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

,

so his rate in jobs/day is 



---

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

,

so her rate in jobs/day is 



We look at the second sentence first:

In 2 days, using 

production = rate × time, Jack's production =  or .

In 4 days, using 

production = rate × time, Jill's production =  or .

Since they finish 1 job, 



Now we look at the first sentence:

Their combined rate is the sum of their rates, so

Their combined rate = 

So in 3 days, using 

production = rate × time, the production =  or .

Since they finish 1 job, 



So the system of equations is



Adding them term by term gives:



Multiply both sides by y



So Jack can do the job in 6 days.

Substituting y=6 in







Multiply through by 3x





So Jill can also do the job in 6 days.

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

Edwin

Since "Jack and Jill together can do a piece of work in 3 days", they do  of the job working together.

The next phrase of the condition says:
    "They can finish the work if Jack works for 2 days and Jill works for 4 days."

You can re-phrase it in this equivalent way:
    They can finish the work if Jack and Jill work together for 2 days and then Jill works for additional 2 days.


OK, very good.
But then, working 2 days together, Jack and Jill will do  of the job.

It means that Jill can complete the remaining  of the job in two days.
Hence, Jill can complete the entire job in 6 days.


    Half of the problem is just solved. Now we can easily complete the rest, too.


Since Jill does the entire job in 6 days, in three days she makes  of the job.
It means that in 3 day Jack makes the remaining  of the job.
In turn, it means that Jack can do the entire work in 6 days.