document.write( "Question 320851: From a pile of 100 pennies, 100 nickles , and 100 dimes . select 21 coins which have a total of exactly $1.00 . how many of each of the three types should be selected? \n" ); document.write( "
Algebra.Com's Answer #229761 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"system%28p%2Bn%2Bd=21%2Cp%2B5n%2B10d=100%29\"\r\n" );
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document.write( "Eliminate n by multiplying the first equation through by -5\r\n" );
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document.write( "\"system%28-5p-5n-5d=-105%2Cp%2B5n%2B10d=100%29\"\r\n" );
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document.write( "Adding these term by term:\r\n" );
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document.write( "\"-4p%2B5d=-5\"\r\n" );
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document.write( "The smallest absolute value of a coefficient is 4,\r\n" );
document.write( "so write all integers in terms of their nearest multiple\r\n" );
document.write( "of 4\r\n" );
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document.write( "\"-4p%2B%284%2B1%29d=-%284%2B1%29\"\r\n" );
document.write( "\"-4p%2B4d%2Bd=-4-1\"\r\n" );
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document.write( "Divide through by 4\r\n" );
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document.write( "\"-p%2Bd%2Bd%2F4=-1-1%2F4\"\r\n" );
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document.write( "Isolate fractional terms:\r\n" );
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document.write( "\"d%2F4%2B1%2F4=p-d-1\"\r\n" );
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document.write( "The right side is an integer, so let that integer be A, \r\n" );
document.write( "set both sides equal to integer A:\r\n" );
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document.write( "\"system%28d%2F4%2B1%2F4=A%2Cp-d-1=A%29\"\r\n" );
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document.write( "Clear the first equation of fractions:\r\n" );
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document.write( "\"system%28d%2B1=4A%2Cp-d-1=A%29\"\r\n" );
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document.write( "Solve the first equation for d\r\n" );
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document.write( "\"d=4A-1\"\r\n" );
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document.write( "Substitute that for d in \"p-d-1=A\"\r\n" );
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document.write( "\"p-d-1=A\"\r\n" );
document.write( "\"p-%284A-1%29-1=A\"\r\n" );
document.write( "\"p-4A%2B1-1=A\"\r\n" );
document.write( "\"p-4A=A\"\r\n" );
document.write( "\"p=5A\"\r\n" );
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document.write( "Subsitutute \"p=5A\" and \"d=4A-1\" in \"p%2Bn%2Bd=21\"\r\n" );
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document.write( "\"p%2Bn%2Bd=21\"\r\n" );
document.write( "\"5A%2Bn%2B4A-1=21\"\r\n" );
document.write( "\"9A%2Bn-1=21\"\r\n" );
document.write( "\"n=22-9A\"\r\n" );
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document.write( "So now we have the numbers of coins in terms of integer A\r\n" );
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document.write( "\"system%28p=5A%2C+n=22-9A%2C+d=4A-1%29\"\r\n" );
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document.write( "10 dimes makes a dollar so there can't be as many as 10 dimes, since\r\n" );
document.write( "we have to have 21 coins, so\r\n" );
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document.write( "\"0%3C=d%3C10\"\r\n" );
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document.write( "\"0%3C=4A-1%3C10\"\r\n" );
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document.write( "Add 1 to all three sides:\r\n" );
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document.write( "\"1%3C=4A%3C11\"\r\n" );
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document.write( "Divide all three sides by 4\r\n" );
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document.write( "\"1%2F4%3C=A%3C11%2F4\"\r\n" );
document.write( "\"1%2F4%3C=A%3C2%263%2F4\"\r\n" );
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document.write( "Since A is an integer then A is either 1 or 2, for\r\n" );
document.write( "they are the only integers between \"1%2F4\" and \"2%263%2F4\"\r\n" );
document.write( "So there will be two solutions,\r\n" );
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document.write( "If A = 1, then \r\n" );
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document.write( "\"system%28p=5A=5%281%29=5%2C+n=22-9A=22-9%281%29=22-9=13%2C+d=4A-1=4%281%29-1=4-1=3%29\"\r\n" );
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document.write( "So that's one solution: 5 pennies, 13 nickels and 3 dimes.\r\n" );
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document.write( "If A = 2, then \r\n" );
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document.write( "\"system%28p=5A=5%282%29=10%2C+n=22-9A=22-9%282%29=22-18=4%2C+d=4A-1=4%282%29-1=8-1=7%29\"\r\n" );
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document.write( "So that's the other solution: 10 pennies, 4 nickels and 7 dimes.\r\n" );
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document.write( "Edwin
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