document.write( "Question 816182: I want to know how many ways I can make $2 using dollars, half dollars, and quarters. I got this answer: \r
\n" ); document.write( "\n" ); document.write( "1. 1 + 1\r
\n" ); document.write( "\n" ); document.write( "2. 8 q
\n" ); document.write( "3. 2 half + 2 half\r
\n" ); document.write( "\n" ); document.write( "4. 1 + 2 half
\n" ); document.write( "5. 1 + 4 quarters
\n" ); document.write( "6. 1 + 1 half + 2 q
\n" ); document.write( "7. 1 dollar + 4 q\r
\n" ); document.write( "\n" ); document.write( "8. 2 half + 4 q
\n" ); document.write( "9. 3 half + 2 q\r
\n" ); document.write( "\n" ); document.write( "I drove myself crazy trying to figure this out, and I want to know if there's a better way to solve this with a chart. What I did was think about, and draw, coins on paper then look relied on just that to figure out how many combinations there were. I think I'm wrong, especially when I used the same method to figure out how many ways I can make $3 using quarters, half dollars, and dollars, and only got 10 different ways (I think). Any help would be sooooooooooooooooo appreciated. Thank you! Jay \n" ); document.write( "

Algebra.Com's Answer #491402 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
Here's an approach which may help.

\n" ); document.write( "The quarters can only come in pairs. Using just dollars, half-dollars and quarters, there is no way to get $2 exactly with an odd number of quarters. (Think about it.)

\n" ); document.write( "Let's look at a rectangular array of q's, where each q represents a quarter:
\n" ); document.write( "

\r\n" );
document.write( "  q   q   q   q\r\n" );
document.write( "  q   q   q   q\r\n" );
document.write( "8q\r\n" );
document.write( "

These 8 quarters add up to $2. Each column of two quarters is equal to a half-dollar. Each row of 4 quarters is equal to a dollar. To find all the combinations: First count the initial array of 8 quarters. Then start replacing columns with h's (for half-dollars).
\n" ); document.write( "

\r\n" );
document.write( "  q   q   q   h      \r\n" );
document.write( "  q   q   q   \r\n" );
document.write( "6q + 1h\r\n" );
document.write( "\r\n" );
document.write( "  q   q   h   h      \r\n" );
document.write( "  q   q      \r\n" );
document.write( "4q + 2h\r\n" );
document.write( "\r\n" );
document.write( "  q   h   h   h      \r\n" );
document.write( "  q         \r\n" );
document.write( "2q + 3h\r\n" );
document.write( "\r\n" );
document.write( "  h   h   h   h               \r\n" );
document.write( "4h\r\n" );
document.write( "

Now repeat the process except if there are a pair of h's replace them with a d:
\n" ); document.write( "

\r\n" );
document.write( "  q   q   q   q\r\n" );
document.write( "  q   q   q   q\r\n" );
document.write( "No pairs of h's\r\n" );
document.write( "\r\n" );
document.write( "  q   q   q   h      \r\n" );
document.write( "  q   q   q   \r\n" );
document.write( "No pairs of h's\r\n" );
document.write( "\r\n" );
document.write( "  q   q   h   h      \r\n" );
document.write( "  q   q      \r\n" );
document.write( "Replace the pair of h's:\r\n" );
document.write( "  q   q   d\r\n" );
document.write( "  q   q    \r\n" );
document.write( "4q + 1d  \r\n" );
document.write( "\r\n" );
document.write( "  q   h   h   h      \r\n" );
document.write( "  q         \r\n" );
document.write( "Replace a pair of h's:\r\n" );
document.write( "  q   h   d      \r\n" );
document.write( "  q        \r\n" );
document.write( "2q + 1h + 1d \r\n" );
document.write( "\r\n" );
document.write( "  h   h   h   h               \r\n" );
document.write( "Replace one pair of h's:\r\n" );
document.write( "  h   h   d\r\n" );
document.write( "2h + 1d\r\n" );
document.write( "Replace the remaining pair of h's:\r\n" );
document.write( "  d   d\r\n" );
document.write( "2d\r\n" );
document.write( "

For other amounts of money, just add more columns of q's. \n" ); document.write( "

\n" );