It takes a painter and his friend 8 hours to paint a room. If the painter was working alone, it would take him 12 hours less than if his friend was working alone. How long does it take the painter to paint the room by himself? Please explain to get brainliest

The painter paint the room by himself in 12h.

Step-by-step explanation:

If we consider that the room is the distance that the painter and his friend need to paint, then each person will have a painting speed and if we sum them they will finish the room in 8 hours. If we call the speed of the painter "x" and the his time "tx", while his friend will be "y" and his time will be "ty". We can apply each case to the average speed formula:

Both working together:

speed = distance / time

x + y = room / 8

The friend alone:

y = room / ty

The painter alone:

x = room / tx

Therefore:

(room/tx) + (room/ty) = room/8

1/tx + 1/ty = 1/8

(tx + ty)/tx*ty = 1/8

8*tx + 8*ty  = tx*ty

Since tx = ty - 12, we have:

8*ty - 96 + 8*ty = ty² - 12*ty

ty² - 12*ty - 16*ty + 96 = 0

ty² - 28*ty + 96 = 0

ty = 4 or ty = 24

Since tx = ty - 12, ty can only be 24, because if it was 4, then the painter time would be negative and that is impossible. So we have:

ty = 24

tx = 24 - 12 = 12

The painter paint the room by himself in 12h.

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