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let x=# of hours it takes trainee to paint the room
standard setup:
cross multiply
x=23 1/3 hours
Hi Allison,
Jane can paint the room in 10 hr; if she and the trainee paint the room in 7 hr, then we know that Jane has painted 70% (7/10) of the room. Therefore the trainee painted 30% of the room in 7 hr.
The trainee can paint 30% of the room in 7 hr. If the trainee has a room all to herself to paint, she takes 7 hr for the first 30%. She can paint another 30% in 7 more hr, another 30% in 7 more hr. This gets us to 30+30+30%, or 90%, in 7+7+7 hr, or 21 hr.
So far the (tired) trainee has painted 90% of the room in 21 hr. I'll leave it to you to figure out how much the trainee needs to finish the room !

Here is another way to think about this problem, using the well-known formula, rate x time = distance, or in this case, rate x time = amount of the job done.

Since we know that Jane can finish the job by herself in 10 hours, that means that her rate is 1/10 of the job per hour.

If Jane and her trainee can do the job together in 7 hours, that means that Jane was able to do rate x time, or 1/10 x 7 = 7/10 of the job in those 7 hours. If she did 7/10 of the job, then her trainee did 3/10 of the job during the 7 hours.

Therefore, to find the trainee's rate, we can set up an equation for the trainee as follows:

rate x time = amount of the job done, or r x 7 = 3/10, which can be written 7r = 3/10. Multiply both sides of the equation by 10 and you get 70r = 3. Then divide by 70 and you find that r = 3/70. The trainee's rate is 3/70 of the job done in 1 hour!

Now, we can go back to our equation and substitute 3/70 for the trainee's rate and 1 for the amount of the job done (1 whole job) to find the amount of time that it would take the trainee to do the job by herself:

rate x time = amt. of job done, or 3/70 x t = 1, which can be written 3t/70 = 1. Multiply both sides of the equation by 70 and you get 3t = 70. Then divide both sides by 3 and you get t = 70/3 = 23 1/3 hours for the trainee to do the job by herself.