Q1: Robert can paint a room in 12 hours. Dennis and Vince can paint the room in 10 hours & 9 hours respectively. If Dennis and Vince work together for 2 hours, How long will it take Robert to finish the job? I think it's 12 (because it says it, but I think the question was how could he do it if he worked together? I'm not sure.), but I'm unsure of my answer so please explain.
How much of the job gets done in one hour by each person?
Robert: 1/12
Dennis: 1/10
Vince: 1/9
So, in 2 hours, Dennis and Vince can do
2/10 + 2/9 = 19/45 of the job
That leaves 26/45 for Robert to do. At his rate of 1/12 per hour, it will take him
(26/45)/(1/12) = 6.933 hours or 6 hr 56 min
To solve this problem, let's start by calculating the rate at which each person can paint.
Robert can paint a room in 12 hours, so his rate is 1/12 of the room per hour.
Similarly, Dennis can paint a room in 10 hours, so his rate is 1/10 of the room per hour.
Vince can paint a room in 9 hours, so his rate is 1/9 of the room per hour.
If Dennis and Vince work together, their combined rate is the sum of their individual rates. Therefore, their combined rate is (1/10 + 1/9) of the room per hour.
Now, let's calculate how much of the room they can paint in 2 hours. To do this, we multiply their combined rate by the time they worked together:
Combined rate = (1/10 + 1/9) = 19/90 of the room per hour
Amount painted in 2 hours = (19/90) * 2 = 19/45 of the room.
So, after working together for 2 hours, Dennis and Vince have painted 19/45 of the room. To find out how much is left for Robert to paint, subtract this amount from the entire room:
Remaining amount of the room = 1 - 19/45 = 26/45 of the room.
Finally, to find out how long it will take Robert to finish the job, we need to calculate his rate and use it to determine the time required to paint the remaining 26/45 of the room.
Robert's rate = 1/12 of the room per hour.
Time required for Robert to finish the remaining 26/45 of the room = (26/45) / (1/12) = (26/45) * (12/1) = 104/15 or approximately 6.93 hours.
Therefore, it will take Robert approximately 6.93 hours or 6 hours and 56 minutes (rounded to the nearest minute) to finish the job.