Question 1039403
a, b, c, the times in days for A, B, and C to do "one whole work".  Use the work rates as  {{{1/T}}} in unit of {{{JOB/DAYS}}}.


{{{system(1/a+1/b=1/12,1/b+1/c=1/16)}}}



A sequence of workers has done one whole job.
{{{(1/a)7+(1/b)5+(1/c)13=1}}}



Goal is to solve for a, b, and c.



Use the first two equations to solve each in terms of b.
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{{{1/a=1/12-1/b}}}
{{{1/a=(b-12)/12b}}}
{{{highlight_green(a=12b/(b-12))}}}
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{{{1/c=1/16-1/b}}}
{{{1/c=(b-16)/16b}}}
{{{highlight_green(c=16b/(b-16))}}}
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Substitute these into the Finished In Thirteen Days equation.
{{{((b-12)/(12b))7+((b-16)/(16b))13=1}}}


Simplify.
{{{7(b-12)/(12b)+13(b-16)/(16b)=1}}}, and simplest denominator IS  2*2*2*2*3*b;


{{{(3*2^4*b)(7(b-12)/(12b)+13(b-16)/(16b))=3*2^4*b}}}


{{{7*3*2^4*b(b-12)/(12b)+13*3*2^4*b(b-16)/(16b)=3*16b}}}


{{{7*4(b-12)+13*3(b-16)=54b}}}


{{{28b-336+39b-624=48b}}}


{{{28b+39b-48b=336+624}}}


{{{19b=960}}}


{{{b=960/19}}}

{{{b=50.53}}}------------Seems strange, maybe correct,  but needs to be rechecked ... Maybe a mistake was made.   If you can understand the process up to here, your work MIGHT be better and without any mistake I made.  You could then continue to go back to the first two equations to solve for a and c.