document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210613>Question 1210613</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Marek and Zaven had 900 coins altogether. Marek spent 25% fewer coins than Zaven. Zaven was left with 50% as many coins as the number he spent. Marek had 75% of the total number of coins the two friends had left in the end. How many coins did Marek have in the end?<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 #854416 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5362)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854416\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> DATA WE HAVE:<BR>\n" );
document.write( "Marek and Zaven had 900 coins altogether.<BR>\n" );
document.write( "That is what they had initially.<BR>\n" );
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document.write( "Zaven was left with 50% (half) as many coins as the number he spent.<BR>\n" );
document.write( "He spent twice as many coins as he was left with. <BR>\n" );
document.write( "For every 1 coin he had left, there were 2 he had spent, and 3 that he originally had.<BR>\n" );
document.write( "That tells me that the number of coins Zaven initially had was a multiple of 3.<BR>\n" );
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document.write( "Marek spent 25% fewer coins than Zaven.<BR>\n" );
document.write( "Marek spent 100%-25%=75% of what Zaven spent.<BR>\n" );
document.write( "If I knew how much Zaven spent I can multiply times 0.75 to find what Marek spent. <BR>\n" );
document.write( "For every 4 coins Zaven spent Marek spent 3. So I could instead divide by 4 and multiply times 3.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "Marek had 75% of the total number of coins the two friends had left in the end.<BR>\n" );
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document.write( "Could Marek have initially had 75% of the total number of coins the two friends had left in the end? It does not seem possible.<BR>\n" );
document.write( "I think what the \"in the end\" part applies to the whole sentence, and what was meant is:<BR>\n" );
document.write( "In the end, Marek had 75% of the total number of coins the two friends had left.<BR>\n" );
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document.write( "AN EASY WAY TO SOLVE IT (guess and check):<BR>\n" );
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document.write( "Fist Guess:<BR>\n" );
document.write( "What if Zaven initially had <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=300coins ALIGN=MIDDLE  ALT=\"300coins\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Then, Zaven must have spent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200coins ALIGN=MIDDLE  ALT=\"200coins\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and must have had left with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=300coins-200coins=100coins ALIGN=MIDDLE  ALT=\"300coins-200coins=100coins\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the end.<BR>\n" );
document.write( "Then, Marek<BR>\n" );
document.write( "must have started with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900coins-300coins=600coins ALIGN=MIDDLE  ALT=\"900coins-300coins=600coins\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "must have spent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.75%2A200coins=150coins ALIGN=MIDDLE  ALT=\"0.75%2A200coins=150coins\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and have had left <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=600coins-150coins=450coins ALIGN=MIDDLE  ALT=\"600coins-150coins=450coins\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the end.<BR>\n" );
document.write( "Then, Zaven and Marek then together had <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100coins%2B450coins=550coins ALIGN=MIDDLE  ALT=\"100coins%2B450coins=550coins\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the end,<BR>\n" );
document.write( "and that would mean that in the end,<BR>\n" );
document.write( "the number of coins Marek had,<BR>\n" );
document.write( "as a fraction and as a percentage of the number of coins Zaven and Marek had together, was<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=450coins%2F%22550+coins%22=9%2F11=%220.818181...%22=%2281.8181...%22 ALIGN=MIDDLE  ALT=\"450coins%2F%22550+coins%22=9%2F11=%220.818181...%22=%2281.8181...%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%25%22 ALIGN=MIDDLE  ALT=\"%22%25%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "We need <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=75 ALIGN=MIDDLE  ALT=\"75\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%25%22 ALIGN=MIDDLE  ALT=\"%22%25%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > as an answer, so we should try to have Marek start with a little less with respect to Zaven.<BR>\n" );
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document.write( "second Guess:<BR>\n" );
document.write( "What if Zaven initially had <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360coins ALIGN=MIDDLE  ALT=\"360coins\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Then, Zaven must have spent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240coins ALIGN=MIDDLE  ALT=\"240coins\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and must have had left with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360coins-240coins=120coins ALIGN=MIDDLE  ALT=\"360coins-240coins=120coins\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the end.<BR>\n" );
document.write( "Then, Marek<BR>\n" );
document.write( "must have started with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900coins-360coins=540coins ALIGN=MIDDLE  ALT=\"900coins-360coins=540coins\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "must have spent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.75%2A240coins=180coins ALIGN=MIDDLE  ALT=\"0.75%2A240coins=180coins\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and have had left <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=540coins-180coins=360coins ALIGN=MIDDLE  ALT=\"540coins-180coins=360coins\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the end.<BR>\n" );
document.write( "Then, Zaven and Marek then together had <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=120coins%2B360coins=480coins ALIGN=MIDDLE  ALT=\"120coins%2B360coins=480coins\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the end,<BR>\n" );
document.write( "and that would mean that in the end,<BR>\n" );
document.write( "the number of coins Marek had,<BR>\n" );
document.write( "as a fraction and as a percentage of the number of coins Zaven and Marek had together, was<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360coins%2F%22480+coins%22=3%2F4=75%2F100=0.75=75%2F100=75 ALIGN=MIDDLE  ALT=\"360coins%2F%22480+coins%22=3%2F4=75%2F100=0.75=75%2F100=75\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%25%22 ALIGN=MIDDLE  ALT=\"%22%25%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "A HYBRID OF THE SOLUTION ABOVE AND jogsarithmetic's solution:<BR>\n" );
document.write( "Let's call the number of coins Zaven had at start, what he spent, and what he had left in the end 3x, 2x, and x, respectively.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%284%2C4%2C%22+%22%2CZaven%2CMarek%2C+Zaven+%2B+Marek%2C%0D%0Ainitially%2C3x%2C900-3x%2C900%2C%0D%0Aspent%2C2x%2C1.5x%2C3.5x%2C%0D%0Afinal%2Cx%2C900-4.5x%2C900-3.5x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then, we have just one variable, and the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28900-4.5x%29%2F%28900-3.5x%29=0.75 ALIGN=MIDDLE  ALT=\"%28900-4.5x%29%2F%28900-3.5x%29=0.75\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900-4.5x=0.75%28900-3.5x%29 ALIGN=MIDDLE  ALT=\"900-4.5x=0.75%28900-3.5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3600-18x=3%28900-3.5x%29 ALIGN=MIDDLE  ALT=\"3600-18x=3%28900-3.5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3600-18x=2700-10.5x ALIGN=MIDDLE  ALT=\"3600-18x=2700-10.5x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3600-2700=7.5x%29 ALIGN=MIDDLE  ALT=\"3600-2700=7.5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=900%2F7.5=120 ALIGN=MIDDLE  ALT=\"x=900%2F7.5=120\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In th e end, the number of coins Marek had was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900-4.5%2A120=900-540=highlight%28360%29 ALIGN=MIDDLE  ALT=\"900-4.5%2A120=900-540=highlight%28360%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Some teachers want problems solved the way it was taught in class, and may not like alternative solutions.<BR>\n" );
document.write( "Some teacher may want to give extra point f=if more that one way to solve the problem is shown. </SPAN>\n" );
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