document.write( "Question 1034450: One alloy is 2 parts iron to 3 parts silver and another alloy is 7 parts iron to 3 parts silver. How much of each should be combined to produce a 30 pound alloy that is 1 part iron to 1 part silver? \n" ); document.write( "

Algebra.Com's Answer #649097 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "One alloy is 2 parts iron to 3 parts silver and another alloy is 7 parts iron to 3 parts silver.
\n" ); document.write( "How much of each should be combined to produce a 30 pound alloy that is 1 part iron to 1 part silver?
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document.write( "Let \"x\" be an amount of the first  alloy to be mixed (in pounds), \r\n" );
document.write( "and \"y\" be an amount of the second alloy to be mixed (in pounds).\r\n" );
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document.write( "So, the \"total mass equation\" is \r\n" );
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document.write( "x + y = 30.    (1)\r\n" );
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document.write( "The first alloy contains 2+3 = 5 parts of equal masses in total.\r\n" );
document.write( "Of them, 2 parts is iron and 3 parts is silver.\r\n" );
document.write( "So, the mass \"x\" of the first alloy contains  = 0.6x of silver.\r\n" );
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document.write( "The second alloy contains 7+3 = 10 parts of equal masses in total.\r\n" );
document.write( "Of them, 7 parts is iron and 3 parts is silver.\r\n" );
document.write( "So, the mass \"y\" of the second alloy contains  = 0.3y of silver.\r\n" );
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document.write( "Now, the \"silver contents equation\" is\r\n" );
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document.write( "0.6x + 0.3y = 0.5*30.   (2)\r\n" );
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document.write( "0.5 on the right side means \"half of mass\" of the combined alloy.\r\n" );
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document.write( "Let's simplify equations (1) and (2) and write it together as a system.\r\n" );
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document.write( "   x +    y = 30.    (1)\r\n" );
document.write( "0.6x + 0.3y = 15.    (2)\r\n" );
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document.write( "To solve it, express x = 30-y from (1) and substitute it into (2). You will get a single equation for y:\r\n" );
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document.write( "0.6*(30-y) + 0.3y = 15.\r\n" );
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document.write( "Simplify and solve it:\r\n" );
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document.write( "18 - 0.6y + 0.3y = 15,\r\n" );
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document.write( "3 = 0.3y,\r\n" );
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document.write( "y = 10.\r\n" );
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document.write( "So, 10 pounds of the 2-nd alloy should be taken, and, hence, \r\n" );
document.write( "the amount of the 1-st alloy is 30 - 10 = 20 pounds.\r\n" );
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document.write( "Check:  20 pounds of the first alloy contain 3 parts of silver of the total 5 parts, i.e. 12 pounds of silver.\r\n" );
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document.write( "        10 pounds of the second alloy contain 3 parts of silver of the total 10 parts, i.e. 3 pounds of silver.\r\n" );
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document.write( "        12+3 = 15 pounds of silver in the combined/final alloy, which is precisely half of its mass.\r\n" );
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document.write( "The problem is solved.\r\n" );
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