document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=157859>Question 157859</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bag of nickels, dimes, quarters and loonies contain 55 coins with a total weight of 130, worth a total of $12.50. If dimes weigh 1, nickels weigh 2, quarters weigh 3, and loonies weigh 4, how many coins of each type are there?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #116322 by </B><A HREF=/tutors/aboutme.mpl?userid=ptaylor><B>ptaylor(2198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ptaylor><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/cgi-bin/show-face.mpl?user=ptaylor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=116322\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x=number of nickels<BR>\n" );
document.write( "y=number of dimes<BR>\n" );
document.write( "z=number of quarters<BR>\n" );
document.write( "w=number of loonies<BR>\n" );
document.write( "(Note: We know before we start that we have only three equations and four unknowns and this means that we may have to do some trial and error to come up with an answer)<BR>\n" );
document.write( "We are told that:\r<BR>\n" );
document.write( "\n" );
document.write( "x+y+z+w=55----------------------------------------------------eq1<BR>\n" );
document.write( "and that: (let's deal in pennies)<BR>\n" );
document.write( "5x+10y+25z+100w=1250 divide each term by 5\r<BR>\n" );
document.write( "\n" );
document.write( "x+2y+5z+20w=250--------------------------------------------eq2\r<BR>\n" );
document.write( "\n" );
document.write( "and that:<BR>\n" );
document.write( "2x+y+3z+4w=130----------------------------------------------------eq3\r<BR>\n" );
document.write( "\n" );
document.write( "subtract eq1 from eq2 and we get:<BR>\n" );
document.write( "y+4z+19w=195--------------------------------------------------------eq1a<BR>\n" );
document.write( "multiply eq2 by 2 and then subtract eq3 from it and we get:<BR>\n" );
document.write( "3y+7z+36w=370--------------------------------------------------------eq2a<BR>\n" );
document.write( "now multiply eq1a by 3 and then subtract eq2a from it and we get:<BR>\n" );
document.write( "5z+21w=215 -------------------------------------------------------------eq3a<BR>\n" );
document.write( "21w=215-5z and<BR>\n" );
document.write( "w=(215-5z)/21-----a relationship between the number of loonies and the number of quarters.<BR>\n" );
document.write( "Now we know several things:<BR>\n" );
document.write( "(1) we cannot have fractions of coins (our answers must be whole numbers)<BR>\n" );
document.write( "(2) we cannot have negative numbers (negative coins not allowed)<BR>\n" );
document.write( "(3) we cannot have more than 12 loonies otherwise we break the bank<BR>\n" );
document.write( "From (3) above, we have the following inequality:<BR>\n" );
document.write( "(215-5z)/21<=12<BR>\n" );
document.write( "If w is:<BR>\n" );
document.write( "12-----no good.  We get negative quarters<BR>\n" );
document.write( "11-----no good. Ditto<BR>\n" );
document.write( "10----BINGO!!!:----BUT NO GOOD<BR>\n" );
document.write( "215-5Z=210; -5Z=-210; Z=42-------------NO GOOD--42 QUARTERS AND 10 LOONIES BREAKS THE BANK, SO LETS KEEP GOING<BR>\n" );
document.write( "9 :-5Z=-26----------------------NO GOOD---FRACTIONS OF QUARTERS<BR>\n" );
document.write( "8 :-5z=-47-----------------------DITTO---FRACTIONS<BR>\n" );
document.write( "7 :-5Z=-68------------------------DITTO---FRACTIONS<BR>\n" );
document.write( "6 :-5Z=-89--------------------------DITTO---FRACTIONS<BR>\n" );
document.write( "5 :-5Z=-110  BINGO!!!! Z=22  THIS MAY WORK.  BUT FIRST LETS SEE IF THERE ARE OTHER POSSIBILITIES<BR>\n" );
document.write( "4 :-5Z=-131--------------FRACTIONS FOR QUARTERS<BR>\n" );
document.write( "3 :-5Z=-152----------------DITTO---FRACTIONS<BR>\n" );
document.write( "2 :-5Z=-173-----------------DITTO FRACTIONS<BR>\n" );
document.write( "1 :-5Z=-194----------------DITTO FRACTIONS\r<BR>\n" );
document.write( "\n" );
document.write( "LOOKS LIKE WE ONLY HAVE ONE SOLUTION:<BR>\n" );
document.write( "w=5 ---LOONIES<BR>\n" );
document.write( "z=22 ---QUARTERS<BR>\n" );
document.write( "Lets plug these into eq1a and see what we get for y:<BR>\n" );
document.write( "y+4z+19w=195<BR>\n" );
document.write( "y+4*22+5*19=195<BR>\n" );
document.write( "y=195-88-95=12 so:<BR>\n" );
document.write( "y=12---DIMES<BR>\n" );
document.write( "Now lets plug these into eq1 and see what we get for x:\r<BR>\n" );
document.write( "\n" );
document.write( "x+y+z+w=55<BR>\n" );
document.write( "x+12+22+5=55<BR>\n" );
document.write( "x=55-39=16<BR>\n" );
document.write( "x=16-----------NICKELS<BR>\n" );
document.write( "CK weights<BR>\n" );
document.write( "12*1+16*2+22*3+5*4=130<BR>\n" );
document.write( "12+32+66+20=130<BR>\n" );
document.write( "130=130<BR>\n" );
document.write( "CK value<BR>\n" );
document.write( "16*5+12*10+22*25+5*100=1250<BR>\n" );
document.write( "80+120+550+500=1250<BR>\n" );
document.write( "1250=1250\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Hope this helps---ptaylor\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );