Question 743397
12 boys and 8 men can finish a piece of work in 5 days.and 8 boys nd 6men can finish it in 7 days . find the time by 1 man alone and 1 boy alone to finish the work.
<pre>
Suppose 1 boy alone can can do 1 job in b days.

Then 1 boy's work rate is 1 job per b days or {{{1_job/b_days}}} or {{{1/b}}}{{{job/day}}}

Then 12 boys' work rate is 12 times {{{1_job/b_days}}} or {{{12/b}}}{{{job/day}}}

and 8 boys' work rate is 8 times {{{1_job/b_days}}} or {{{8/b}}}{{{job/day}}}

Suppose 1 man alone can can do 1 job in m days.

Then 1 man's work rate is 1 job per m days or {{{1_job/m_days}}} or {{{1/m}}}{{{job/day}}}

Then 8 men's work rate is 8 times {{{1_job/m_days}}} or {{{8/m}}}{{{job/day}}}

and 6 men's work rate is 6 times {{{1_job/m_days}}} or {{{6/m}}}{{{job/day}}}
</pre>
>>12 boys and 8 men can finish a piece of work in 5 days<<
<pre>
So the combined work rate of 12 boys and 8 men is 1 job per 5 days or {{{1_job/5_days}}} or {{{1/5}}}{{{job/day}}}

So we get one equation from

{{{(matrix(8,1,
12, "boys'",work, rate,in, jobs,per,day))}}}{{{""+""}}}{{{(matrix(8,1,
8, "men's",work, rate,in, jobs,per,day))}}}{{{""=""}}}{{{(matrix(14,1,
the, combined,work,rate,of,12,boys,and,8,men,in, jobs,per,day))}}}

{{{12/b}}}{{{""+""}}}{{{8/m}}}{{{""=""}}}{{{1/5}}}
<pre>
>>8 boys and 6 men can finish it in 7 days<<
</pre>
So the combined work rate of 8 boys and 6 men is 1 job per 7 days or {{{1_job/5_days}}} or {{{1/5}}}{{{job/day}}}

So we get the other equation from

{{{(matrix(8,1,
8, "boys'",work, rate,in, jobs,per,day))}}}{{{""+""}}}{{{(matrix(8,1,
6, "men's",work, rate,in, jobs,per,day))}}}{{{""=""}}}{{{(matrix(14,1,
the, combined,work,rate,of,8,boys,and,6,men,in, jobs,per,day))}}}

{{{8/b}}}{{{""+""}}}{{{6/m}}}{{{""=""}}}{{{1/7}}} 

So we have the system of equations

{{{system(12/b+8/m=1/5,8/b+6/m=1/7)}}}

IMPORTANT:  DO NOT CLEAR OF FRACTIONS:

To eliminate the terms in m, multiply the first equation 
through by 3, and multiply the second equation through
by -4:

{{{system(36/b+24/m=3/5,-32/b-24/m=-4/7)}}}
 
Adding those equations term by term gives

{{{4/b=3/5-4/7}}}

NOW we can clear of fractions.  Multiply both sides
by the LCD of 35b

{{{140=21b-20b}}}

{{{140=b}}}

So each boy will take 140 days.

Substitute 140 for b in

{{{12/b+8/m=1/5}}}
{{{12/120+8/m=1/5}}}
{{{1/10+8/m=1/5}}}

Multiply both sides by 10m

{{{m+80=2m}}}
{{{80=m}}}

So each man will take 80 days.

Edwin</pre>