Question 1207751
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You have the correct set up so far.


I'll show two ways of solving that equation.
Here is one way.
1/10 + 1/x = 1/6
30x(1/10 + 1/x) = 30x(1/6)
3x + 30 = 5x
30 = 5x-3x
30 = 2x
2x = 30
x = 30/2
<font color=red>x = 15 hours</font> is the amount of time April takes when working alone.


Or you can solve it like this.
1/10 + 1/x = 1/6
x/(10x) + 10/(10x) = 1/6
(x+10)/(10x) = 1/6
6(x+10) = 10x
6x+60 = 10x
60 = 10x-6x
60 = 4x
4x = 60
x = 60/4
<font color=red>x = 15 hours</font>


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Another approach


Let's say hypothetically that the total surface area to paint all four rooms is 600 square feet.


Patrick, when working alone, can paint all 600 square feet in 10 hours.
His unit rate is 600/10 = 60 sq ft per hour.
Formula:  rate = amountDone/time


When Patrick and April work together, they get the job done in 6 hours. 
I'll assume neither person slows down the other.
Their combined unit rate is 600/6 = 100 sq ft per hour.


Therefore, April's unit rate must be 100-60 = 40 sq ft per hour when she works alone.


Let's determine how long April will take when she does the job alone.
rate*time = amountDone
time = amountDone/rate
time = 600/40
time = <font color=red>15 hours</font>


The hypothetical 600 sq ft figure I came up with isn't special at all. Turns out you can replace it with any other positive number to get the same final answer marked in red.
I picked 600 since 10*6 = 60, and then I tacked a zero at the end to make the area appear a a bit more realistic pertaining to those four rooms.
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