document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1134856>Question 1134856</A>: <I><SPAN STYLE=\"word-break: break-all;\">  maria can paint the walls of an apartment in 8 hours. After she has worked 3 hours, Pat joins her and they finish the job in 2 hours more. How long would it take Pat to do the whole job without the aid of Maria?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #752358 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=752358\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let m = the rate that maria paints the apartment.<BR>\n" );
document.write( "let p = the rate that pat paints the apartment.\r<BR>\n" );
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document.write( "rate * time = quantity of work completed.\r<BR>\n" );
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document.write( "maria can paint the apartment in 8 hours.\r<BR>\n" );
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document.write( "that leads to m * 8 = 1, where...\r<BR>\n" );
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document.write( "m is the rate that maria works.<BR>\n" );
document.write( "1 is the quantity of work completed which is 1 painted apartment.\r<BR>\n" );
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document.write( "from m * 8 = 1, solve for m to get m = 1/8.\r<BR>\n" );
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document.write( "this means maria can paint 1/8 of the apartment in 1 hour.\r<BR>\n" );
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document.write( "after she had worked 3 hours, maria has painted 3/8 of the apartment.\r<BR>\n" );
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document.write( "that means that 5/8 of the apartment still needs to be painted.\r<BR>\n" );
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document.write( "when they work together, the rate that maria works and pat works is additive.\r<BR>\n" );
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document.write( "together they finish the apartment in 2 hours.\r<BR>\n" );
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document.write( "that leads to (m + p) * 2 = 5/8 which becomes (1/8 + p) * 2 = 5/8 because maria'a rate is equal to 1/8.\r<BR>\n" );
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document.write( "simplify this equation to get 1/8 * 2 + 2 * p = 5/8.\r<BR>\n" );
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document.write( "simplify further to get 2/8 + 2 * p = 5/8.\r<BR>\n" );
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document.write( "subtract 2/8 from both sides of this equaiton to get 2 * p = 3/8\r<BR>\n" );
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document.write( "solve for p to get p = (3/8) / 2 = 3/16\r<BR>\n" );
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document.write( "that's the rate that pat can paint the apartment at.\r<BR>\n" );
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document.write( "confirm by replacing p with 3/16 in the equation of (1/8 + p) * 2 = 5/8\r<BR>\n" );
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document.write( "equation becomes (1/8 + 3/16) * 2 = 5/8\r<BR>\n" );
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document.write( "simplify to get 2/8 + 6/16 = 5/8\r<BR>\n" );
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document.write( "since 6/16 is the same as 3/8, you get 2/8 + 3/8 = 5/8 which is true, confirming the determination that pat's rate of painting the apartment is correct.\r<BR>\n" );
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document.write( "to find how long it would take pat to do the work itself, go back to rate * time = quantity formula and replace p with 3/6 and q with 1 to get 3/16 * time = 1<BR>\n" );
document.write( "solve for time to get time = 1 * 16/3 = 16/3 = 5 and 1/3 hours.\r<BR>\n" );
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document.write( "it would take pat 5 and 1/3 hours to completely paint the apartment by herself.\r<BR>\n" );
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document.write( "that's your solution.\r<BR>\n" );
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