Algebra.Com's Answer #703091 by ikleyn(52864)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "It takes Nicolai one and a half hours to paint the walls of a room and two hours to paint the ceiling.
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document.write( "Elena needs exactly one hour to paint the walls of the same room and one hour to paint the ceiling.
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document.write( "If Nicolai and Elena work together, what is the shortest possible time in minutes in which they can paint the walls and the ceiling of that room?\r
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document.write( "My answer is 76, but the correct answer is apparently 72. Can someone please explain the working? Thank you!
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document.write( "The solution by the other tutor is INCORRECT.
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document.write( "The correct answer is 72 minutes, and I will show below WHY. And HOW.\r
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document.write( "1. Let us assign Nicolai to paint the walls, and let us assign Elena to paint the ceiling.\r\n" );
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document.write( " After 1 hour working, Elena will complete painting ceiling.\r\n" );
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document.write( " During this 1 hour Nicolai will complete painting of 2/3 of walls.\r\n" );
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document.write( " Then Elena will switch for painting the walls with Nikolai.\r\n" );
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document.write( " Their combined rate of work painting the walls is  =  =  =  of the walls surface per minute.\r\n" );
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document.write( " Hence, it will take  =  = 12 minutes for them to complete the job painting the remaining  of the wall surface.\r\n" );
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document.write( " Last calculation 60 minutes + 12 minutes = 72 minutes gives you the ANSWER.\r\n" );
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document.write( "2. In n.1 I simply \"guessed\" the right solution.\r\n" );
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document.write( " Now I want to show you how to derive it MATHEMATICALLY.\r\n" );
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document.write( " In the general case, let \"w\" and \"c\" be two real numbers between 0 and 1: 0 <= w, c <= 1.\r\n" );
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document.write( " Let us assign Nicolai to paint w-th part of the wall area and c-th part of the ceiling.\r\n" );
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document.write( " Accordingly, Elene will paint (1-w)-th part of the wall area and (1-c)-th part the ceiling.\r\n" );
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document.write( " Our task is to chose the values w and c in a way to minimize the total time.\r\n" );
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document.write( " The necessary condition for it is THIS:\r\n" );
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document.write( " Nicolai and Elena should complete their personal assignments SIMULTANEOUSLY. (*)\r\n" );
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document.write( " I will leave this statement without the proof (because it is self-evident).\r\n" );
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document.write( " It is self-evident, but EXTREMELY helpful.\r\n" );
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document.write( "3. Now let see what does it mean in a quantitative sense.\r\n" );
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document.write( " The time for Nicolai to complete his assignment is 90*w + 120*c minutes (90 is 90 minutes = 1.5 hours and 120 = 120 minutes = 2 hours).\r\n" );
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document.write( " The time for Elena to complete her assignment is 60*(1-w) + 60*(1-c) minutes (60 is 60 minutes = 1 hour and 60 = 60 minutes = 1 hour).\r\n" );
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document.write( " The condition (*) gives you an equation\r\n" );
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document.write( " 90*w + 120*c = 60*(1-w) + 60*(1-c),\r\n" );
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document.write( " which you can simplify to the form\r\n" );
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document.write( " 5w + 6c = 4. (1)\r\n" );
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document.write( " So, you see that the optimal solution must obey to the condition (1) to provide the minimal time.\r\n" );
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document.write( "4. Now, the time that Nicolai will spend for his personal assignment is \r\n" );
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document.write( " N = 90w + 120c = 90w + 108c + 12c = 18*(5w + 6c) + 12c = 18*4 + 12c = 72 + 12c.\r\n" );
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document.write( " And you see that N is always greater than 72, if c > 0.\r\n" );
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document.write( " So, to provide minimal time 72 minutes for Nicolai, the value of \"c\" must be ZERO.\r\n" );
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document.write( " Thus we get this solution w =  , c =  , which I \"guessed\" in n.1.\r\n" );
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document.write( " Now it is DERIVED with mathematical precision.\r\n" );
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document.write( "SOLVED.\r
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