document.write( "Question 1207307: Hi
\n" ); document.write( "Alan has 13 pieces of $2 $5 and $10 notes in his wallet. If the amount of money he has is $67 how many pieces of each note does he have \n" ); document.write( "

Algebra.Com's Answer #845114 by ikleyn(52777) $\"\"$ $\"About$

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\n" ); document.write( "Alan has 13 pieces of $2 $5 and $10 notes in his wallet.
\n" ); document.write( "If the amount of money he has is $67 how many pieces of each note does he have ?
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\n" ); document.write( "\n" ); document.write( "        In this my post, I'd like to present a regular solution to this problem.\r
\n" ); document.write( "\n" ); document.write( "        Not the most witty and not the most slow, but something the most REGULAR, \r
\n" ); document.write( "\n" ); document.write( "        which is in between of these extremes.\r
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document.write( "Let the numbers of pieces of $2, $5, and $10 notes be, respectively, x, y, and z. \r\n" );
document.write( "Then you have this system of two equations in three unknowns\r\n" );
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document.write( "     x  + y +  z  = 13,     (1)\r\n" );
document.write( "    2x + 5y + 10z = 67.     (2)\r\n" );
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document.write( "You should find a solution / (solutions) in integer non-negative numbers.\r\n" );
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document.write( "Multiply first equation by 2\r\n" );
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document.write( "    2x + 2y +  2z = 26,     (1')\r\n" );
document.write( "    2x + 5y + 10z = 67.     (2)\r\n" );
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document.write( "Subtract equation (1') from equation (2).  You will get\r\n" );
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document.write( "         3y +  8z = 41.     (3)\r\n" );
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document.write( "This equation is in two unknowns, but you need to solve it in integer non-negative numbers.\r\n" );
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document.write( "Under this restriction, this equation is so called linear Diophantine equation.\r\n" );
document.write( "A standard way to solve it is \"trial and error\".\r\n" );
document.write( "In other words, you should try several integer positive values of z, \r\n" );
document.write( "and find relevant values of y. Those values of z that provide non-negative integer y will be the solutions.\r\n" );
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document.write( "From equation (3), the candidates for z to try are z = 1, 2, 3, 4, and 5 - - - only five values,\r\n" );
document.write( "so it is not a catastrophic job to try all five.\r\n" );
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document.write( "To facilitate this job, people usually make a Table such as I provide below\r\n" );
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document.write( "    z        41-8z              is y = 41-8z           Integer solution\r\n" );
document.write( "                                a multiple of 3 ?      y = \r\n" );
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document.write( "    1   41-8*1 = 41- 8 = 33           Yes                  11\r\n" );
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document.write( "    2   41-8*2 = 41-16 = 25           No\r\n" );
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document.write( "    3   41-8*3 = 41-24 = 17           No\r\n" );
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document.write( "    4   41-8*4 = 41-32 =  9           Yes                   3\r\n" );
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document.write( "    5   41-8*5 = 41-40 =  1           No\r\n" );
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document.write( "From the Table, you see that there are exactly two solutions to equation (3) for y and z.\r\n" );
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document.write( "One solution is        (y,z) = (11,1).  The corresponding value of x, from (1), is x= 13 - y - z = 13 - 11 - 1 = 1.\r\n" );
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document.write( "The other solution is  (y,z) =  (3,4).  The corresponding value of x, from (1), is x= 13 - y - z = 13 - 3 - 4 = 6.\r\n" );
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document.write( "At this point, the problem is just solved, in full.\r\n" );
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document.write( "ANSWER.  There are two solutions.\r\n" );
document.write( "         One solution is       (one $2 note,  eleven $5 notes and one  $10 note).\r\n" );
document.write( "         The other solution is (six $2 notes, three  $5 notes and four $10 notes).\r\n" );
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document.write( "You can easily CHECK it on your own that these numbers satisfy to all problem's requirements.\r\n" );
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\n" ); document.write( "\n" ); document.write( "In this problem, making \"trial and errors\" was dowable procedure.\r
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\n" ); document.write( "\n" ); document.write( "In other similar problems, if the number of \"trials and errors\" is great, you may use
\n" ); document.write( "the tools like Excel to facilitate such a job and to make a calculation Table quickly.\r
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\n" ); document.write( "\n" ); document.write( "In any case, what is described in my post, is traditionally considered as a standard procedure
\n" ); document.write( "to solve similar problems.\r
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\n" ); document.write( "\n" ); document.write( "Again, as a reminder, originally you have only two equations for three unknowns.
\n" ); document.write( "But an additional restriction of having non-negative integer solutions reduces
\n" ); document.write( "the possible number of solutions from infinity to a finite number.\r
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