document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1119320>Question 1119320</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A boy has 180 coins altogether . There are nickels dimes and quarters. Number of dimes is half the number of nickels and quarters. Value of coins is 16. How many of each kind of coin does he have<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #734808 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=734808\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The statement \"number of dimes is half the number of nickels and quarters\" is ambiguous.  It could mean the numbers of nickels and quarters are each 2x and the number of dimes is x; or it could mean the number of nickels and quarters together is 2x and the number of dimes is x.<br><BR>\n" );
document.write( "A little analysis shows that the first interpretation leads to an inconsistent set of constraints; so the second interpretation must be correct.<br><BR>\n" );
document.write( "In that case, the total number of coins is 3x; since the number of dimes is x, we know 1/3 of the coins are dimes.  1/3 of 180 is 60; we know the number of dimes is 60.<br><BR>\n" );
document.write( "After accounting for 60 dimes worth a total of $6, we have 120 coins remaining, all nickels or quarters, with a total value of $10.<br><BR>\n" );
document.write( "A traditional algebraic approach from here would be to let n be the number of nickels; then an equation that ways the total value of the nickels and quarters is $10 would be<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%28n%29%2B25%28120-n%29+=+1000 ALIGN=MIDDLE  ALT=\"5%28n%29%2B25%28120-n%29+=+1000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That equation is solved relatively easily.<br><BR>\n" );
document.write( "However, I like to solve the problem from here in a less formal manner, using some mental arithmetic, like this....<br><BR>\n" );
document.write( "(1) If all 120 remaining coins were nickels, the value would be $6; that is $4 short of the actual remaining value of $10.<br><BR>\n" );
document.write( "(2) Exchanging a nickel for a quarter keeps the total number of coins the same but increases the total value by 20 cents.<br><BR>\n" );
document.write( "(3) To make up the $4 \"shortfall\" in the value of the nickels and quarters, the number of times I need to replace a nickel with a quarter is 400/20 = 20.<br><BR>\n" );
document.write( "(4) Therefore the number of quarters is 20 and the number of nickels is 120-20 = 100.<br><BR>\n" );
document.write( "Typing all those words of explanation makes that seem like a very long process.  But if you are good with mental arithmetic, it takes far less time than writing and solving a formal algebraic equation. </SPAN>\n" );
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