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

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): You can put this solution on YOUR website!
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

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