document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1024735>Question 1024735</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Daniel and Lauren are painters. Daniel is faster than Lauren by 2 hours. Occasionally they work with Ian, who paints very quickly, as fast as Daniel and Lauren together. One day, they had to paint a room. Lauren and Daniel started to paint first for 1 hour, then Ian arrived an all 3 of them finished painting after another 3.5 hours. How long will it take each of them to paint the room individually? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #640104 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20056)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=640104\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Daniel and Lauren are painters. Daniel is faster than Lauren <BR>\n" );
document.write( "by 2 hours. Occasionally they work with Ian, who paints very <BR>\n" );
document.write( "quickly, as fast as Daniel and Lauren together. One day, they <BR>\n" );
document.write( "had to paint a room. Lauren and Daniel started to paint first<BR>\n" );
document.write( "for 1 hour, then Ian arrived an all 3 of them finished painting <BR>\n" );
document.write( "after another 3.5 hours. How long will it take each of them <BR>\n" );
document.write( "to paint the room individually?<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Make this chart to include all 5 situations mentioned:\r\n" );
document.write( "\r\n" );
document.write( "               Number\r\n" );
document.write( "              of rooms                Painting\r\n" );
document.write( "              painted      Time         rate\r\n" );
document.write( "             or fraction  painting       in\r\n" );
document.write( "              thereof     in hours    rooms/hr\r\n" );
document.write( "------------------------------------------------\r\n" );
document.write( "Daniel         \r\n" );
document.write( "Lauren          \r\n" );
document.write( "Ian             \r\n" );
document.write( "D&L          \r\n" );
document.write( "D&L&I      \r\n" );
document.write( "</PRE><BR>\n" );
document.write( "Daniel is faster than Lauren by 2 hours.<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Let the time it takes Lauren to paint 1 room be L hours.\r\n" );
document.write( "Therefore the time it takes Daniel to paint 1 room is L-2.\r\n" );
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document.write( "So we fill in 1 each for the number of rooms each can paint\r\n" );
document.write( "and L and L-2 for their times\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "               Number\r\n" );
document.write( "              of rooms                Painting\r\n" );
document.write( "              painted      Time         rate\r\n" );
document.write( "             or fraction  painting       in\r\n" );
document.write( "              thereof     in hours    rooms/hr\r\n" );
document.write( "------------------------------------------------\r\n" );
document.write( "Daniel          1           L-2        \r\n" );
document.write( "Lauren          1            L           \r\n" );
document.write( "Ian             \r\n" );
document.write( "D&L         \r\n" );
document.write( "D&L&I\r\n" );
document.write( "\r\n" );
document.write( "We fill in their rates in rooms/hour by dividing rooms by\r\n" );
document.write( "hours:\r\n" );
document.write( "\r\n" );
document.write( "               Number\r\n" );
document.write( "              of rooms                Painting\r\n" );
document.write( "              painted      Time         rate\r\n" );
document.write( "             or fraction  painting       in\r\n" );
document.write( "              thereof     in hours    rooms/hr\r\n" );
document.write( "------------------------------------------------\r\n" );
document.write( "Daniel          1           L-2        1/(L-2) \r\n" );
document.write( "Lauren          1            L           1/L\r\n" );
document.write( "Ian             \r\n" );
document.write( "D&L         \r\n" );
document.write( "D&L&I   \r\n" );
document.write( "</PRE><BR>\n" );
document.write( "...Ian, who paints...as fast as Daniel and Lauren together<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "So his rate is the sum of their rates, so we add their rates\r\n" );
document.write( "and put that for Ian's rate and 1 for the number of rooms\r\n" );
document.write( "he could paint at this rate.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "               Number\r\n" );
document.write( "              of rooms                Painting\r\n" );
document.write( "              painted      Time         rate\r\n" );
document.write( "             or fraction  painting       in\r\n" );
document.write( "              thereof     in hours    rooms/hr\r\n" );
document.write( "------------------------------------------------\r\n" );
document.write( "Daniel          1           L-2        1/(L-2) \r\n" );
document.write( "Lauren          1            L           1/L\r\n" );
document.write( "Ian             1                     1/(L-2)+1/L\r\n" );
document.write( "D&L          \r\n" );
document.write( "D&L&I\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "Lauren and Daniel started to paint first for 1 hour,<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "So we add their rates  and put that for the rate for\r\n" );
document.write( "D&L and also put 1 hour for the time for D&L\r\n" );
document.write( "     \r\n" );
document.write( "\r\n" );
document.write( "               Number\r\n" );
document.write( "              of rooms                Painting\r\n" );
document.write( "              painted      Time         rate\r\n" );
document.write( "             or fraction  painting       in\r\n" );
document.write( "              thereof     in hours    rooms/hr\r\n" );
document.write( "------------------------------------------------\r\n" );
document.write( "Daniel          1           L-2        1/(L-2) \r\n" );
document.write( "Lauren          1            L           1/L\r\n" );
document.write( "Ian             1                     1/(L-2)+1/L\r\n" );
document.write( "D&L                          1        1/(L-2)+1/L\r\n" );
document.write( "D&L&I     \r\n" );
document.write( "\r\n" );
document.write( "Next we get the fraction of a room that they painted\r\n" );
document.write( "during that 1 hour, by multiplying their rate by\r\n" );
document.write( "their time.\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "then Ian arrived an all 3 of them finished painting after another 3.5 hours<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "So we fill in 3.5 hours for the time for D&L&I, and add the rates for\r\n" );
document.write( "Ian and D&L.  They are the same rates so we just multiply it by 2: \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "               Number\r\n" );
document.write( "              of rooms                Painting\r\n" );
document.write( "              painted      Time         rate\r\n" );
document.write( "             or fraction  painting       in\r\n" );
document.write( "              thereof     in hours    rooms/hr\r\n" );
document.write( "------------------------------------------------\r\n" );
document.write( "Daniel          1           L-2        1/(L-2) \r\n" );
document.write( "Lauren          1            L           1/L\r\n" );
document.write( "Ian             1                     1/(L-2)+1/L\r\n" );
document.write( "D&L          1/(L-2)+1/L     1        1/(L-2)+1/L\r\n" );
document.write( "D&L&I                       3.5     2(1/(L-2)+1/L)\r\n" );
document.write( "\r\n" );
document.write( "Then we fill in the fraction of the room they\r\n" );
document.write( "painted after Ian joined in, by multiplying the\r\n" );
document.write( "rate on the bottom row by the time:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "               Number\r\n" );
document.write( "              of rooms                Painting\r\n" );
document.write( "              painted      Time         rate\r\n" );
document.write( "             or fraction  painting       in\r\n" );
document.write( "              thereof     in hours    rooms/hr\r\n" );
document.write( "------------------------------------------------\r\n" );
document.write( "Daniel          1           L-2        1/(L-2) \r\n" );
document.write( "Lauren          1            L           1/L\r\n" );
document.write( "Ian             1                     1/(L-2)+1/L\r\n" );
document.write( "D&L          1/(L-2)+1/L     1        1/(L-2)+1/L\r\n" );
document.write( "D&L&I   2[3.5(1/(L-2)+1/L]  3.5     2(1/(L-2)+1/L) \r\n" );
document.write( "\r\n" );
document.write( "Then the equation comes from adding the fractions of\r\n" );
document.write( "a room painted in the last two lines and setting that\r\n" );
document.write( "equal to 1 room:\r\n" );
document.write( "\r\n" );
document.write( "      1/(L-2)+1/L + 2[3.5(1/(L-2)+1/L)] = 1\r\n" );
document.write( "\r\n" );
document.write( "           1/(L-2)+1/L + 7(1/(L-2)+1/L) = 1\r\n" );
document.write( "\r\n" );
document.write( "                         8(1/(L-2)+1/L) = 1\r\n" );
document.write( "\r\n" );
document.write( "                            8/(L-2)+8/L = 1\r\n" );
document.write( "\r\n" );
document.write( "Multiply through by the LCD of L(L-2)\r\n" );
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document.write( "                            8L + 8(L-2) = L(L-2)\r\n" );
document.write( "                           8L + 8L - 16 = L² - 2L\r\n" );
document.write( "                               16L - 16 = L² - 2L    \r\n" );
document.write( "                                      0 = L² - 18L + 16\r\n" );
document.write( "\r\n" );
document.write( "Solve that by the quadratic formula: Get L = 17.062 hrs. \r\n" );
document.write( "approximately\r\n" );
document.write( "\r\n" );
document.write( "(we also get L = 0.938 which is extraneous.)\r\n" );
document.write( "\r\n" );
document.write( "So it takes Lauren 17.062 hrs to paint 1 room.\r\n" );
document.write( "\r\n" );
document.write( "Since it takes 2 hours less for Daniel to paint 1 room,  \r\n" );
document.write( "\r\n" );
document.write( "So it takes Daniel 15.062 hrs to paint 1 room.\r\n" );
document.write( "\r\n" );
document.write( "Finally we must calculate Ian's time to paint 1 room. \r\n" );
document.write( "\r\n" );
document.write( "Ian's time is 1 room divided by his rate:\r\n" );
document.write( "\r\n" );
document.write( "1 ÷ [1/(L-2)+1/L] = \r\n" );
document.write( "1 ÷ [1/(17.063-2)+1/17.063] =\r\n" );
document.write( "\r\n" );
document.write( "1 ÷ 0.125\r\n" );
document.write( "\r\n" );
document.write( "8\r\n" );
document.write( "\r\n" );
document.write( "So it takes Ian 8 hours to paint 1 room.\r\n" );
document.write( "\r\n" );
document.write( "Edwin</PRE> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );