document.write( "Question 561946: How MANY WAYS can you use 21 coins to make $1? i have tried 2 WAYS so fAR....
\n" ); document.write( " ONE WAY......2 QUARTERS, 3 DIMES,1 NICKEL,15 PENNIES\r
\n" ); document.write( "\n" ); document.write( " ANOTHER WAY.......7 DIMES, 4 NICKELS,10 PENNIES \n" ); document.write( "

Algebra.Com's Answer #364432 by KMST(5328) $\"\"$ $\"About$

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With a systematic analysis I find 4 solutions and can be pretty sure that I have all the solutions.
\n" ); document.write( "Let's define variables
\n" ); document.write( " $\"q\"$ = number of quarters
\n" ); document.write( " $\"d\"$ = number of dimes
\n" ); document.write( " $\"n\"$ = number of nickels
\n" ); document.write( " $\"p\"$ = number of pennies
\n" ); document.write( "There are 21 coins, so $\"q%2Bd%2Bn%2Bp=21\"$
\n" ); document.write( "The value adds to $1.00, so $\"25q%2B10d%2B5n%2Bp=100\"$ --> $\"5q%2B2d%2Bn%2Bp%2F5=20\"$
\n" ); document.write( "So $\"p\"$ must be a multiple of 5, meaning that it must be 0, 5, 10, 15, or 20.
\n" ); document.write( "It cannot be 0, because the 21 coins would add up to at least $1.05 (21 nickels).
\n" ); document.write( "It cannot be 20 pennies, because the remaining coin would have to be worth $0.80.
\n" ); document.write( "FIVE PENNIES
\n" ); document.write( "Wirh 5 pennies the remaining 16 coins add up to 95 cents, so we have
\n" ); document.write( " $\"25q%2B10d%2B5n=95\"$ --> $\"5q%2B2d%2Bn=19\"$ and
\n" ); document.write( " $\"q%2Bd%2Bn=16\"$
\n" ); document.write( "Combining both equations we get $\"4q%2Bd=3\"$ .
\n" ); document.write( "The only integer solution is $\"q=0\"$ , $\"d=3\"$
\n" ); document.write( "Substituting in $\"q%2Bd%2Bn=16\"$
\n" ); document.write( "we get $\"3%2Bn=16\"$ --> $\"n=13\"$
\n" ); document.write( "First solution: no quarters, 3 dimes, 13 nickels, 5 pennies
\n" ); document.write( "TEN PENNIES
\n" ); document.write( "With 10 pennies the remaining 11 coins add up to 90 cents, so we have
\n" ); document.write( " $\"25q%2B10d%2B5n=90\"$ --> $\"5q%2B2d%2Bn=18\"$ and
\n" ); document.write( " $\"q%2Bd%2Bn=11\"$
\n" ); document.write( "Combining both equations we get $\"4q%2Bd=7\"$ .
\n" ); document.write( "The integer solutions are $\"q=1\"$ , with $\"d=3\"$
\n" ); document.write( "and $\"q=0\"$ , with $\"d=7\"$
\n" ); document.write( "For $\"q=1\"$ , with $\"d=3\"$ ,
\n" ); document.write( " $\"q%2Bd%2Bn=11\"$ means $\"4%2Bn=11\"$ --> $\"n=7\"$
\n" ); document.write( "For $\"q=0\"$ , with $\"d=7\"$ ,
\n" ); document.write( " $\"q%2Bd%2Bn=11\"$ means $\"7%2Bn=11\"$ --> $\"n=4\"$
\n" ); document.write( "Two more solutions:
\n" ); document.write( "1 quarter, 3 dimes, 7 nickels, 10 pennies
\n" ); document.write( "no quarters, 7 dimes, 4 nickels, 10 pennies
\n" ); document.write( "FIFTEEN PENNIES
\n" ); document.write( "With 15 pennies the remaining 6 coins add up to 85 cents, so we have
\n" ); document.write( " $\"25q%2B10d%2B5n=85\"$ --> $\"5q%2B2d%2Bn=17\"$ and
\n" ); document.write( " $\"q%2Bd%2Bn=6\"$
\n" ); document.write( "Combining both equations we get $\"4q%2Bd=11\"$ .
\n" ); document.write( "The integer solutions are $\"q=2\"$ , with $\"d=3\"$ ,
\n" ); document.write( " $\"q=1\"$ , with $\"d=7\"$ ,
\n" ); document.write( "and $\"q=0\"$ , with $\"d=11\"$
\n" ); document.write( "However, only the first solution works with $\"q%2Bd%2Bn=6\"$
\n" ); document.write( "and gives us $\"5%2Bn=6\"$ --> $\"n=1\"$
\n" ); document.write( "One more solutions:
\n" ); document.write( "2 quarters, 3 dimes, 1 nickel, 15 pennies \n" ); document.write( "

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