document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197111>Question 1197111</A>: <I><SPAN STYLE=\"word-break: break-all;\"> abbie paints twice as fast as beth and three times as fast as cathie. if it takes them 60 minutes to paint a living room with all three working together,how long would it take each woman working alone? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830228 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=830228\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Abbie paints twice as fast as Beth<BR>\n" );
document.write( "This means Beth takes twice as long as Abbie\r<BR>\n" );
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document.write( "Abbie paints three times as fast as Cathie<BR>\n" );
document.write( "So Cathie takes three times as long as Abbie\r<BR>\n" );
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document.write( "x = amount of time, in minutes, Abbie needs to do the job by herself<BR>\n" );
document.write( "2x = amount of time, in minutes, Beth needs to do the job by herself<BR>\n" );
document.write( "3x = amount of time, in minutes, Cathie needs to do the job by herself\r<BR>\n" );
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document.write( "The reciprocal of each item represents the unit rate<BR>\n" );
document.write( "1/x = unit rate for Abbie<BR>\n" );
document.write( "1/(2x) = unit rate for Beth<BR>\n" );
document.write( "1/(3x) = unit rate for Cathie<BR>\n" );
document.write( "Each unit rate is in \"jobs per minute\". \r<BR>\n" );
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document.write( "Those fractional unit rates add to this<BR>\n" );
document.write( "(1/x) + (1/(2x)) + (1/(3x))<BR>\n" );
document.write( "(6/(6x)) + (3/(6x)) + (2/(6x))<BR>\n" );
document.write( "(6+3+2)/(6x)<BR>\n" );
document.write( "11/(6x)\r<BR>\n" );
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document.write( "If the girls work together, without getting in each others' way, then their combined unit rate is 11/(6x) jobs per minute.\r<BR>\n" );
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document.write( "Multiply this unit rate with the total time they use (60 min) and set this equal to 1 to represent getting 1 full job done.\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "(unit rate)*(time) = amount done<BR>\n" );
document.write( "(11/(6x))*(60 min) = 1 full job<BR>\n" );
document.write( "(11/(6x))*(60) = 1<BR>\n" );
document.write( "110/x = 1<BR>\n" );
document.write( "110 = x*1<BR>\n" );
document.write( "x = 110\r<BR>\n" );
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document.write( "Abbie needs 110 minutes to get the job done on her own.<BR>\n" );
document.write( "Beth needs 2x = 2*110 = 220 minutes to get the job done on her own.<BR>\n" );
document.write( "Cathie needs 3x = 3*110 = 330 minutes to get the job done on her own.\r<BR>\n" );
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document.write( "Another approach:\r<BR>\n" );
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document.write( "a = Abbie's rate <BR>\n" );
document.write( "b = Beth's rate<BR>\n" );
document.write( "c = Cathie's rate\r<BR>\n" );
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document.write( "Abbie paints twice as fast as Beth, so,<BR>\n" );
document.write( "a = 2b<BR>\n" );
document.write( "Also, Abbie paints three times as fast as Cathie<BR>\n" );
document.write( "a = 3c\r<BR>\n" );
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document.write( "Both equations mentioned are equal to 'a', which means we can equate the right hand sides<BR>\n" );
document.write( "2b = 3c<BR>\n" );
document.write( "Divide both sides by 2 allowing us to solve for b<BR>\n" );
document.write( "b = 3c/2 = 1.5c\r<BR>\n" );
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document.write( "So we can say<BR>\n" );
document.write( "a = 3c<BR>\n" );
document.write( "b = 1.5c\r<BR>\n" );
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document.write( "The ratio a:b:c can be expressed as 3c:1.5c:c<BR>\n" );
document.write( "Multiply all parts by 2 to get 6c:3c:2c<BR>\n" );
document.write( "Every part of this ratio involves the variable c in some fashion.\r<BR>\n" );
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document.write( "The coefficients add to 6+3+2 = 11<BR>\n" );
document.write( "We can think of it like saying<ul><li>Abbie does 6 parts</li><li>Beth does 3 parts</li><li>Cathie does 2 parts</li></ul>When the girls work together,<ul><li>Abbie does 6/11 of the job</li><li>Beth does 3/11 of the job</li><li>Cathie does 2/11 of the job</li></ul><BR>\n" );
document.write( "Let's say the job is to paint 3300 square feet of wall<BR>\n" );
document.write( "I picked some large multiple of 11. Feel free to pick something else. <BR>\n" );
document.write( "I picked a multiple of 11 so that multiplying with the fractions leads to a whole number.\r<BR>\n" );
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document.write( "Let's now subdivide the work like so<ul><li>Abbie's portion is (6/11)*3300 = 1800 sq ft</li><li>Beth's portion is (3/11)*3300 = 900 sq ft</li><li>Cathie's portion is (2/11)*3300 = 600 sq ft</li></ul>Note that 1800:900:600 reduces to 6:3:2 after dividing all three parts by 300.\r<BR>\n" );
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document.write( "Since it took 60 minutes for each woman to do their portions of the wall, this means<ul><li>Abbie's unit rate is 1800/60 = 30 sq ft per min</li><li>Beth's unit rate is 900/60 = 15 sq ft per min</li><li>Cathie's unit rate is 600/60 = 10 sq ft per min</li></ul>The ratio 30:15:10 is equivalent to 6:3:2 after dividing all three parts by 5.\r<BR>\n" );
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document.write( "Now let's have them work alone.<BR>\n" );
document.write( "If Abbie works alone, then she needs to paint the entire 3300 sq ft. Her unit rate is 30 sq ft per min. <BR>\n" );
document.write( "Therefore, she needs 3300/30 = 110 minutes.<BR>\n" );
document.write( "I'm using the idea that <BR>\n" );
document.write( "time = (amount done)/(rate of work)\r<BR>\n" );
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document.write( "If Beth works alone, then she needs 3300/15 = 220 minutes\r<BR>\n" );
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document.write( "If Cathie works alone, then she needs 3300/10 = 330 minutes\r<BR>\n" );
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document.write( "<font color=red>Answers:</font><BR>\n" );
document.write( "Abbie needs 110 minutes if she works alone<BR>\n" );
document.write( "Beth needs 220 minutes if she works alone<BR>\n" );
document.write( "Cathie needs 330 minutes if she works alone\r<BR>\n" );
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document.write( "Footnotes:<BR>\n" );
document.write( "110 min = 60 min + 50 min = 1 hr + 50 min<BR>\n" );
document.write( "220 min = 180 min + 40 min = 3 hr + 40 min<BR>\n" );
document.write( "330 min = 300 min + 30 min = 5 hr + 30 min<BR>\n" );
document.write( "180 min = 180/60 = 3 hrs<BR>\n" );
document.write( "300 min = 300/60 = 5 hrs<BR>\n" );
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