you can paint a room in 8 hours. your friend and you can paint a room in 5 hours together. how long does it take your friend alone?
Your rate = room/8
friends rate = room/x
combined rate = room/8 + room/x
= room(x+8)/(8x)
then
5 = room / [room(x+8)/(8x)]
5 = 8x/(x+8)
8x = 5x + 40
3x = 40
x = 40/3 = 13 1/3
friend alone could do it in 13 hours and 20 minutes
To find out how long it takes your friend to paint the room alone, we can use the concept of work rates.
Let's assume that the work rate of you alone is "a" and the work rate of your friend alone is "b" (measured in rooms per hour).
We know that you can paint a room in 8 hours, so your work rate is 1 room per 8 hours, or a = 1/8.
Additionally, we know that you and your friend can paint a room together in 5 hours. So, the combined work rate is 1 room per 5 hours, or (a + b) = 1/5.
To find the work rate of your friend alone (b), we subtract your work rate (a) from the combined work rate (a + b):
b = (a + b) - a
Now, let's substitute the known values:
b = (1/5) - (1/8)
To simplify the equation, we need to find a common denominator for the fractions:
b = (8/40) - (5/40)
Now, we can subtract the fractions:
b = 3/40
The work rate of your friend alone is 3/40 (rooms per hour).
To find how long it takes your friend to paint the room alone, we can invert the work rate:
Time taken = 1 / work rate
Time taken = 1 / (3/40)
Now, let's divide 1 by 3/40:
Time taken = 1 * (40/3)
Time taken = 40 / 3
Therefore, it takes your friend approximately 13.33 hours to paint the room alone.