document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210372>Question 1210372</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If painter A can paint a wall in one hour and painter B can paint the same wall in two hours, how long does it take for painter A and B to paint the same wall together?<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 #852123 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52776)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852123\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "If painter A can paint a wall in one hour and painter B can paint the same wall in two hours, <BR>\n" );
document.write( "how long does it take for painter A and B to paint the same wall together?<BR>\n" );
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document.write( "This problem is so simple that I will solve it mentally using reasoning, ONLY.\r<BR>\n" );
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document.write( "Working together, they will spend equal time.\r\n" );
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document.write( "Since painter A works twice as fast as painter B, painter A will make twice greater job than painter B.\r\n" );
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document.write( "Hence, painter A will make 2/3 of the job, while painter B will make 1/3 of the job.\r\n" );
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document.write( "To make 2/3 of the job, painter A needs 2/3 of an hour, i.e. 40 minutes.\r\n" );
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document.write( "Hence, working together, both painters will complete the job in 2/3 of an hour, or in 40 minutes.\r\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "Simple problems should be solved with simple methods, isn't it ?\r<BR>\n" );
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document.write( "This problem is interesting and deserves a thorough discussion.\r<BR>\n" );
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document.write( "From one hand side, it is a typical joint-work word problem.\r<BR>\n" );
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document.write( "But a typical joint-work word problem usually tells about small pieces of a whole job<BR>\n" );
document.write( "that can be done by participants in a certain time interval.\r<BR>\n" );
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document.write( "Therefore, been a typical problem, this given problem, from the other hand side, is unusual, <BR>\n" );
document.write( "because it tells about grand pieces of a whole job, like the whole wall, which can be painted in one hour.\r<BR>\n" );
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document.write( "And if a student is non-experienced, this immediately puts the student in a dead end.\r<BR>\n" );
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document.write( "Some tutors may come explaining that a student can work with grand pieces exactly in the same<BR>\n" );
document.write( "way as he/she works with small pieces.\r<BR>\n" );
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document.write( "But I do not believe that an average/typical beginner student will be satisfied and will accept such an explanation.<BR>\n" );
document.write( "Such explanations can be good for tutors or for teachers (because formally they complete their job doing this way), <BR>\n" );
document.write( "but a student will unlikely be satisfied and unlikely will accept such a solution.\r<BR>\n" );
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document.write( "Again, explanations with grand pieces are good for teacher and for tutors, especially if they want to shock a student. <BR>\n" );
document.write( "Yes, the shocking method is sometimes a good method for teaching, especially for advanced students.<BR>\n" );
document.write( "But an average young student will remain unsatisfied. His soul will ask for a more standard solution/explanation <BR>\n" );
document.write( "in terms he/she is accustomed to: in terms of \"small pieces\".\r<BR>\n" );
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document.write( "Here is an example on how to build/(to create) such a solution/explanation for this given problem.\r<BR>\n" );
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document.write( "Let's make our reasoning not in terms of hours, but in terms of minutes.\r<BR>\n" );
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document.write( "Then the problem can be equivalently re-phrased this way\r<BR>\n" );
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document.write( "    If painter A can paint a wall in 60 minutes and painter B can paint the same wall in 120 minutes,\r\n" );
document.write( "    how long does it take for painters A and B to paint the same wall together?\r\n" );
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document.write( "Surely, now this modified problem looks like an absolutely standard even for a beginner. <BR>\n" );
document.write( "Now we pronounce a standard mantra \r<BR>\n" );
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document.write( "    painter A makes 1/60 of the job in one minute,\r\n" );
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document.write( "    painter B makes 1/120 of the job in one minute.\r\n" );
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document.write( "    And working together, they will make  \r\n" );
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document.write( "         <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F60 ALIGN=MIDDLE  ALT=\"1%2F60\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F120 ALIGN=MIDDLE  ALT=\"1%2F120\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F120 ALIGN=MIDDLE  ALT=\"2%2F120\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F120 ALIGN=MIDDLE  ALT=\"1%2F120\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F120 ALIGN=MIDDLE  ALT=\"3%2F120\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F40 ALIGN=MIDDLE  ALT=\"1%2F40\"   DISABLED_style= \"max-width:90%; height:auto;\" >  of the job per minute.\r\n" );
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document.write( "   From it, we conclude immediately that the necessary time \r\n" );
document.write( "   to complete the job working together is 40 minutes.\r\n" );
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document.write( "This way, the problem is solved within the usual/traditional framework.\r<BR>\n" );
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document.write( "I wrote these explanations to show that sometime it is useful <BR>\n" );
document.write( "to take a look to a problem from the other/unexpected point/angle of view.\r<BR>\n" );
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document.write( "XXXXXXXXXXXX &nbsp;&nbsp;S e v e r a l  &nbsp;&nbsp;d a y s  &nbsp;&nbsp;l a t e r &nbsp;&nbsp;XXXXXXXXXXXX\r<BR>\n" );
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document.write( "And now I invite you to laugh with me at the AI solution to a similar problem.\r<BR>\n" );
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document.write( "Today, 8/13/2025, I posted this similar problem to Google \r<BR>\n" );
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document.write( "    If painter A can paint a wall in half an hour and painter B can paint the same wall in half an hour,\r\n" );
document.write( "    how long does it take for painters A and B to paint the same wall together?\r\n" );
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document.write( "Below is copy-paste of the AI response\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- - - - - - - S T A R T -  - - - - - - - -\r<BR>\n" );
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document.write( "AI Overview\r<BR>\n" );
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document.write( "If painter A can paint a wall in 30 minutes (1/2 hour) and painter B can paint the same wall in 30 minutes (1/2 hour), <BR>\n" );
document.write( "together they would take 15 minutes (1/4 hour) to paint the wall.\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "Rate of painting: Painter A paints 1/2 of the wall in 1 minute, and painter B also paints 1/2 of the wall in 1 minute.\r<BR>\n" );
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document.write( "Combined rate: Together, they paint 1/2 + 1/2 = 1 wall in one minute.\r<BR>\n" );
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document.write( "Time to paint the whole wall: Since they paint 1 wall in 1 minute, they can paint the whole wall together in 1/(1/2 + 1/2) = 1/1 = 15 minutes.\r<BR>\n" );
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document.write( "https://www.google.com/search?q=If+painter+A+can+paint+a+wall+in+half+an+hour+and+painter+B+can+paint+the+same+wall+in+half+an+hour%2C+how+long+does+it+take+for+painter+A+and+B+to+paint+the+same+wall+together%3F&rlz=1C1CHBF_enUS1071US1071&oq=If+painter+A+can+paint+a+wall+in+half+an+hour+and+painter+B+can+paint+the+same+wall+in+half+an+hour%2C+how+long+does+it+take+for+painter+A+and+B+to+paint+the+same+wall+together%3F&gs_lcrp=EgZjaHJvbWUyBggAEEUYOdIBCTE5MDZqMGoxNagCCLACAfEFFpskuz7iUCjxBRabJLs-4lAo&sourceid=chrome&ie=UTF-8\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- - - - - - - F I N I S H -  - - - - - - - -\r<BR>\n" );
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document.write( "8/13/2025\r<BR>\n" );
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document.write( "Notice that the answer of the AI is correct, but his intermediate reasoning and calculations are wrong.\r<BR>\n" );
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document.write( "In other words, in this case, the AI, without having a prototype of the solution / (without having a TEMPLATE)<BR>\n" );
document.write( "behaves exactly like a dumb student: he can't figure out what to do and how to get the answer correctly.<BR>\n" );
document.write( "This only confirms my reasoning above about the average beginner student.\r<BR>\n" );
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document.write( "The Google AI and such a student behave identically and suffer identically.<BR>\n" );
document.write( "Proved experimentally.\r<BR>\n" );
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document.write( "Ha - ha - ha.\r<BR>\n" );
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document.write( "This story has interesting continuation.\r<BR>\n" );
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document.write( "On September 2, 2025, I submitted this problem to Google AI again.\r<BR>\n" );
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document.write( "This time it produced correct, right, perfect and ideal solution.\r<BR>\n" );
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document.write( "It tells me that AI needs very much to have an adequate, attentive and qualified teacher/tutor/mentor.\r<BR>\n" );
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document.write( "Then the AI learns and increases its level and becomes adequate.\r<BR>\n" );
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document.write( "I am glad to see this progress.\r<BR>\n" );
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