If painter A can paint a room in three hours by themselves, and painter B can paint the same room in four hours, and painter C can paint the same room in six hours, how long will it take for all three to paint the room together if they do not get in each others way? Thanks

This problem can be solved by working out the fraction of work done by each worker in one hour.

For A, fraction of work done per hour is 1/3.
For B, 1/4.
For C, 1/6.

If they all work together, in one hour they would have done
(1/3+1/4+1/6)
= (4+3+2)/12
= 9/12
= 3/4
of the work.

So if they paint together, it would take them 1/(3/4) = 4/3 days.

To find out how long it will take for all three painters to paint the room together, we can calculate their combined work rate.

First, let's determine the individual work rates of each painter. The work rate is the inverse of the time taken to complete the job.

Painter A's work rate: 1 room / 3 hours = 1/3 room per hour
Painter B's work rate: 1 room / 4 hours = 1/4 room per hour
Painter C's work rate: 1 room / 6 hours = 1/6 room per hour

Now, add up their individual rates to get the combined work rate:

Combined work rate = 1/3 + 1/4 + 1/6

To add these fractions, we need to find a common denominator. The least common multiple of 3, 4, and 6 is 12.

Combined work rate = (1/3) * (4/4) + (1/4) * (3/3) + (1/6) * (2/2)
= 4/12 + 3/12 + 2/12
= 9/12

Therefore, the combined work rate of all three painters is 9/12 room per hour.

To find out how long it will take for all three painters to finish painting the room together, we can use the equation:

Time = 1 / Combined work rate

Time = 1 / (9/12)
Time = 12/9
Time = 4/3

So, it will take all three painters 4/3 hours to paint the room together, or approximately 1 hour and 20 minutes.