Question 140876
A,B,and C can finish a job in six days. If B and C work together, the job will take 9 days; if A and C work together, the job will take 8 days. In how many days can each man working alone do the job?
:
Let the completed job = 1
write 3 equations
{{{6/A}}} + {{{6/B}}} + {{{6/C}}} = 1
:
{{{9/B}}} + {{{9/C}}} = 1
:
{{{8/A}}} + {{{8/C}}} = 1
:
Use elimination: mult the eq1 by 3, and eq2 by 2:
{{{18/A}}} + {{{18/B}}} + {{{18/C}}} = 3
{{{0/A}}} +  {{{18/B}}} + {{{18/C}}} = 2
----------------------------------------- Subtraction eliminates B & C, Find A
{{{18/A}}} = 1
A = 18 days by himself
:
Use elimination: mult the eq1 by 4, and eq3 by 3:
{{{24/A}}} + {{{24/B}}} + {{{24/C}}} = 4
{{{24/A}}} +  {{{0/B}}} + {{{24/C}}} = 3
----------------------------------------- Subtraction eliminates A & C, Find B
24/B = 1
B = 24 days by himself
:
Use eq2, substitute 24 for B, find C
{{{9/24}}} + {{{9/C}}} = 1
Multiply equation by 24C
9C + 24(9) = 24C
216 = 24C - 9C
216 = 15C
C = {{{216/15}}}
C = 14.4 days by himself 
;
Check solutions in eq1
{{{6/18}}} + {{{6/24}}} + {{{6/14.4}}} = 
.333 + .250 + 4.17 = 1