document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1104867>Question 1104867</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi<BR>\n" );
document.write( "michael had 80% more money than alvin.vanessa had 40% less than alvin. Michael and alvin gave vanessa some money in the ratio of 3:2 respectively. In the end Vanessa had twice as much money as before. Given that Michael had $144 more than Vanessa in the end, how much money did each of them have at first.\r<BR>\n" );
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document.write( "thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #719658 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=719658\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> took a while, but i think i have it.\r<BR>\n" );
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document.write( "M = amount of money that michael has to start with.<BR>\n" );
document.write( "A = amount of money that alvin has to start with.<BR>\n" );
document.write( "V = amount of money that vanessa has to start with.<BR>\n" );
document.write( "X = amount of money that michael gives to vanessa.<BR>\n" );
document.write( "Y = amount of money that alvin gives to vanessa.\r<BR>\n" );
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document.write( "M = 1.8 * A (michael has 80% more money than alvin).<BR>\n" );
document.write( "V = .6 * A (vanessa has 40% less money than alvin).\r<BR>\n" );
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document.write( "X/Y = 3/2<BR>\n" );
document.write( "from this, you can derive X = 3/2 * Y and Y = 2/3 * X.\r<BR>\n" );
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document.write( "you are given that michael and alvin give vanessa some money in the ratio of 3/2, and that, after they give her the money, she has twice as much as she had originally.\r<BR>\n" );
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document.write( "the equation for that becomes:<BR>\n" );
document.write( "2V = V + X + Y is the equation that models this.<BR>\n" );
document.write( "from this equation, you can derive V = X + Y.\r<BR>\n" );
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document.write( "you are given that, after michael has given vanessa some money, that michael has 144 dollars more than vanessa has.\r<BR>\n" );
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document.write( "M - X = 2V + 144.\r<BR>\n" );
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document.write( "since 2V is equal to V + X + Y, then you get:\r<BR>\n" );
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document.write( "M - X = V + X + Y + 144.\r<BR>\n" );
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document.write( "since V = .6 * A, you get:\r<BR>\n" );
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document.write( "M - X = .6 * A + X + Y + 144.\r<BR>\n" );
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document.write( "since M is equal to 1.8 * A, you get:\r<BR>\n" );
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document.write( "1.8 * A - X = .6 * A + X + Y + 144.\r<BR>\n" );
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document.write( "add X to both sides of this equation and subtract .6 * A from both sides of this equation to get:\r<BR>\n" );
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document.write( "1.2 * A = 2 * X + Y + 144.\r<BR>\n" );
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document.write( "since Y is equal to 2/3 * X, this equation becomes:\r<BR>\n" );
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document.write( "1.2 * A = 2 * X + 2/3 * X + 144.\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "1.2 * A = 8/3 * X + 144.\r<BR>\n" );
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document.write( "you know that V = X + Y and that Y = 2/3 * X.<BR>\n" );
document.write( "from this you can derive that V = 5/3 * X.<BR>\n" );
document.write( "from this you can derive that X = 3/5 * V.<BR>\n" );
document.write( "you know that V = .6 * A.<BR>\n" );
document.write( ".6 is equivalent to 3/5.<BR>\n" );
document.write( "therefore you can derive that X = 3/5 * V which becomes X = 3/5 * 3/5 * A which results in X = 9/25 * A.\r<BR>\n" );
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document.write( "your equation of 1.2 * A = 8/3 * X + 144 becomes:\r<BR>\n" );
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document.write( "1.2 * A = 8/3 * 9/25 * A + 144.\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "1.2 * A = 72/75 * A + 144.\r<BR>\n" );
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document.write( "subtract 72/75 * A from both sides of this equation to get:\r<BR>\n" );
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document.write( "1.2 * A - 72/75 * A = 144\r<BR>\n" );
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document.write( "1.2 is equal to 90/75, therefore you get:\r<BR>\n" );
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document.write( "90/75 * A - 72/75 * A = 144.\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "18/75 * A = 144.\r<BR>\n" );
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document.write( "solve for A to get:\r<BR>\n" );
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document.write( "A = 75 / 18 * 144 = 600.\r<BR>\n" );
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document.write( "M = 1.8 * A = 1080.<BR>\n" );
document.write( "V = .6 * A = 360.\r<BR>\n" );
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document.write( "your solution should be:\r<BR>\n" );
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document.write( "M = 1080 (what michael had to start with)<BR>\n" );
document.write( "V = 360 (what vanessa had to start with)<BR>\n" );
document.write( "A = 600 (what alvin had to start with)\r<BR>\n" );
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document.write( "you also know that V = X + Y which led you to V = 5/3 * X.<BR>\n" );
document.write( "from this, you derived that X = 3/5 * V<BR>\n" );
document.write( "this makes X = 3/5 * 360 = 216.\r<BR>\n" );
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document.write( "you know that Y = 2/3 * X.<BR>\n" );
document.write( "this makes Y = 144.\r<BR>\n" );
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document.write( "you now have:\r<BR>\n" );
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document.write( "M = 1080<BR>\n" );
document.write( "V = 360<BR>\n" );
document.write( "A = 600<BR>\n" );
document.write( "X = 216<BR>\n" );
document.write( "Y = 144.\r<BR>\n" );
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document.write( "you already have your solution.\r<BR>\n" );
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document.write( "that is:\r<BR>\n" );
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document.write( "M = 1080<BR>\n" );
document.write( "V = 360<BR>\n" );
document.write( "A = 600\r<BR>\n" );
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document.write( "the calculation of X and Y is to confirm the solution is correct.\r<BR>\n" );
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document.write( "you are given that michael has 144 more than vanessa after vanessa was given the money by michael and alvin.\r<BR>\n" );
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document.write( "she had double what he had originally, then she had 720.\r<BR>\n" );
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document.write( "360 + 216 + 144 = 720 which is 2 * 360, so that part is confirmed.\r<BR>\n" );
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document.write( "michael had 1080 to start.<BR>\n" );
document.write( "after he gave vanessa 216, he had 864 left.\r<BR>\n" );
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document.write( "864 - 720 = 144, so the part that he had 144 more than vanessa after all the exchanges of money were complete is also confirmed.\r<BR>\n" );
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document.write( "the solution is confirmed to be good. </SPAN>\n" );
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