document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201833>Question 1201833</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Introduce variables and write two equations<BR>\n" );
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document.write( "A collection of dimes and quarters is worth $15.25. There are 103 coins in total. How many of each kind of coin is there? </SPAN>\n" );
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document.write( "Answer: <BR>\n" );
document.write( "<font color=red>70 dimes<BR>\n" );
document.write( "33 quarters</font>\r<BR>\n" );
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document.write( "70 dimes + 33 quarters = 70*10 + 33*25 = 1525 cents = 15.25 dollars\r<BR>\n" );
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document.write( "Work Shown:\r<BR>\n" );
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document.write( "d = number of dimes<BR>\n" );
document.write( "q = number of quarters<BR>\n" );
document.write( "These variables are placeholders for nonnegative whole numbers 0,1,2,3,...\r<BR>\n" );
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document.write( "d+q = 103 coins total<BR>\n" );
document.write( "q = 103-d<BR>\n" );
document.write( "It will be used in a <font color=blue>substitution</font> step later\r<BR>\n" );
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document.write( "10d = value of the dimes only (cents)<BR>\n" );
document.write( "25q = value of the quarters only (cents)<BR>\n" );
document.write( "10d+25q = total value (cents)<BR>\n" );
document.write( "10d+25q = 1525<BR>\n" );
document.write( "10d+25( q ) = 1525<BR>\n" );
document.write( "10d+25( <font color=blue size=4>q</font> ) = 1525<BR>\n" );
document.write( "10d+25( <font color=blue size=4>103-d</font> ) = 1525   ... <font color=blue>substitution</font><BR>\n" );
document.write( "10d+25*103+25*(-d) = 1525<BR>\n" );
document.write( "10d+2575-25d = 1525<BR>\n" );
document.write( "-15d+2575 = 1525<BR>\n" );
document.write( "-15d = 1525-2575<BR>\n" );
document.write( "-15d = -1050<BR>\n" );
document.write( "d = -1050/(-15)<BR>\n" );
document.write( "d = <font color=red>70 dimes</font>\r<BR>\n" );
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document.write( "q = 103-d<BR>\n" );
document.write( "q = 103-70<BR>\n" );
document.write( "q = <font color=red>33 quarters</font>\r<BR>\n" );
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document.write( "Another approach:\r<BR>\n" );
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document.write( "The 25 cents in $15.25 tells us that there's at least one quarter. <BR>\n" );
document.write( "It is impossible to reach this subtotal amount with dimes alone.\r<BR>\n" );
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document.write( "If we had 1 quarter and 103-1 = 102 dimes, then we have: <BR>\n" );
document.write( "1 quarter + 102 dimes = 1*25 + 102*10 = 1045 cents = $10.45<BR>\n" );
document.write( "We come up short of the goal $15.25\r<BR>\n" );
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document.write( "Let's replace 1 dime with 1 quarter.<BR>\n" );
document.write( "Difference in coin value = 25-10 = 15 cents<BR>\n" );
document.write( "Each time we do this replacement of \"1 dime for 1 quarter\", the total value goes up by 15 cents aka $0.15<BR>\n" );
document.write( "This is the net change.\r<BR>\n" );
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document.write( "Because we want the $10.45 to go to $15.25, we must increase by $4.80<BR>\n" );
document.write( "(since 15.25-10.45 = 4.80 )\r<BR>\n" );
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document.write( "n = number of increases of 15 cents each<BR>\n" );
document.write( "0.15n = total increase needed<BR>\n" );
document.write( "0.15n = 4.80<BR>\n" );
document.write( "n = 4.80/0.15<BR>\n" );
document.write( "n = 32<BR>\n" );
document.write( "We need 32 increases of 15 cents each to go from $10.45 to $15.25<BR>\n" );
document.write( "Meaning we need to do the replacement of \"1 dime for 1 quarter\" a total of 32 times.\r<BR>\n" );
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document.write( "Add 32 to the quarter count, and subtract 32 from the dime count.<BR>\n" );
document.write( "1+32 = <font color=red>33 quarters</font><BR>\n" );
document.write( "102-32 = <font color=red>70 dimes</font><BR>\n" );
document.write( "which are the final answers.\r<BR>\n" );
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document.write( "Check:<BR>\n" );
document.write( "70 dimes + 33 quarters<BR>\n" );
document.write( "= 70*10 + 33*25<BR>\n" );
document.write( "= 700 cents + 825 cents<BR>\n" );
document.write( "= 1525 cents<BR>\n" );
document.write( "= 1525/100<BR>\n" );
document.write( "= 15.25 dollars<BR>\n" );
document.write( "The answers are confirmed.<BR>\n" );
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