SOLUTION: A painter can paint a room in 12 hours. An apprentice, who can paint the room in 20 hours is added to the workforce. If they work together at the rates indicated, the number of hou

Question 1198277: A painter can paint a room in 12 hours. An apprentice, who can paint the room in 20 hours is added to the workforce. If they work together at the rates indicated, the number of hours needed to paint the room will be
A) 6.5
B) 7.5
C) 8.5
D) 9
E) 12
I have already tried multiple solutions, all of which have sadly failed.
Hope you can help😁

Found 3 solutions by josgarithmetic, greenestamps, Alan3354:
Answer by josgarithmetic(39627)   (Show Source): You can put this solution on YOUR website!
The painter rate
An apprentice rate
Combined Rate both:

Solve the combined rate, and find its reciprocal. That is how many hours if both do the job together.
Answer by greenestamps(13206)   (Show Source): You can put this solution on YOUR website!

The response from the other tutor describes the standard method for solving "working together" problems like this; but it is so brief that it might not be much help to a student who is trying to learn how to solve the problem.

Here is an expanded explanation of the method.

The painter can paint the room in 12 hours, so he can paint 1/12 of the room in 1 hour.

The apprentice can paint the room in 20 hours, so he can paint 1/20 of the room in 1 hour.

Let x be the number of hours it takes the two of them to paint the room together.

In x hours, the painter paints x/12 of the room and the apprentice paints x/20 of the room.

Since x is the number of hours it takes the two together to paint the room, the job is complete (the fraction of the job that is complete is "1") when the painter has painted x/12 of the room and th apprentice has painted x/20 of the room:

Multiply by the least common denominator (60) to clear fractions:

ANSWER: B) 7.5 hours

And here is an alternative method that many students prefer because it avoids working with fractions.

Consider the least common multiple of the two given times, which is 60 hours.

In 60 hours, the painter could paint 60/12 = 5 of those rooms; the apprentice could paint 60/20 = 3 of them. So together in 60 hours the two of them could paint 5+3 = 8 of the rooms; and that means the time it would take the two of them together to paint the one room is 60/8 = 7.5

Again, of course, the answer is 7.5 hours.

Notice that the actual numbers used in the two different methods are the same....

Try both methods with other similar problems and find what "works" best for you.

Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
A painter can paint a room in 12 hours. An apprentice, who can paint the room in 20 hours is added to the workforce. If they work together at the rates indicated, the number of hours needed to paint the room will be
----------------------
12*20/(12+20) = 240/32 = 7.5 hours
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