3)/C}}} = 1 >> {{{4/(3A)}}} + {{{4/(3B)}}} + {{{4/(3C)}}} = 1 >> {{{4/A}}} + {{{4/B}}} + {{{4/C}}} = 3
:
convert 1 & 5/7 hrs to {{{12/7}}}
{{{(12/7)/A}}} + {{{(12/7)/B}}} = 1 >>  {{{12/(7A)}}} + {{{12/(7B)}}} = 1 >> {{{12/(A)}}} + {{{12/(B)}}} = 7
:
Multiply 1st equation by 3 and subtract the 2nd equation
{{{12/A}}} + {{{12/B}}} + {{{12/C}}} = 9
{{{12/A}}} + {{{12/B}}} + {{{0/C}}} =  7
---------------------eliminates A & B, find C
{{{12/C}}} = 2
12 = 2C
C = 6 hrs alone 
:
The 3rd equation we can use decimal: 2 & 2/5 hrs = 2.4 hrs
{{{2.4/B}}} + {{{2.4/C}}} = 1
Replace C with 6, find B
{{{2.4/B}}} + {{{2.4/6}}} = 1
{{{2.4/B}}} + .4 = 1
{{{2.4/B}}} = 1 -.4
{{{2.4/B}}} = .6
.6B = 2.4
B = {{{2.4/.6}}}
B = 4 hrs alone
:
Find A, using the 2nd equation
{{{12/A}}} + {{{12/4}}} = 7
{{{12/A}}} + 3 = 7
{{{12/A}}} = 7 - 3
{{{12/A}}} = 4 
4A = 12
A = 3 hrs alone 
:
:
Check solution in the 1st equation using a calc
{{{(4/3)/3}}} + {{{(4/3)/4}}} + {{{(4/3)/6}}} =
:
:
:  
A and B working together can do a job in 6 hours. A become ill after 3 hours of working with B, and B finished the job, continuing to work alone in 8 hours. How long would it take each working alone to do the job?
:
Working the same together equation:
{{{6/A}}} + {{{6/B}}} = 1
:
Working when one gets sick equation: (B works a total of 11 hrs)
{{{3/A}}} + {{{11/B}}} = 1
:
Multiply the above equation by 3 and subtract the 1st equation
{{{6/A}}} + {{{22/B}}} = 2
{{{6/A}}} + {{{6/B}}} = 1
-----------------------------eliminates A
16B = 1
B = 16 hrs alone
:
Find A using the 1st equation
{{{6/A}}} + {{{6/16}}} = 1 
{{{6/A}}} = {{{16/16}}} - {{{6/16}}}
{{{6/A}}} = {{{10/16}}}
Cross multiply
10A = 6 * 16
10A = 96
A = 9.6 hrs alone
;
:
Check solution on a calc using the "sick equation"
{{{3/9.6}}} + {{{11/16}}} = 1