document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=285536>Question 285536</A>: <I><SPAN STYLE=\"word-break: break-all;\">  I need help figuring out this problem:\r<BR>\n" );
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document.write( "Melanie can paint a certain barn by herself in x days. Her helper melissa can paint the same barn by herself in 2x days. write a rational expression for the fraction of the barn that they complete in one day by working together. Evaluate the expression for x=5.\r<BR>\n" );
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document.write( "would it be written as 2x+5x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #206958 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=206958\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Melanie paints the barn in x days.<BR>\n" );
document.write( "Melissa paints the barn in 2x days.\r<BR>\n" );
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document.write( "The formula to use is rate * time = units\r<BR>\n" );
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document.write( "The number of units is equal to 1 (the barn)\r<BR>\n" );
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document.write( "Melanie can paint the barn in x days, so the formula for her would be:\r<BR>\n" );
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document.write( "r * x = 1\r<BR>\n" );
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document.write( "For Melanie, r = 1/x which is the amount of the barn she can paint in one day.\r<BR>\n" );
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document.write( "Melissa can paint the barn in 2x days, so the formula for her would be:\r<BR>\n" );
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document.write( "r * 2x = 1\r<BR>\n" );
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document.write( "For Melissa, r = 1/2x which is the amount of the barn she can paint in one day.\r<BR>\n" );
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document.write( "When they work together, their rates are combined (added together).\r<BR>\n" );
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document.write( "Formula for them working together would be:\r<BR>\n" );
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document.write( "(1/x + 1/2x) * T = 1\r<BR>\n" );
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document.write( "Their combined rate is (1/x + 1/2x).\r<BR>\n" );
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document.write( "Since 1/x is equivalent to 2/2x, then their combined rate becomes:\r<BR>\n" );
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document.write( "(2/2x + 1/2x) * T = 1 which becomes:\r<BR>\n" );
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document.write( "3/2x * T = 1\r<BR>\n" );
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document.write( "Their combined rate is 3/2x which means that they can paint 3/2x of the barn in one day.\r<BR>\n" );
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document.write( "If x = 5, then the following happens:\r<BR>\n" );
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document.write( "Melanie's rate is 1/5.\r<BR>\n" );
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document.write( "Melissa's rate is 1/10.\r<BR>\n" );
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document.write( "Their combined rate is 1/5 + 1/10 = 3/10.\r<BR>\n" );
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document.write( "It would take Melanie 5 days to paint the barn working alone because 5 * 1/5 = 1.\r<BR>\n" );
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document.write( "It would take Melissa 10 days to paint the barn working alone because 10 * 1/5 = 1\r<BR>\n" );
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document.write( "It would take both of them 10/3 days to paint the barn working together because 10/3 * 3/10 = 1\r<BR>\n" );
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document.write( "In 10/3 days, Melanie has painted 1/5 * 10/3 = 2/3 of the barn.\r<BR>\n" );
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document.write( "In 10/3 days, Melissa has painted 1/10 * 10/3 = 1/3 of the barn.\r<BR>\n" );
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document.write( "Together they have painted 2/3 + 1/3 of the barn = 1 of the barn = the whole barn.\r<BR>\n" );
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document.write( "Your answer is:\r<BR>\n" );
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document.write( "Their combined rate is 1/x + 1/2x = 2/2x + 1/2x = 3/2x.\r<BR>\n" );
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document.write( "If x is equal to 5, then their combined rate is 1/5 + 1/10 = 2/10 + 1/10 = 3/10.\r<BR>\n" );
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document.write( "It would not be 2x + 5x.\r<BR>\n" );
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document.write( "In this type of problem, always think of the basic formula and how it fits the problem.\r<BR>\n" );
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document.write( "The basic formula is rate * time = units.\r<BR>\n" );
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document.write( "If Melanie can paint the barn in x days, then time is x.\r<BR>\n" );
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document.write( "The formula becomes rate * x = units.\r<BR>\n" );
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document.write( "To get her rate, you have to divide both sides of this equation by x to get rate = units / x which puts the x in the denominator.\r<BR>\n" );
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document.write( "Since units is 1, then we get rate = 1/x for Melanie.\r<BR>\n" );
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document.write( "For Melissa, we got rate = 1/2x because she takes twice as long as Melanie.\r<BR>\n" );
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document.write( "When they asked you what happens when x = 5, you needed to replace x with 5 in the equations as I did above.\r<BR>\n" );
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document.write( "1/x + 2/x became 1/5 + 1/10.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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