document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=919228>Question 919228</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bracelet that is supposed to be 18-Karat Gold weighs 238 grams.  Gold weighs 19.3 grams per cubic centimeter.  Copper weights 9 grams per cubic centimeter.   The volume of the bracelet is 15 cubic centimeters of gold and copper. <BR>\n" );
document.write( "Is the bracelet 18-Karat Gold? - I found out that it wasn't, but i have to use the substitution method to figure it out, but I don't know how to.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Thanks in advance (: </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #557596 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=557596\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> WHAT YOUR TEACHER/TEXTBOOK THINKS:<BR>\n" );
document.write( "18K gold is a mixture of 18 parts by volume of gold and 6 parts by volume of something else (or 75 % gold by volume and 25% other stuff by volume). Volumes mixed add, and so do masses.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >= volume of gold in cubic centimeters<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" >= volume of copper in cubic centimeters<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19.3a ALIGN=MIDDLE  ALT=\"19.3a\"   DISABLED_style= \"max-width:90%; height:auto;\" >= mass of gold in grams<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9c ALIGN=MIDDLE  ALT=\"9c\"   DISABLED_style= \"max-width:90%; height:auto;\" >= mass of copper in grams<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%2Bc=15%2C19.3a%2B9c=238%29 ALIGN=MIDDLE  ALT=\"system%28a%2Bc=15%2C19.3a%2B9c=238%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is their system.<BR>\n" );
document.write( "Their solution by substitution could be<BR>\n" );
document.write( "solving <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bc=15 ALIGN=MIDDLE  ALT=\"a%2Bc=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bc=15 ALIGN=MIDDLE  ALT=\"a%2Bc=15\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=15-a ALIGN=MIDDLE  ALT=\"c=15-a\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and substituting into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19.3a%2B9c=238 ALIGN=MIDDLE  ALT=\"19.3a%2B9c=238\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19.3a%2B9%2815-a%29=238 ALIGN=MIDDLE  ALT=\"19.3a%2B9%2815-a%29=238\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19.3a%2B9%2815-a%29=238 ALIGN=MIDDLE  ALT=\"19.3a%2B9%2815-a%29=238\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19.3a%2B135-9a=238 ALIGN=MIDDLE  ALT=\"19.3a%2B135-9a=238\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19.3a-9a=238-135 ALIGN=MIDDLE  ALT=\"19.3a-9a=238-135\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10.3a=103 ALIGN=MIDDLE  ALT=\"10.3a=103\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=103%2F10.3 ALIGN=MIDDLE  ALT=\"a=103%2F10.3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=10 ALIGN=MIDDLE  ALT=\"a=10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since 10 cubic centimeters is less that 75% of 15 cubic centimeters,<BR>\n" );
document.write( "which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.75%2A15=11.25 ALIGN=MIDDLE  ALT=\"0.75%2A15=11.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > cubic centimeters,<BR>\n" );
document.write( "your teacher textbook concludes that it is nor 18k gold, but less than 18k.<BR>\n" );
document.write( "The ratio <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2F15=2%2F3=16%2F24 ALIGN=MIDDLE  ALT=\"10%2F15=2%2F3=16%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the ratio that corresponds to 16k gold.<BR>\n" );
document.write( "Unfortunately 18K gold is 75% by weight, not by volume, so the problem has uglier numbers and the conclusion is different.<BR>\n" );
document.write( " <BR>\n" );
document.write( "A CHEMIST'S POINT OF VIEW<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= grams of gold in the bracelet<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >= grams of copper in the bracelet<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=238 ALIGN=MIDDLE  ALT=\"x%2By=238\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "18K gold is supposed to be 75% gold (25% something else) by weight,<BR>\n" );
document.write( "and in an alloy, the volumes of the metals mixed add up to the volume of the mix.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F19.3 ALIGN=MIDDLE  ALT=\"x%2F19.3\"   DISABLED_style= \"max-width:90%; height:auto;\" >= cubic centimeters of gold in the bracelet<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2F9 ALIGN=MIDDLE  ALT=\"y%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >= cubic centimeters of copper in the bracelet<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F19.3%2By%2F9=15 ALIGN=MIDDLE  ALT=\"x%2F19.3%2By%2F9=15\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%2B19.3y=2605.5 ALIGN=MIDDLE  ALT=\"9x%2B19.3y=2605.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "My system of equations (with the way I defined x and y) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%2By=238%2Cx%2F19.3%2By%2F9=15%29 ALIGN=MIDDLE  ALT=\"system%28x%2By=238%2Cx%2F19.3%2By%2F9=15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To solve by substitution, I solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=238 ALIGN=MIDDLE  ALT=\"x%2By=238\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=238 ALIGN=MIDDLE  ALT=\"x%2By=238\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=238-y ALIGN=MIDDLE  ALT=\"x=238-y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and then substitute that expression for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%2B19.3y=2605.5 ALIGN=MIDDLE  ALT=\"9x%2B19.3y=2605.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%2B19.3y=2605.5 ALIGN=MIDDLE  ALT=\"9x%2B19.3y=2605.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%28238-y%29%2B19.3y=2605.5 ALIGN=MIDDLE  ALT=\"9%28238-y%29%2B19.3y=2605.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2142-9y%2B19.3y=2605.5 ALIGN=MIDDLE  ALT=\"2142-9y%2B19.3y=2605.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10.3y=2605.5-2142 ALIGN=MIDDLE  ALT=\"10.3y=2605.5-2142\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10.3y=463.5 ALIGN=MIDDLE  ALT=\"10.3y=463.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=463.5%2F10.3 ALIGN=MIDDLE  ALT=\"y=463.5%2F10.3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=45 ALIGN=MIDDLE  ALT=\"y=45\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded)<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=238-45=193 ALIGN=MIDDLE  ALT=\"x=238-45=193\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The percent copper is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=43%2F238%2A100=18.9 ALIGN=MIDDLE  ALT=\"43%2F238%2A100=18.9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "CONCLUSION:<BR>\n" );
document.write( "Since 18K gold is supposed to be 75% gold (25% something else) by weight, the bracelet has more gold than 18K gold. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );