SOLUTION: 6 people worked for 9 days to complete two fifths of a project. If three joined the original 6 people, how many days were needed to complete the project?

You can do it in your head.  I'll show you that way and then I'll show you
how to do it by algebra:

IN YOUR HEAD.
When the 3 joined them, there were one and a half times as many people,
but the remaining ths of the job is one and a half times the ths they had
already completed, so it would take them the same time, or 9 days.

BY ALGEBRA:
Let x = the number of days needed to complete the remaining  of the project.

The 6 people's combined rate is ths of project per 9 days or  or 

 or  or 

Since 3 people is half of 6 people, the three who joined them's combined rate
is half that or , since half as many people work at only half the rate 
of 6 people.

Therefore the combined rate of all 9 people is  or  or 

After the three joined them after ths of the project was completed, there was 
only ths of the project left to do, so we set up this proportion:

1 project is to 15 days as ths of a project is to x days.

          

Cross multiply:

          x = ·15
          x = 9

So 9 more days were needed to complete the project.

Edwin