document.write( "Question 624085: Sarah has a collection of dimes, nickels, and quarters worth $15.75. She has 10 more dimes than nickels and twice as many quarters as dimes. How many coins of each type does she have? \n" ); document.write( "
Algebra.Com's Answer #392554 by math-vortex(648) $\"\"$ $\"About$
You can put this solution on YOUR website!
\r\n" );
document.write( "Hi, there--\r\n" );
document.write( "\r\n" );
document.write( "The Problem:\r\n" );
document.write( "Sarah has a collection of dimes, nickels, and quarters worth $15.75.  She has 10 more dimes than nickels and twice as many quarters as dimes.  How many coins of each type does she have?\r\n" );
document.write( "\r\n" );
document.write( "A Solution:\r\n" );
document.write( "Let d be the number of dimes.\r\n" );
document.write( "Let n be the number of nickels.\r\n" );
document.write( "Let q be the number of quarters.\r\n" );
document.write( "\r\n" );
document.write( "Now we need three equations using the information from the problem.\r\n" );
document.write( "Sarah has ten more dimes that nickels.\r\n" );
document.write( "d = n + 10\r\n" );
document.write( "\r\n" );
document.write( "She has twice as many quarters as dimes.\r\n" );
document.write( "q = 2d\r\n" );
document.write( "\r\n" );
document.write( "The collection of coins is worth $15.75.\r\n" );
document.write( "The value of the dimes is .10 times the number of dimes, or 0.10d.\r\n" );
document.write( "The value of the nickels is .05 times the number of nickels, or 0.05n.\r\n" );
document.write( "The value of the quarters is 0.25q.\r\n" );
document.write( "The values of the three types of coins equals $15.75, so\r\n" );
document.write( "0.10d + 0.05n + 0.25q = 15.75\r\n" );
document.write( "\r\n" );
document.write( "Now we have three equations in three variables. We can solve the system for d, n, and q.\r\n" );
document.write( "d = n + 10\r\n" );
document.write( "q = 2d\r\n" );
document.write( "0.10d + 0.05n + 0.25q = 15.75\r\n" );
document.write( "\r\n" );
document.write( "Rewrite the first equation in a \"n=...\" form. (Subtract 10 from both sides.)\r\n" );
document.write( "d = n + 10\r\n" );
document.write( "n = d - 10\r\n" );
document.write( "\r\n" );
document.write( "Substitute d - 10 for n in the third equation\r\n" );
document.write( "0.10d + 0.05n + 0.25q = 15.75\r\n" );
document.write( "0.10d + 0.05(d-10) + 0.25q = 15.75\r\n" );
document.write( "\r\n" );
document.write( "We see in the second equation that q and 2d are equivalent. Substitute 2 * d for q in the \r\n" );
document.write( "third equation.\r\n" );
document.write( "0.10d + 0.05(d-10) + 0.25(2d) = 15.75\r\n" );
document.write( "\r\n" );
document.write( "Solve this equation for d.\r\n" );
document.write( "0.10d + 0.05d - 0.50 + 0.50d = 15.75\r\n" );
document.write( "0.65d - 0.50 = 15.75\r\n" );
document.write( "0.65d = 15.75 + 0.50\r\n" );
document.write( "0.65d = 16.25\r\n" );
document.write( "d = 25\r\n" );
document.write( "\r\n" );
document.write( "There are 25 dimes. \r\n" );
document.write( "There are twice as many quarters as dimes, so there must be 50 quarters. \r\n" );
document.write( "There are ten more dimes than nickels, so there must be 15 nickels.\r\n" );
document.write( "\r\n" );
document.write( "We need to make sure tis combination of coins equals $15.75.\r\n" );
document.write( "25 dimes are worth 25*0.10= $2.50.\r\n" );
document.write( "50 quarters are worth 50*0.25=$12.50.\r\n" );
document.write( "15 nickels are worth 15*0.05 = $0.75.\r\n" );
document.write( "\r\n" );
document.write( "$2.50 + 12.50 + 0.75 = $15.75\r\n" );
document.write( "\r\n" );
document.write( "That's it! Feel free to email me if you have questions about the solution.\r\n" );
document.write( "\r\n" );
document.write( "Ms.Figgy\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
document.write( "
\n" ); document.write( "
\n" );