Question 1067383
What we have to find is
{{{a}}}= number of days that it would take worker A to complete the job by himself (or herself),
and
{{{b}}}= number of days needed for B to complete the job working alone.
In one day of work, the fraction of the job that
A would complete is {{{1/a}}} ,
and the fraction of the job B would complete
is {{{1/b}}} .
Working together, they would complete
{{{1/a+1/b}}} of the job each day,
and that would be {{{1/8}}} of the job.
(You could also say that
{{{8*(1/a+1/b)=1}}} ),
but either way, you end up with
{{{1/a+1/b=1/8}}} .
{{{6*(1/8)=6/8=3/4}}} is the fraction of the job
that A and B complete, working together,
during the first 6 days.
{{{6*(1/b)}}} is the fraction of the whole job
that B does all alone, during the next 6 days.
After that, the fraction of the job that has been completed is
{{{1=3/4+6(1/b)}}} or {{{1=3/4+6/b}}} .
We solve that to find {{{b}}} :
Multiplying both sides of the equal sign times {{{4b}}}
we get the equivalent equation
{{{4b=3b+24}}}
{{{4b-3b=24}}}
{{{highlight(b=24)}}} .
Now, substituting the value for {{{b}}}
into {{{1/a+1/b=1/8}}} ,
we get {{{1/a+1/24=1/8}}} ,
which we solve for {{{a}}} .
First, we multiply both sides of the equal sign
times {{{24a}}} to get the equivalent equation
{{{24+a=24a/8}}}
{{{24+a=3a}}}
{{{24=3a-a}}}
{{{24=2a}}}
{{{a=24/2}}}
{{{highlight(a=12)}}} .