Question

I think the answer is 2/5

Could you maybe explain your reasoning a little bit?

Let the amount of work done by Ruan per minute =W
Then the amount of work done by Leonardo per minute=2W
60 minutes x W + 40 minutes x 2W =1 fully painted house 
W =1/140 - portion of the house painted by Ruan in one minute.
W =2/140 =1/70 -portion of the house painted by Leonardo in one minute
30 minutes/140 + 10 minutes/70 =P - portion of the house painted by both men by 9:30 am.
P = 5/14 =35.71% portion of the house that was painted by 9:30 am.
30 minutes/140 + 30 minutes / 70 =P -The remaining portion of the house that will be painted between 9:30 am and 10:00 am
P =9/14 =64.29% portion of the house that will be finished between 9:30 am and 10:00 am.

I get 5/14 which is the same as our guest.

Ruan began painting a room at 9:00 AM. Leonardo, who can paint twice as fast as Ruan, started helping Ryan at 9:20 AM, and they worked together until the room was fully painted at 10:00am. What fraction of the room had been painted by 9:30 AM? Express your answer as a common fraction in lowest terms and clearly show/explain your reasoning.

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Here is my logic   

Let Ruan paint one unit of area every minute.

then Leonado is twice as fast so he will paint 2 units of area ever minute.

In the whole hour Ruan will paint  60 units of area

Leonado only paints for 40 minutes so he will paint a total of 80 units of area.

So in the hour they paint 60+80 = 140 units of area and this is the whole room.

Now how much is painted by 9:30am

Ruan has painted for 30 minutes so he has painted  30 units of area.

Leonado has painted for 10 minutes so he has painted 20 units of area.

So in the first half hour  30+20 = 50 units of area is painted.

So that is    \(\frac{50}{140}=\frac{5}{14}\)       of the room is painted by 9:30

Ruan began painting a room at 9:00 AM. Leonardo, who can paint twice as fast as Ruan, started helping Ryan at 9:20 AM, and they worked together until the room was fully painted at 10:00am. What fraction of the room had been painted by 9:30 AM? Express your answer as a common fraction in lowest terms and clearly show/explain your reasoning.

Let the fraction of the room that Ruan can paint in one minute be  (1/x)

Let the faction of the room that Leonardo can paint in one minute  be 2 (1/x) =  (2/x)

So...fraction of the room painted in one minute * minutes on the job  =  part of the job done

Ruan paints for 60 minutes when the job is completed and  Leonardo paints for 40 minutes when the job is done....so we have this equation :

(1/x)(60) + (2/x)(40)  = 1

60/x  + 80/x  = 1

[60 + 80] / x   = 1

140 / x  = 1    ⇒   x  =  140

So....Ruan can paint (1/140) of the room in one minute 

So...in 30 minutes, Ruan paints  (1/140)(30)  =  30 / 140 of the room

And in  10 minutes, Leonardo paints  (2 / 140)(10) = 20 / 140 of the room

So...together....they paint   (30 + 20 ) / 140   =  50 / 140    = 5 / 14  of the room by 9:30 AM

Thanks for all the answers guys! I really appreciate the help. Isn't it cool how many different solutions this problem has?