SOLUTION: A painter and his son can paint a room together in 8 hours. If the father works alone for 3 hours and then is joined by his son, the two together can complete the job in 6 hours. H

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Question 471087: A painter and his son can paint a room together in 8 hours. If the father works alone for 3 hours and then is joined by his son, the two together can complete the job in 6 hours. How long will it take each person alone to paint the room?
Answer by Edwin McCravy(20054) (Show Source):
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A painter and his son can paint a room together in 8 hours. If the father works alone for 3 hours and then is joined by his son, the two together can complete the job in 6 hours. How long will it take each person alone to paint the room?

Make this chart: Rooms painted or Rate fraction hours in thereof worked rooms/hr Father alone for x hours Son alone for y hours Father & Son working together 8 hours Father alone for 3 hours Father & Son working together 6 hours Let x = the number of hours it would take the father alone to paint 1 room Let y = the number of hours it would take the son alone to paint 1 room Father & Son working together take 8 hours to paint 1 room Father worked alone for 3 hours Father and son worked together for 6 hours Fill those in the chart Rooms painted or Rate fraction hours in thereof worked rooms/hr Father alone for x hours 1 x Son alone for y hours 1 y Father & Son working together 8 hours 1 8 Father alone for 3 hours 3 Father & Son working together 6 hours 6 Fill in the rates in rooms/hr for the first three rows by dividing rooms by hours Rooms painted or Rate fraction hours in thereof worked rooms/hr Father alone for x hours 1 x 1/x Son alone for y hours 1 y 1/y Father & Son working together 8 hours 1 8 1/8 Father alone for 3 hours 3 Father & Son working together 6 hours 6 Now we can fill in the rate for the father alone for 3 hours as the same rate 1/x as father alone for x hours. Similarly, we can fill in the rate for the father & Son for 3 hours as the same rate 1/8 as father & son for 8 hours. Rooms painted or Rate fraction hours in thereof worked rooms/hr Father alone for x hours 1 x 1/x Son alone for y hours 1 y 1/y Father & Son working together 8 hours 1 8 1/8 Father alone for 3 hours 3 1/x Father & Son working together 6 hours 6 1/8 Now we can fill in the fractions of a room painted in the last two cases by multiplying the rate in rooms/hr by the time in hours: Rooms painted or Rate fraction hours in thereof worked rooms/hr Father alone for x hours 1 x 1/x Son alone for y hours 1 y 1/y Father & Son working together 8 hours 1 8 1/8 Father alone for 3 hours 3/x 3 1/x Father & Son working together 6 hours 6/8 6 1/8 The system of two equations comes from: Solve the second equation for x Reduce 6/8 to 3/4 Clear of fractions by multiplying thru by LCD of 4x 12 + 3x = 4x 12 = x Substitute in the first equation: Clear of fractions by multiplying thru by LCD of 24y 2y + 24 = 3y 24 = y So it takes the father 12 hours to paint a romm and it takes the son 24 hours to paint a room. Edwin