Question 1058038
Jack and Jill together can do a piece of work in 3 days. 
They can finish the work if Jack works for 2 days and Jill 
works for 4 days. Find the time required for each to do the 
work.
<pre>
Let the time for Jack to do the job = x days

So Jack's rate in jobs per day is

1 job per x days or

{{{matrix(1,2,1,job)/matrix(1,2,x,days)}}},

so his rate in jobs/day is 

{{{matrix(1,2,1/x,jobs/day)}}}

---

Let the time for Jill to do the job = y days

So Jill's rate in jobs per day is

1 job per y days or

{{{matrix(1,2,1,job)/matrix(1,2,y,days)}}},

so her rate in jobs/day is 

{{{matrix(1,2,1/y,jobs/day)}}}

We look at the second sentence first:
</pre>
They can finish the work if Jack works for 2 days 
and Jill works for 4 days.
<pre>
In 2 days, using 

production = rate × time, Jack's production = {{{2*(1/x)}}} or {{{2/x}}}.

In 4 days, using 

production = rate × time, Jill's production = {{{4*(1/y)}}} or {{{4/y}}}.

Since they finish 1 job, 

{{{2/x+4/y}}}{{{""=""}}}{{{1}}}

Now we look at the first sentence:
</pre>
Jack and Jill together can do a piece of work in 3 days.
<pre>
Their combined rate is the sum of their rates, so

Their combined rate = {{{1/x+1/y}}}

So in 3 days, using 

production = rate × time, the production = {{{3*(1/x+1/y)}}} or {{{3/x+3/y}}}.

Since they finish 1 job, 

{{{3/x+3/y}}}{{{""=""}}}{{{1}}}

So the system of equations is

{{{system(2/x+4/y=1,3/x+3/y=1)}}

Don't clear of fractions.  Instead leave them
as they are and use elimination.

Multiply the first equation through by 3 and
the second equation through by -2

{{{system(6/x+12/y=3,-6/x-6/y=-2)}}}

Adding them term by term gives:

{{{6/y=1}}}

Multiply both sides by y

{{{6=y}}}

So Jack can do the job in 6 days.

Substituting y=6 in

{{{2/x+4/y=1}}}

{{{2/x+4/6=1}}}

{{{2/x+2/3=1}}}

Multiply through by 3x

{{{6+2y=3x}}}

{{{6=y}}}

So Jill can also do the job in 6 days.

They can each do the job working alone in 6 days.

Edwin</pre>