document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199919>Question 1199919</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A piggy bank contains only nickels, dimes, and quarters. There is a total of 105 coins and the total value of those coins is $10.75.\r<BR>\n" );
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document.write( "Find a general solution of the number of each coin.<BR>\n" );
document.write( "Use the general solutions from part 1 to list 3 specific solutions (for n, d, & q).\r<BR>\n" );
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document.write( "n=nickel<BR>\n" );
document.write( "d=dime<BR>\n" );
document.write( "q=a quarter </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833901 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=833901\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "n = number of nickels<BR>\n" );
document.write( "d = number of dimes<BR>\n" );
document.write( "q = number of quarters<BR>\n" );
document.write( "These are nonnegative integers {0,1,2,3,etc}\r<BR>\n" );
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document.write( "Given info: <font color=blue>There is a total of 105 coins and the total value of those coins is $10.75</font>\r<BR>\n" );
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document.write( "Break that into 2 pieces<BR>\n" );
document.write( "Given info: <font color=blue>There is a total of 105 coins</font> and <font color=blue>the total value of those coins is $10.75</font>\r<BR>\n" );
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document.write( "The first part lets us say<BR>\n" );
document.write( "n+d+q = 105\r<BR>\n" );
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document.write( "and the second fact lets us say<BR>\n" );
document.write( "0.05n+0.10d+0.25q = 10.75<BR>\n" );
document.write( "which is equivalent to<BR>\n" );
document.write( "5n+10d+25q = 1075<BR>\n" );
document.write( "after moving the decimal point two spots to the right for each item. <BR>\n" );
document.write( "We multiplied both sides by 100.\r<BR>\n" );
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document.write( "We have this system of equations<BR>\n" );
document.write( "n+d+q = 105<BR>\n" );
document.write( "5n+10d+25q = 1075\r<BR>\n" );
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document.write( "We have 2 equations, but 3 unknowns.<BR>\n" );
document.write( "Unfortunately there isn't enough information to determine the values of n,d,q.\r<BR>\n" );
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document.write( "The next section goes into greater detail as to why this is the case.\r<BR>\n" );
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document.write( "n+d+q = 105 solves to n = 105-d-q\r<BR>\n" );
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document.write( "Substitute that into the other equation<BR>\n" );
document.write( "5n+10d+25q = 1075<BR>\n" );
document.write( "5(105-d-q)+10d+25q = 1075<BR>\n" );
document.write( "525-5d-5q+10d+25q = 1075<BR>\n" );
document.write( "525+5d+20q = 1075<BR>\n" );
document.write( "5(105+d+4q) = 1075<BR>\n" );
document.write( "105+d+4q = 1075/5<BR>\n" );
document.write( "105+d+4q = 215<BR>\n" );
document.write( "d = 215-105-4q<BR>\n" );
document.write( "d = 110-4q\r<BR>\n" );
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document.write( "If q = 10, then,<BR>\n" );
document.write( "d = 110-4q<BR>\n" );
document.write( "d = 110-4*10<BR>\n" );
document.write( "d = 70<BR>\n" );
document.write( "and,<BR>\n" );
document.write( "n = 105-d-q<BR>\n" );
document.write( "n = 105-70-10<BR>\n" );
document.write( "n = 25<BR>\n" );
document.write( "Showing that (n, d, q) = (25, 70, 10) is one ordered triple solution.<BR>\n" );
document.write( "Check:<BR>\n" );
document.write( "n+d+q = 25+70+10 = 105 coins total<BR>\n" );
document.write( "5n+10d+25q = 5*25+10*70+25*10 = 1075 cents total value<BR>\n" );
document.write( "1075 cents = 1075/100 = $10.75\r<BR>\n" );
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document.write( "If q = 11, then,<BR>\n" );
document.write( "d = 110-4q<BR>\n" );
document.write( "d = 110-4*11<BR>\n" );
document.write( "d = 66<BR>\n" );
document.write( "and,<BR>\n" );
document.write( "n = 105-d-q<BR>\n" );
document.write( "n = 105-66-11<BR>\n" );
document.write( "n = 28<BR>\n" );
document.write( "Showing that (n, d, q) = (28, 66, 11) is another ordered triple solution.<BR>\n" );
document.write( "Check:<BR>\n" );
document.write( "n+d+q = 28+66+11 = 105 coins total<BR>\n" );
document.write( "5n+10d+25q = 5*28+10*66+25*11 = 1075 cents total value\r<BR>\n" );
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document.write( "Try out other values of q to see what happens.\r<BR>\n" );
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document.write( "Here are all the ordered triple solutions (n,d,q) such that we focus on positive integers only.<BR>\n" );
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document.write( "(n, d, q) = (1, 102, 2)\r\n" );
document.write( "(n, d, q) = (4, 98, 3)\r\n" );
document.write( "(n, d, q) = (7, 94, 4)\r\n" );
document.write( "(n, d, q) = (10, 90, 5)\r\n" );
document.write( "(n, d, q) = (13, 86, 6)\r\n" );
document.write( "(n, d, q) = (16, 82, 7)\r\n" );
document.write( "(n, d, q) = (19, 78, 8)\r\n" );
document.write( "(n, d, q) = (22, 74, 9)\r\n" );
document.write( "(n, d, q) = (25, 70, 10)\r\n" );
document.write( "(n, d, q) = (28, 66, 11)\r\n" );
document.write( "(n, d, q) = (31, 62, 12)\r\n" );
document.write( "(n, d, q) = (34, 58, 13)\r\n" );
document.write( "(n, d, q) = (37, 54, 14)\r\n" );
document.write( "(n, d, q) = (40, 50, 15)\r\n" );
document.write( "(n, d, q) = (43, 46, 16)\r\n" );
document.write( "(n, d, q) = (46, 42, 17)\r\n" );
document.write( "(n, d, q) = (49, 38, 18)\r\n" );
document.write( "(n, d, q) = (52, 34, 19)\r\n" );
document.write( "(n, d, q) = (55, 30, 20)\r\n" );
document.write( "(n, d, q) = (58, 26, 21)\r\n" );
document.write( "(n, d, q) = (61, 22, 22)\r\n" );
document.write( "(n, d, q) = (64, 18, 23)\r\n" );
document.write( "(n, d, q) = (67, 14, 24)\r\n" );
document.write( "(n, d, q) = (70, 10, 25)\r\n" );
document.write( "(n, d, q) = (73, 6, 26)\r\n" );
document.write( "(n, d, q) = (76, 2, 27)\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "I used computer software to quickly generate this list. <BR>\n" );
document.write( "This is to do so efficiently and to avoid error. <BR>\n" );
document.write( "It's also to ensure that the list is fully exhaustive.\r<BR>\n" );
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document.write( "Once again, there isn't enough information to nail down the single exact value of each n, d, and q value.<BR>\n" );
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