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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 9:40am. 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.

 Dec 27, 2020
 #1
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First set r to the speed Ryan paints the room.

So the equation is

40r+20(2r)=1

80r=1

This would work if the speed of paint the room was consent.

So

20r/80r=1/4

This is the amount of the room before Leonardo started helping.

Then 80r-20r=60r

60r/2=30r

30r/80r+1/4=3/8+1/4=3/8+2/8=5/8

So 5/8 of the room was painted by 9:30.

 Dec 27, 2020
 #2
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Call the part of  room  that  Ryan can paint in the first 20 minutes,  1/x

 

When Leonardo  joins in   the part that is painted between 9:20 and 9:40 (20 minutes) is   (1/x) + (2/x)   =  3/x

 

So  in 40 minutes  the whole job is done....so

 

(1/x + 3/x)  =  4/ (x)   = 1    ⇒   x  = 4

 

So....the part painted  between 9  and 9:30   is

 

(1/4)  +  ( 1/2) (3/4)   =

 

(1/4)  + 3/8  =

 

2/8  + 3/8  = 

 

5/8

 

 

cool cool cool

 Dec 27, 2020

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