document.write( "Question 821805: harry has $2.25 in nickles, dimes, and quarters. If he had twice as many nickels, half as many dimes and the same number of quarters, he would have $2.50. If he has 27 coins altogether, then how many of each does he have? \n" ); document.write( "

Algebra.Com's Answer #494445 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
Let n = the number of nickels (at the start)
\n" ); document.write( "Let d = the number of dimes (at the start)
\n" ); document.write( "Let q = the number of quarters (at the start)

\n" ); document.write( "Since each nickel has a value of 5 cents, n nickels will have a value of 5n.
\n" ); document.write( "Since each dime has a value of 10 cents, d dimes will have a value of 10d.
\n" ); document.write( "Since each quarter has a value of 25 cents, q quarters will have a value of 25q.

\n" ); document.write( "From \"harry has $2.25 in nickles, dimes, and quarters.\" we can write the equation:
\n" ); document.write( "5n + 10d + 25q = 225 (Note that we are using cents as our unit of money.)
\n" ); document.write( "From \"If he had twice as many nickels, half as many dimes and the same number of quarters, he would have $2.50.\" we can write the equation:
\n" ); document.write( "5(2n) + 10(1/2)d + 25q = 250
\n" ); document.write( "which simplifies to:
\n" ); document.write( "10n + 5d + 25q = 250
\n" ); document.write( "From \"he has 27 coins altogether\" we can write the equation:
\n" ); document.write( "n + d + q = 27

\n" ); document.write( "So we have a system of three equations in three variables:
\n" ); document.write( "5n + 10d + 25q = 225
\n" ); document.write( "10n + 5d + 25q = 250
\n" ); document.write( "n + d + q = 27

\n" ); document.write( "I don't know what methods you may have learned which could be used to solve such a system. I am going to use a matrix and Gaussian elimination. (Note: algebra.com doesn't do matrices very well. So I am only going to show the rectangular array of numbers (without the usual brackets to indicate a matrix).)
\n" ); document.write( "

\r\n" );
document.write( "     5    10    25   225\r\n" );
document.write( "    10     5    25   250\r\n" );
document.write( "     1     1     1    27\r\n" );
document.write( "Replaced row 1 with 1/5 times row 1.\r\n" );
document.write( "     1     2     5    45\r\n" );
document.write( "    10     5    25   250\r\n" );
document.write( "     1     1     1    27\r\n" );
document.write( "Added -10 times row 1 to row 2.\r\n" );
document.write( "      1      2      5     45\r\n" );
document.write( "      0    -15    -25   -200\r\n" );
document.write( "      1      1      1     27\r\n" );
document.write( "Added -1 times row 1 to row 3.\r\n" );
document.write( "      1      2      5     45\r\n" );
document.write( "      0    -15    -25   -200\r\n" );
document.write( "      0     -1     -4    -18\r\n" );
document.write( "Replaced row 2 with -1/15 times row 2.\r\n" );
document.write( "      1      2      5     45\r\n" );
document.write( "      0      1    5/3   40/3\r\n" );
document.write( "      0     -1     -4    -18\r\n" );
document.write( "Added -2 times row 2 to row 1.\r\n" );
document.write( "      1      0    5/3   55/3\r\n" );
document.write( "      0      1    5/3   40/3\r\n" );
document.write( "      0     -1     -4    -18\r\n" );
document.write( "Added 1 times row 2 to row 3.\r\n" );
document.write( "       1       0     5/3    55/3\r\n" );
document.write( "       0       1     5/3    40/3\r\n" );
document.write( "       0       0    -7/3   -14/3\r\n" );
document.write( "Replaced row 3 with -3/7 times row 3.\r\n" );
document.write( "      1      0    5/3   55/3\r\n" );
document.write( "      0      1    5/3   40/3\r\n" );
document.write( "      0      0      1      2\r\n" );
document.write( "Added -5/3 times row 3 to row 1.\r\n" );
document.write( "      1      0      0     15\r\n" );
document.write( "      0      1    5/3   40/3\r\n" );
document.write( "      0      0      1      2\r\n" );
document.write( "Added -5/3 times row 3 to row 2.\r\n" );
document.write( "    1    0    0   15\r\n" );
document.write( "    0    1    0   10\r\n" );
document.write( "    0    0    1    2\r\n" );
document.write( "

which translates into:
\n" ); document.write( "n = 15
\n" ); document.write( "d = 10
\n" ); document.write( "q = 2 \n" ); document.write( "

\n" );