Question 4363
Let A = number of days it would take Allan to do the job alone,
Let B = number of days it would take Bobby to do the job alone, and
Let C = number of days it would take Charlie to do the job alone.



{{{1/A}}} = part of the job that Allan can do in 1 day,
{{{1/B}}} = part of the job that Bobby can do in 1 day, and
{{{1/C}}} = part of the job that Charlie can do in 1 day.


Also given in the problem, 
{{{1/42}}} = part of the job that Allan and Bobby can do in 1 day,
{{{1/31}}} = part of the job that Bobby and Charlie can do in 1 day, and
{{{1/20}}} = part of the job that Allan and Charlie can do in 1 day.


There are three unknowns, so there must be three equations to solve:


{{{1/A + 1/B = 1/42}}}  = Equation #1
{{{1/B + 1/C = 1/31}}}  = Equation #2
{{{1/A + 1/C = 1/20}}}  = Equation #3


To eliminate one of these unknowns, Subract Equation #2 - Equation #3, which eliminates the {{{1/C}}}

{{{1/B-1/A = 1/31 - 1/20}}}


To this equation, now add Equation #1, which eliminates the {{{1/A}}}
{{{1/B-1/A = 1/31 - 1/20}}}
{{{1/A + 1/B = 1/42}}}  


{{{1/B + 1/B = 1/31 - 1/20 + 1/42}}}


Finding a least common denominator for denominators like 31, 20, and 42 is harder than just using the product of the denominators, which is {{{31*20*42}}}= 26040.  The result is as follows:


{{{2/B = (1/31) * ((20*42)/(20*42)) - (1/20) * ((31*42)/(31*42)) + (1/42) *((31*20)/(31*20))}}}


{{{ 2/B = (20*42 - 31*42 + 31*20) / (31*20*42) }}}

{{{ 2/B = (840- 1302 + 620) / (31*20*42) }}}

{{{ 2/B = (158) / (31*20*42) }}}


Divide both sides by 2, giving

{{{ 1/B = (79) / (31*20*42) }}}


To find B, take the reciprocal of both sides:


{{{B= (31*20*42)/79 }}}, which is approximately 330 days.


To find A,

{{{1/A = 1/42 - 1/B}}}

{{{1/A = 1/42 - 79/(31*20*42)}}}
{{{1/A = ((31*20) - 79)/ (31*20*42)}}}

{{{1/A = 541/ (31*20*42) }}}

{{{A = (31*20*42)/541}}} , which is approximately 48 days.

To find C

{{{1/C= 1/31 -1/B}}}

{{{1/C= 1/31-79/(31*20*42)}}}
{{{1/C= ((42*20)- 79)/ (31*20*42)}}}

{{{1/C = 761/ (31*20*42) }}}

{{{C = (31*20*42)/761}}} , which is approximately 34 days.


[NOTE:  Where did you get this problem??  If I had known when I started this problem that it would have been this complicated, I would have left it for Kenny!!  There is probably an easier way to do this.  Ask him!!]


R^2 at SCC