document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=208973>Question 208973</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the ratio a:c if a:b= 1:2, b:d = 4:5, d:c = 3:1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #157994 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=157994\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the ratio a:c if a:b= 1:2, b:d = 4:5, d:c = 3:1<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "x : y is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2F+y ALIGN=MIDDLE  ALT=\"x+%2F+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the same as x divided by y<BR>\n" );
document.write( "see reference: http://www.icoachmath.com/sitemap/Ratio.html<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "a:b means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fb+=+1%2F2 ALIGN=MIDDLE  ALT=\"a%2Fb+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b%2F2 ALIGN=MIDDLE  ALT=\"a+=+b%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+2%2Aa ALIGN=MIDDLE  ALT=\"b+=+2%2Aa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "b:d means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Fd+=+4%2F5 ALIGN=MIDDLE  ALT=\"b%2Fd+=+4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+4d%2F5 ALIGN=MIDDLE  ALT=\"b+=+4d%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+5b%2F4 ALIGN=MIDDLE  ALT=\"d+=+5b%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "d:c means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2Fc+=+3%2F1 ALIGN=MIDDLE  ALT=\"d%2Fc+=+3%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+3c ALIGN=MIDDLE  ALT=\"d+=+3c\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+d%2F3 ALIGN=MIDDLE  ALT=\"c+=+d%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "now that you have these translations out of the way you can solve.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "a:c means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fc ALIGN=MIDDLE  ALT=\"a%2Fc\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "-----<BR>\n" );
document.write( "to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fc ALIGN=MIDDLE  ALT=\"a%2Fc\"   DISABLED_style= \"max-width:90%; height:auto;\" > you need to find relationships in terms of a and c and then solve.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "in order to get a relationship from a to c, you need to go through b and d because there is no direct relationship between a and c.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "start with the relationship between a and b.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+b%2F2 ALIGN=MIDDLE  ALT=\"a+=+b%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "get the relationship from b to d<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+4d%2F5 ALIGN=MIDDLE  ALT=\"b+=+4d%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "replace b with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4d%2F5 ALIGN=MIDDLE  ALT=\"4d%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+=+%284d%2F5%29%2F2 ALIGN=MIDDLE  ALT=\"+a+=+%284d%2F5%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+4d%2F10 ALIGN=MIDDLE  ALT=\"a+=+4d%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "get the relationship from d to c<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+3c ALIGN=MIDDLE  ALT=\"d+=+3c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "replace d with 3c to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+%284%2A3c%29%2F10 ALIGN=MIDDLE  ALT=\"a+=+%284%2A3c%29%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+12c%2F10 ALIGN=MIDDLE  ALT=\"a+=+12c%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+6c%2F5 ALIGN=MIDDLE  ALT=\"a+=+6c%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+6c%2F5 ALIGN=MIDDLE  ALT=\"a+=+6c%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the same as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fc+=+6%2F5 ALIGN=MIDDLE  ALT=\"a%2Fc+=+6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the same as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3Ac+=+6%3A5 ALIGN=MIDDLE  ALT=\"a%3Ac+=+6%3A5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "to confirm, let's try to find c directly and then try to find c indirectly.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "let a = 6<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+6c+%2F+5 ALIGN=MIDDLE  ALT=\"a+=+6c+%2F+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then c must equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5a%2F6 ALIGN=MIDDLE  ALT=\"5a%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2A6%29%2F6 ALIGN=MIDDLE  ALT=\"%285%2A6%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we know that a = 6 and c = 5 solving directly.<BR>\n" );
document.write( "------<BR>\n" );
document.write( "let's find c indirectly.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we start off with a = 6<BR>\n" );
document.write( "we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Aa ALIGN=MIDDLE  ALT=\"2%2Aa\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A6 ALIGN=MIDDLE  ALT=\"2%2A6\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2Ab%29%2F4 ALIGN=MIDDLE  ALT=\"%285%2Ab%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2A12%29%2F4 ALIGN=MIDDLE  ALT=\"%285%2A12%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15 ALIGN=MIDDLE  ALT=\"15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2F3 ALIGN=MIDDLE  ALT=\"d%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2F3 ALIGN=MIDDLE  ALT=\"15%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+6 ALIGN=MIDDLE  ALT=\"a+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+5 ALIGN=MIDDLE  ALT=\"c+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > solving indirectly.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the formulas check out and the relationship you are looking for is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3Ac+=+6%3A5 ALIGN=MIDDLE  ALT=\"a%3Ac+=+6%3A5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "is there an easier way to do this?<BR>\n" );
document.write( "yes there is<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the shortcut method is as follows:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3Ac ALIGN=MIDDLE  ALT=\"a%3Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%3Ab%29+%2A+%28b%3Ad%29+%2A+%28d%3Ac%29 ALIGN=MIDDLE  ALT=\"%28a%3Ab%29+%2A+%28b%3Ad%29+%2A+%28d%3Ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you know those ratios are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3A2 ALIGN=MIDDLE  ALT=\"1%3A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%3A5 ALIGN=MIDDLE  ALT=\"4%3A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%3A1 ALIGN=MIDDLE  ALT=\"3%3A1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you know this is the same as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F5 ALIGN=MIDDLE  ALT=\"4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F1 ALIGN=MIDDLE  ALT=\"3%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so you have:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fc ALIGN=MIDDLE  ALT=\"a%2Fc\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Fb%29+%2A+%28b%2Fd%29+%2A+%28d%2Fc%29 ALIGN=MIDDLE  ALT=\"%28a%2Fb%29+%2A+%28b%2Fd%29+%2A+%28d%2Fc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29+%2A+%284%2F5%29+%2A+%283%2F1%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29+%2A+%284%2F5%29+%2A+%283%2F1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2A4%2A3%29%2F%282%2A5%2A1%29 ALIGN=MIDDLE  ALT=\"%281%2A4%2A3%29%2F%282%2A5%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F10 ALIGN=MIDDLE  ALT=\"12%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F5 ALIGN=MIDDLE  ALT=\"6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "your ratio for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fc+=+6%2F5 ALIGN=MIDDLE  ALT=\"a%2Fc+=+6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );