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 #845140 by MathTherapy(10552) $\"\"$ $\"About$

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document.write( "Hi\r\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\r\n" );
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document.write( "Let number of $2 and $5 notes be T, and F, respectively\r\n" );
document.write( "Since there are 13 notes, number of $10 notes = 13 - T - F\r\n" );
document.write( "We then get the total-amount equation as: 2T + 5F + 10(13 - T - F) = 67\r\n" );
document.write( "                                         2T + 5F + 130 - 10T - 10F = 67\r\n" );
document.write( "                                                         - 8T - 5F = 67 - 130  \r\n" );
document.write( "                                                           8T + 5F = 63\r\n" );
document.write( "Now, as 3 UNKNOWNS/VARIABLES are being sought, but an equation with 2 variables is derived, we use the\r\n" );
document.write( "DIOPHANTINE-EQUATIION method to solve. \r\n" );
document.write( "8T + 5F = 63 ----- eq (i)\r\n" );
document.write( "Because MULTIPLES of 5 have units digits of either 0 or 5, we solve eq (i) in terms of F, so our final\r\n" );
document.write( "equation will have a denominator of 5\r\n" );
document.write( "8T + 5F = 63 then becomes: \r\n" );
document.write( "Now, looking at the above equation, we see that, substituting values of T, from 1, up to 7 - NOT 8,\r\n" );
document.write( "because 63 - 8(8) will produce a NEGATIVE number of $5 bills (UNACCEPTABLE!!) in the numerator. Now, \r\n" );
document.write( "again looking at , we see that the numerator 63 - 8T MUST produce a number that is a \r\n" );
document.write( "MILTIPLE of 5, but 5's MULTIPLES ONLY end in 0 or 5. Therefore, the numbers from 1 - 7 that MUST be \r\n" );
document.write( "multiplied by the 8 in 8T, MUST PRODUCE a number that, when SUBTRACTED from 63, gives a MULTIPLE of\r\n" );
document.write( "5 (the denominator) with units digit of either 0 or 5. For this numerator-MULTPLE of 5 to end\r\n" );
document.write( "in 0, a MULTIPLE of 8 would need to end with a units digit of 3 (63 - ?3 = ?0). But, there is NO\r\n" );
document.write( "MULTIPLE of 8 that end in 3. AS a matter of fact, 8 is an even number and when multiplied by any \r\n" );
document.write( "other number (odd or even), another EVEN number will ensue. So, the only other multiples of 8 would\r\n" );
document.write( "be those that end in 8 (have a units digit of 8). When checked, digits from 1 - 7, produce only 2\r\n" );
document.write( "multiples of 8 that end with an 8. These are 1 (8 * 1 = 8), and 6 (8 * 6 = 48). Let's now apply these \r\n" );
document.write( "values of T to the DIOPHANTINE equation, \r\n" );
document.write( "    With T = 1, we get:                        With T = 6, we get:    \r\n" );
document.write( "Number of $5 notes, or                      Number of $5 notes, or \r\n" );
document.write( "As such, number of $10 notes = 13 - 1 - 11 = 1            As such, number of $10 notes = 13 - 6 - 3 = 4  \r\n" );
document.write( "This gives us:            This gives us:

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