document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=553995>Question 553995</A>: <I><SPAN STYLE=\"word-break: break-all;\"> \" An alloy containing copper and silver in the ratio 1 : 2 is mixed with another alloy containing copper and silver in the ration 2 : 3. In what ratio must the two alloys be mixed to produce a new alloy containing copper and silver in the ratio 13 : 22? \" \r<BR>\n" );
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document.write( "* I have tried it many times and I still got the answer 5:4 because<BR>\n" );
document.write( "Let x = the # of first alloy units<BR>\n" );
document.write( "Let y = the # of second alloy units\r<BR>\n" );
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document.write( "You need 13 copper, so since the first gives you 1 unit while the second gives you 2, you have<BR>\n" );
document.write( "1x + 2y = 13\r<BR>\n" );
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document.write( "Similarly, you need 22 silver and the first gives you 2 while the second gives you 3, so you have<BR>\n" );
document.write( "2x + 3y = 22\r<BR>\n" );
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document.write( "Solve this system of equations so you get<BR>\n" );
document.write( "x = 13 - 2y<BR>\n" );
document.write( "2 (13 - 2y) + 3y = 22<BR>\n" );
document.write( "26 - 4y + 3y = 22<BR>\n" );
document.write( "- y = -4<BR>\n" );
document.write( "y = 4\r<BR>\n" );
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document.write( "x + 2(4) = 13<BR>\n" );
document.write( "x = 5\r<BR>\n" );
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document.write( "therefore, the ratio will be x:y which is alloy 1: alloy 2, or 5:4<BR>\n" );
document.write( "But still I still I'm wrong , Please help me about this.<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 #361154 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=361154\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> it took a while but i think i've got it.<BR>\n" );
document.write( "i needed to put it into numbers before i could solve it with letters.<BR>\n" );
document.write( "here's how i solved it with letters.<BR>\n" );
document.write( "when the copper to silver ratio is 1/2, this means that the alloy is 1/3 copper and 2/3 silver.<BR>\n" );
document.write( "when the copper to silver ratio is 2/3, this means that the alloy is 2/5 copper and 3/5 silver.<BR>\n" );
document.write( "when the copper to silver ratio is 13/22, this means that the alloy is 13/35 copper and 22/35 silver.<BR>\n" );
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document.write( "let x equal the amount of the first alloy and lety = the amount of the second alloy and let z equal the amount of the final alloy.<BR>\n" );
document.write( "you get 2 equations that need to be solved simultaneously.<BR>\n" );
document.write( "the first equation is:<BR>\n" );
document.write( "x + y = z<BR>\n" );
document.write( "the second equation is:<BR>\n" );
document.write( "(1/3)x + (2/5)y = (13/35)z<BR>\n" );
document.write( "from the first equation, solve for y to get:<BR>\n" );
document.write( "y = z - x)<BR>\n" );
document.write( "substitute for y in the second equation to get:<BR>\n" );
document.write( "(1/3)x + (2/5)(z-x) = (13/35)z<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "(1/3)x + (2/5)z - (2/5)x = (13/35)z<BR>\n" );
document.write( "multiply both sides of this equation by (3*5*35) to get rid of the denominators to get:<BR>\n" );
document.write( "175x + 210z - 210x = 195z<BR>\n" );
document.write( "subtract 210z from both sides of this equation to get:<BR>\n" );
document.write( "175x - 210x = 195z - 210z<BR>\n" );
document.write( "combine like terms to get:<BR>\n" );
document.write( "-35x = -15z<BR>\n" );
document.write( "substitute (x+y) for z to get:<BR>\n" );
document.write( "-35x = -15(x+y)<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "-35x = -15x -15y<BR>\n" );
document.write( "add 35x and 15y to both sides of this equation to get:<BR>\n" );
document.write( "15y = 35x - 15x<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "15y = 20x<BR>\n" );
document.write( "divide both sides of this equation by 15 to get:<BR>\n" );
document.write( "y = 20x/15<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "y = 4x/3<BR>\n" );
document.write( "this is the ratio you are looking for.<BR>\n" );
document.write( "let's see if it works.<BR>\n" );
document.write( "if x = 3, then y = 4<BR>\n" );
document.write( "you have 3 units of the first solution and 4 unis of the second solution to yield 7 units of the combined solution.<BR>\n" );
document.write( "the first solution yields 1 unit of copper and 2 units of silver.<BR>\n" );
document.write( "the second solution yields 1.6 units of copper and 2.4 units of silver.<BR>\n" );
document.write( "the ratio of copper to silver in the first solution is 1/2.<BR>\n" );
document.write( "the ratio of copper in the second solution is 1.6/2.4 = 2/3.<BR>\n" );
document.write( "the combined solution yields 1 + 1.6 = 2.6 units of copper and 2 + 2.4 = 4.4 units of silver.<BR>\n" );
document.write( "the ratio of copper to silver in the combined solution is 2.6/4.4 = 13/22.<BR>\n" );
document.write( "everything checks out so the answer appears to be good.<BR>\n" );
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