document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1064127>Question 1064127</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You have two solutions; solution 1 contains 10% chemical X, and solution 2 contains 50% chemical X.\r<BR>\n" );
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document.write( "(a)You mix 5 liters of solutions 1 with 3 liters of solution 2. What percent chemical X is the resulting solution?<BR>\n" );
document.write( "(b)You want to make 12 liters of solution that contains 20% chemical X. how many liters of solutions 1 and 2 should you mix together?<BR>\n" );
document.write( "(c)To make a solution that is 40% chemical X, what ratio of solution 1 to solution 2 should you mix? By ratio, we mean something like x parts solution 1 to y parts solution 2. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #679181 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=679181\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x equal the amount of liters in solution 1.<BR>\n" );
document.write( "let y equal the amount of liters in solution 2.\r<BR>\n" );
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document.write( "then:\r<BR>\n" );
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document.write( ".1 * x = amount of chemical X in solution 1.<BR>\n" );
document.write( ".5 * y = amount of chemical X in solution 2.\r<BR>\n" );
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document.write( "(a) You mix 5 liters of solutions 1 with 3 liters of solution 2. What percent chemical X is the resulting solution?\r<BR>\n" );
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document.write( "when x = 5 and y = 3, you get x + y = 8, and you get .1 * x + .5 * y = .1 * 5 + .5 * 3 = .5 + 1.5 = 2.\r<BR>\n" );
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document.write( "you get a total of 8 liters, 2 of which are chemical X.\r<BR>\n" );
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document.write( "the percent of chemical X in the resulting solution is 2/8 = 25%.\r<BR>\n" );
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document.write( "(b)You want to make 12 liters of solution that contains 20% chemical X. how many liters of solutions 1 and 2 should you mix together?\r<BR>\n" );
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document.write( "x + y = 12<BR>\n" );
document.write( ".1 * x + .5 * y = .2 * (x + y)\r<BR>\n" );
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document.write( "since x + y = 12, these formulas become:\r<BR>\n" );
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document.write( "x + y = 12<BR>\n" );
document.write( ".1 * x + .5 * y = .2 * 12 = 2.4\r<BR>\n" );
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document.write( "since x + y = 12, this means that x must be equal to 12 - y.\r<BR>\n" );
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document.write( "replace x in the equation of .1 * x + .5 * y = 2.4 with 12 - y to get:\r<BR>\n" );
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document.write( ".1 * (12 - y) + .5 * y = 2.4\r<BR>\n" );
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document.write( "simplify to get 1.2 - .1 * y + .5 * y = 2.4\r<BR>\n" );
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document.write( "combine like terms to get 1.2 + .4 * y = 2.4\r<BR>\n" );
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document.write( "subtract 1.2 from both sides of the equation to get .4 * y = 1.2\r<BR>\n" );
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document.write( "divide both sides of the equation by .4 to get y = 1.2 / .4 = 3.\r<BR>\n" );
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document.write( "since x + y = 12, this means that x must be equal to 9.\r<BR>\n" );
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document.write( "you get x = 9 and y = 3\r<BR>\n" );
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document.write( "the amount of chemical X is therefore .1 * 9 + .5 * 3 = .9 + 1.5 = 2.4.\r<BR>\n" );
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document.write( "2.4 / 12 = 20%.\r<BR>\n" );
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document.write( "you need 9 liters of solution 1 and 3 liters of solution 2 to get a solution that is 20% composed of chemical X.\r<BR>\n" );
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document.write( "what you had here are two equations that needed to be solved simultaneously.\r<BR>\n" );
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document.write( "those equation were:\r<BR>\n" );
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document.write( "x + y = 12<BR>\n" );
document.write( ".1 * x + .5 * y = 2.4\r<BR>\n" );
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document.write( "if you had decided to solve these by elimination, you would have multiplied the second equation by 10 and left the second equation as is to get:\r<BR>\n" );
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document.write( "x + y = 12<BR>\n" );
document.write( "x + 5y = 24\r<BR>\n" );
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document.write( "you would then have subtracted the first equation from the second to get:\r<BR>\n" );
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document.write( "4y = 12\r<BR>\n" );
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document.write( "you would then have solved for y to get y = = 3\r<BR>\n" );
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document.write( "that would then have led to x = 9.\r<BR>\n" );
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document.write( "same answer.\r<BR>\n" );
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document.write( "first solution was by substitution.<BR>\n" );
document.write( "second solution was by elimination.\r<BR>\n" );
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document.write( "(c)To make a solution that is 40% chemical X, what ratio of solution 1 to solution 2 should you mix? By ratio, we mean something like x parts solution 1 to y parts solution 2.\r<BR>\n" );
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document.write( "you want .1 * x + .5 * y = .4 * (x + y)\r<BR>\n" );
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document.write( "simplify this to get:\r<BR>\n" );
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document.write( ".1 * x + .5 * y = .4 * x + .4 * y\r<BR>\n" );
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document.write( "subtract .1 * x from both sides of the equation and subtract .4 * y from both sides of the equation to get:\r<BR>\n" );
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document.write( ".5 * y - .4 * y = .4 * x - .1 * x\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( ".1 * y = .3 * x\r<BR>\n" );
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document.write( "divide both sides of the equation by y and divide both sides of the equation by .3 and you get:\r<BR>\n" );
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document.write( ".1 / .3 = x / y\r<BR>\n" );
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document.write( "simplify to get 1/3 = x/y\r<BR>\n" );
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document.write( "that's your ratio.\r<BR>\n" );
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document.write( "x/y = 1/3\r<BR>\n" );
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document.write( "you will need one x for every three y.\r<BR>\n" );
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document.write( "this means 1 liter of 10% solution for every 3 liters of 50% solution.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "assume x = 1 and y = 3.<BR>\n" );
document.write( "your resulting solution will be 4 liters total and the amount of chemical X will be .1 * 1 + .5 * 3 = 1.6.<BR>\n" );
document.write( "1.6 / 4 = 40%.\r<BR>\n" );
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document.write( "assume x = 2 and y = 6.<BR>\n" );
document.write( "your resulting solution will be 8 liters total and the amount of chemical X will be .1 * 2 + .5 * 6 = 3.2.<BR>\n" );
document.write( "3.2 / 8 = 40%\r<BR>\n" );
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document.write( " </SPAN>\n" );
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