document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=172994>Question 172994</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A painter can paint a room three hours less than an apprentice.  Together they can pain the room in two hours. How long will it take the apprentice working alone to paint the room?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #127860 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=127860\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> formula to work with is:<BR>\n" );
document.write( "rate * time = number of units<BR>\n" );
document.write( "the number of units is 1 room.<BR>\n" );
document.write( "it takes the painter x - 3 hours to paint a room.<BR>\n" );
document.write( "it take the apprentice x hours to paint the same room.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "let Rp = rate of the painter.<BR>\n" );
document.write( "let Ra = rate of the apprentice.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since rate * time = number of units,<BR>\n" );
document.write( "for the painter:<BR>\n" );
document.write( "Rp * (x-3) = 1<BR>\n" );
document.write( "for the apprentice:<BR>\n" );
document.write( "Ra * (x) = 1<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "if they are both working together, it takes 2 hours to paint the room.<BR>\n" );
document.write( "working together their rates are additive.<BR>\n" );
document.write( "since rate * time = number of units,<BR>\n" );
document.write( "for both:<BR>\n" );
document.write( "2 * (Ra + Rp) = 1<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "for the apprentice:<BR>\n" );
document.write( "Ra * x = 1<BR>\n" );
document.write( "which means:<BR>\n" );
document.write( "Ra = 1/x<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "for the painter:<BR>\n" );
document.write( "Rp * (x-3) = 1<BR>\n" );
document.write( "which means:<BR>\n" );
document.write( "Rp = 1/(x-3)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "substituting for Ra and Rp, the formula:<BR>\n" );
document.write( "2 * (Ra + Rp) = 1<BR>\n" );
document.write( "becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+%2A+%281%2Fx+%2B+1%2F%28x-3%29%29+=+1 ALIGN=MIDDLE  ALT=\"2+%2A+%281%2Fx+%2B+1%2F%28x-3%29%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fx+%2B+2%2F%28x-3%29+=+1 ALIGN=MIDDLE  ALT=\"2%2Fx+%2B+2%2F%28x-3%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "if we multiply both sides of the equation by x*(x-3), we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28x-3%29+%2B+2%2Ax+=+x%2A%28x-3%29 ALIGN=MIDDLE  ALT=\"2%2A%28x-3%29+%2B+2%2Ax+=+x%2A%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax+-+6+%2B+2%2Ax+=+x%2A%28x-3%29 ALIGN=MIDDLE  ALT=\"2%2Ax+-+6+%2B+2%2Ax+=+x%2A%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ax+-+6+=+x%5E2+-+3%2Ax ALIGN=MIDDLE  ALT=\"4%2Ax+-+6+=+x%5E2+-+3%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "if we subtract 4*x from both sides of the equation, and if we add 6 to both sides of the equation, we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+x%5E2+-+7%2Ax+%2B+6 ALIGN=MIDDLE  ALT=\"0+=+x%5E2+-+7%2Ax+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is the same as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+7%2Ax+%2B+6+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+7%2Ax+%2B+6+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "this factors out to be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%2A%28x-6%29+=+0 ALIGN=MIDDLE  ALT=\"%28x-1%29%2A%28x-6%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which give us:<BR>\n" );
document.write( "x = 1<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "x = 6<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "x cannot be 1, since x-3 would be negative which is not possible.<BR>\n" );
document.write( "x must be 6.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "if x is 6, then<BR>\n" );
document.write( "x-3 is 3.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since x * Ra = 1<BR>\n" );
document.write( "then Ra must be 1/6<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since (x-3)*Rp = 1<BR>\n" );
document.write( "then Rp must be 1/3<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "to prove this is true, we substitute in the original equations to see if they are true.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "first equation:<BR>\n" );
document.write( "x * Ra = 1<BR>\n" );
document.write( "6 * 1/6 = 1 is true.<BR>\n" );
document.write( "second equation:<BR>\n" );
document.write( "(x-3)*Rp = 1<BR>\n" );
document.write( "(6-3)*1/3 = 1<BR>\n" );
document.write( "3*1/3 = 1 is true.<BR>\n" );
document.write( "third equation:<BR>\n" );
document.write( "2 * (Ra + Rp) = 1<BR>\n" );
document.write( "2 * (1/6 + 1/3) = 1<BR>\n" );
document.write( "2 * (1/6 + 2/6) = 1<BR>\n" );
document.write( "2 * (3/6) = 1<BR>\n" );
document.write( "6/6 = 1 which is true.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "answer of x = 6 looks good.<BR>\n" );
document.write( "apprentice takes 6 hours to paint a room.<BR>\n" );
document.write( "painter takes 6-2 = 3 hours to paint the same room.<BR>\n" );
document.write( "together they can paint the room in 2 hours.<BR>\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );