document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=293528>Question 293528</A>: <I><SPAN STYLE=\"word-break: break-all;\"> It takes Larry 8 hours to paint his house.  Larry started to paint the first 3 hours.  His friend Patrick joined him and they both finished it in 2 hours.  How long will it take Patrick to paint the house if he was to do it alone?<BR>\n" );
document.write( "I have tried using fractions and the two equation method.  However, what gets me is the 3 hours Larry started to paint by himself. Help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #211871 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=211871\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It takes Larry 8 hours to paint his house.<BR>\n" );
document.write( "Larry paints for 3 hours by himself.<BR>\n" );
document.write( "His friend Patrick joins him and they finish in 2 hours.<BR>\n" );
document.write( "How long would it take Patrick to paint the house himself?\r<BR>\n" );
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document.write( "Larry would take 8 hours to paint the house by himself.\r<BR>\n" );
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document.write( "His rate of painting is 1/8 of the house per hour.\r<BR>\n" );
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document.write( "This is based on the formula R * T = U\r<BR>\n" );
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document.write( "The number of units is equal to R is equal to 1.\r<BR>\n" );
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document.write( "The time is equal to 8 hours.\r<BR>\n" );
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document.write( "Plug those numbers into the equation to get R * 8 = 1\r<BR>\n" );
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document.write( "Divide both sides of that equation by 8 to get R = 1/8.\r<BR>\n" );
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document.write( "Plug 1/8 into that formula to get 1/8 * 8 = 1 so this rate is good.\r<BR>\n" );
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document.write( "in 3 hours Larry has painted 3/8 of the house.\r<BR>\n" );
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document.write( "That leaves 5/8 of the house left to do.\r<BR>\n" );
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document.write( "Larry and Patrick work together to finish the rest of the house in 2 hours.\r<BR>\n" );
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document.write( "Since they are working together their rates are additive.\r<BR>\n" );
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document.write( "Larry's rate is 1/8 of the house per hour.\r<BR>\n" );
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document.write( "Patrick's rate of painting the house per hour is represented by x.\r<BR>\n" );
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document.write( "The formula is Number of Units = Rate per Unit * Time\r<BR>\n" );
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document.write( "The number of units is equal to 1 if they had to do the whole house.\r<BR>\n" );
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document.write( "Since they only have to do 5/8 of the house, then the number of units is 5/8.\r<BR>\n" );
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document.write( "Since they are working together, their rates are combined.\r<BR>\n" );
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document.write( "The combines rate is (1/8 + x)\r<BR>\n" );
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document.write( "the amount of time they took is 2 hours.\r<BR>\n" );
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document.write( "The formula becomes:\r<BR>\n" );
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document.write( "5/8 = (1/8 + x) * 2\r<BR>\n" );
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document.write( "Simplify by removing parentheses to get:\r<BR>\n" );
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document.write( "5/8 = 2/8 + 2*x\r<BR>\n" );
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document.write( "Subtract 2/8 from both sides of the equation to get:\r<BR>\n" );
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document.write( "5/8 - 2/8 = 2*x\r<BR>\n" );
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document.write( "Combine like terms to get:\r<BR>\n" );
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document.write( "3/8 = 2*x\r<BR>\n" );
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document.write( "Divide both sides of the equation by 2 to get:\r<BR>\n" );
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document.write( "3/16 = x\r<BR>\n" );
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document.write( "Since x is Patrick's rate, this means that Patrick can paint 3/16 of the house in 1 hour.\r<BR>\n" );
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document.write( "This means that the formula for Patrick to do the whole house himself would be:\r<BR>\n" );
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document.write( "R * T = U where R = 3/16 and U = 1.\r<BR>\n" );
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document.write( "This formula becomes (3/16) * T = 1\r<BR>\n" );
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document.write( "Divide both sides of this equation by 3/16 to get T = 1/(3/16) which is the same as T = 16/3 which simplifies to T = 5 and 1/3 hours.\r<BR>\n" );
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document.write( "It would take Patrick 5 and 1/3 hours to paint the house by himself.\r<BR>\n" );
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document.write( "The key to this type of problem is the number of units that are left to be done.\r<BR>\n" );
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