document.write( "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? \n" ); document.write( "

Algebra.Com's Answer #323134 by Edwin McCravy(20054) $\"\"$ $\"About$

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

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