document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=992764>Question 992764</A>: <I><SPAN STYLE=\"word-break: break-all;\"> X can do a certain job in 10 hours and Y can do it in 12 hours. After X and Y have been working together for 1 hour, Z joins them, and the three completed the job in 3 more hours, How long will it take Z to do the job alone? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #612269 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=612269\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> rate of x is equal to a<BR>\n" );
document.write( "rate of y is equal to b<BR>\n" );
document.write( "rate of z is equal to c\r<BR>\n" );
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document.write( "rate * time = quantity\r<BR>\n" );
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document.write( "the total quantity required to complete is 1 job.<BR>\n" );
document.write( "partial quantities are just shown as q until they are solved for.<BR>\n" );
document.write( "you'll see how that works below.\r<BR>\n" );
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document.write( "x can do the job in 10 hours.\r<BR>\n" );
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document.write( "for x, rate * time = quantity becomes a * 10 = 1<BR>\n" );
document.write( "solve for a to get:<BR>\n" );
document.write( "a = 1/10<BR>\n" );
document.write( "that's the rate that x works at.<BR>\n" );
document.write( "quantity is 1 because we are looking for the rate that x works at to complete the whole job working alone.\r<BR>\n" );
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document.write( "for y, rate * time = quantity becomes b * 12 = 1<BR>\n" );
document.write( "solve for b to get:<BR>\n" );
document.write( "b = 1/12<BR>\n" );
document.write( "that's the rate that b works at.<BR>\n" );
document.write( "quantity is 1 because we are looking for the rate that y works at to complete the whole job working alone.\r<BR>\n" );
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document.write( "x and y work on the job for 1 hour.<BR>\n" );
document.write( "their rates are additive.<BR>\n" );
document.write( "rate * time = quantity<BR>\n" );
document.write( "(a + b) * t = q<BR>\n" );
document.write( "a is the rate that x works at which is equal to 1/10<BR>\n" );
document.write( "b is the rate that y works at which is equal to 1/12<BR>\n" );
document.write( "t is 1 hour<BR>\n" );
document.write( "q is unknown<BR>\n" );
document.write( "formula becomes:<BR>\n" );
document.write( "(1/10 + 1/12) * 1 = q<BR>\n" );
document.write( "here we are looking for how much of the job is completed by x and y working together without z.<BR>\n" );
document.write( "that's why the quantity is shown as q, rather than 1.<BR>\n" );
document.write( "solve for q and you will get that q = 11/60 of the job.<BR>\n" );
document.write( "x and y working together for 1 hour will complete 11/60 of the job in that hour.\r<BR>\n" );
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document.write( "what remains of the job to still be done is 1 - 11/60 = 60/60 - 11/60 = 49/60.\r<BR>\n" );
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document.write( "z joins them and all three finish the job in 3 additional hours.\r<BR>\n" );
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document.write( "since their rates when working together are additive, the formula to use would be:\r<BR>\n" );
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document.write( "(a + b + c) * t = q.\r<BR>\n" );
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document.write( "all three working together for 3 hours will complete 49/60 of the job.\r<BR>\n" );
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document.write( "we have:<BR>\n" );
document.write( "a = 1/10<BR>\n" );
document.write( "b = 1/12<BR>\n" );
document.write( "c = c because we don't know what it is yet.<BR>\n" );
document.write( "t = 3 hours<BR>\n" );
document.write( "q = 49/60\r<BR>\n" );
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document.write( "formula becomes:\r<BR>\n" );
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document.write( "(1/10 + 1/12 + c) * 3 = 49/60\r<BR>\n" );
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document.write( "combine 1/10 + 1/12 and you get 22/120 which simplifies to 11/60.\r<BR>\n" );
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document.write( "formula becomes:\r<BR>\n" );
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document.write( "(11/60 + c) * 3 = 49/60\r<BR>\n" );
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document.write( "simplify the left side of the equation to get:\r<BR>\n" );
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document.write( "33/60 + 3 * c = 49/60\r<BR>\n" );
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document.write( "subtract 33/60 from both sides of the equation to get:\r<BR>\n" );
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document.write( "3 * c = 16/60\r<BR>\n" );
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document.write( "divide both sides of the equation by 3 to get:\r<BR>\n" );
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document.write( "c = 16/180 which simplifies to 8/90.\r<BR>\n" );
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document.write( "the rate of z is 8/90 of the job in 1 hour.\r<BR>\n" );
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document.write( "check your original equation to see if this is correct.\r<BR>\n" );
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document.write( "(1/10 + 1/12 + 8/90) * 3 = 49/60\r<BR>\n" );
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document.write( "do the math to get:\r<BR>\n" );
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document.write( "49/60 = 49/60<BR>\n" );
document.write( "this confirms the solution is correct.\r<BR>\n" );
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document.write( "you now know the rate of z which is equal to 8/90\r<BR>\n" );
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document.write( "working alone, the rate * time = distance formula becomes:\r<BR>\n" );
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document.write( "8/90 * t = 1\r<BR>\n" );
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document.write( "8/90 is the rate of z<BR>\n" );
document.write( "t is the time that we want to solve for<BR>\n" );
document.write( "1 is the portion of the job to be completed which is the whole job.\r<BR>\n" );
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document.write( "solve for t to get t = 1 / (8/90) = 1 * (90/8) = 11.25 hours.\r<BR>\n" );
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document.write( "z can do the job alone in 11.25 hours.\r<BR>\n" );
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document.write( "that's your solution.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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