document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=384381>Question 384381</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how many gallons of a 15% sugar solution must be mixed with 5 gallons of a 40% sugar solution to make a 30% sugar solution </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #272110 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=272110\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are multiple solutions to this problem because you have 1 equation in 2 unknowns to work with.\r<BR>\n" );
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document.write( "Let x = number of gallons of 15% sugar solution.<BR>\n" );
document.write( "Let y = number of gallons of 30% sugar solution.\r<BR>\n" );
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document.write( "The formula is:\r<BR>\n" );
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document.write( ".15*x + .4*5 = .3*y\r<BR>\n" );
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document.write( "Simplify to get:\r<BR>\n" );
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document.write( ".15*x + 2 = .3*y\r<BR>\n" );
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document.write( "Divide both sides of this equation by .3 to get:\r<BR>\n" );
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document.write( ".5*x + 6.666666667 = y\r<BR>\n" );
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document.write( "This is the same as:\r<BR>\n" );
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document.write( "y = .5*x + 6.6666666667\r<BR>\n" );
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document.write( "For every value of x, you have a corresponding value of y that will satisfy the equation.\r<BR>\n" );
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document.write( "You can graph this equation.\r<BR>\n" );
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document.write( "It looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28600%2C600%2C-1%2C20%2C-10%2C20%2C.5%2Ax+%2B+6+%2B+%282%2F3%29%29 ALIGN=MIDDLE  ALT=\"graph+%28600%2C600%2C-1%2C20%2C-10%2C20%2C.5%2Ax+%2B+6+%2B+%282%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I'll pick 3 points at x = 5, 10, 15 to show you that the relationship holds for all values of x.\r<BR>\n" );
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document.write( "When x = 5, y = 9.166666667: .15*x + 2 = 2.75 and .3*y = 2.75. <BR>\n" );
document.write( "When x = 10, y = 11.66666667: .15*x + 2 = 3.5 and .3*y = 3.5.<BR>\n" );
document.write( "When x = 15, y = 14.166666667: .15*x + 2 = 4.25 and .3*y = 4.25.\r<BR>\n" );
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document.write( "The equation holds for all values of x.\r<BR>\n" );
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document.write( "The answer to your question is that:\r<BR>\n" );
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document.write( "x gallons of a 15% solution will produce .5*x + 6.66666667 gallons of a 30% solution.\r<BR>\n" );
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document.write( "This comes from the equation of y = .5*x + 6.66666667\r<BR>\n" );
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document.write( "The x is the number of gallons of 15% solution.<BR>\n" );
document.write( "The y is the number of gallons of 30% solution.\r<BR>\n" );
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document.write( "If the question had said:\r<BR>\n" );
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document.write( "How many gallons of a 15% solution plus 5 gallons of a 40% solution are required to make 15 gallons of a 30% solution, then you could have solved the problem for an exact solution.\r<BR>\n" );
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document.write( "In that case, you get y = 15 and the equation of y = .5*x + 6.6666667 becomes:\r<BR>\n" );
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document.write( "15 = .5*x + 6.6666667\r<BR>\n" );
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document.write( "Now you have 1 equation in one unknown that can be solved.\r<BR>\n" );
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document.write( "You subtract 6.66666667 from both sides of the equation to get:\r<BR>\n" );
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document.write( "8.33333333 = .5*x\r<BR>\n" );
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document.write( "You divide both sides of the equation by .5 to get:\r<BR>\n" );
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document.write( "16.66666667 = x\r<BR>\n" );
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document.write( "Your answer, in that case, is that 16.66666667 gallons of a 15% solution plus 5 gallons of a 40% solution are required to make 15 gallons of a 30% solution.\r<BR>\n" );
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document.write( ".15*16.66666667 = 2.5<BR>\n" );
document.write( ".4*5 = 2<BR>\n" );
document.write( "2.5 + 2 = 4.5 gallons of solution from the 15% and 40% solution.\r<BR>\n" );
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document.write( ".3*15 = 4.5 gallons  of solution from the 30% solution.\r<BR>\n" );
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document.write( "Answer is confirmed as correct.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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