document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196915>Question 1196915</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose a is to b as c is to d;that is, a/b =c/d<BR>\n" );
document.write( "a) Show that b is to a as d is to c.<BR>\n" );
document.write( "b) Show that a-b is to b as c-d is to d.<BR>\n" );
document.write( "c) Show that a is to a+b as c is to c+d.<BR>\n" );
document.write( "d) Show that a+b is to c+d as b is to d. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830321 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=830321\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "a is to b as c is to d<BR>\n" );
document.write( "a/b = c/d<BR>\n" );
document.write( "ad = bc ..... multiply both sides by bd<BR>\n" );
document.write( "d/c = b/a ... divide both sides by ac<BR>\n" );
document.write( "b/a = d/c<BR>\n" );
document.write( "b is to a as d is to c\r<BR>\n" );
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document.write( "The claim (a) is proven to be true. <BR>\n" );
document.write( "We can reverse the order of 'a' vs b, as long as we reverse c and d as well.\r<BR>\n" );
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document.write( "To avoid division by zero errors, we'll have a,b,c,d as nonzero values.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "I'll start with what we want to show which is<BR>\n" );
document.write( "(a-b)/b = (c-d)/d<BR>\n" );
document.write( "and we'll work to reaching<BR>\n" );
document.write( "a/b = c/d<BR>\n" );
document.write( "In a sense, we're working backwards and we'll reverse the order later.\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "(a-b)/b = (c-d)/d<BR>\n" );
document.write( "a/b-b/b = c/d-d/d .... break up the fraction<BR>\n" );
document.write( "a/b-1 = c/d-1  ....... reduce<BR>\n" );
document.write( "a/b = c/d   .... add 1 to both sides\r<BR>\n" );
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document.write( "Now reverse the order of the steps shown<BR>\n" );
document.write( "This reversal is possible since each step can be undone, aka there's an inverse to it.<BR>\n" );
document.write( "For instance, the \"add 1 to both sides\" step can be undone with \"subtract 1 from both sides\".\r<BR>\n" );
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document.write( "So this is what the steps look like when going forward<BR>\n" );
document.write( "a/b = c/d<BR>\n" );
document.write( "a/b-1 = c/d-1<BR>\n" );
document.write( "a/b-b/b = c/d-d/d<BR>\n" );
document.write( "(a-b)/b = (c-d)/d<BR>\n" );
document.write( "This can be thought of as the main set of steps, while the previous paragraph is like supplementary steps or something you'd have on a separate sheet of scratch paper. Though if I was the teacher, I'd probably want to see the student's entire thought process (assuming you use this reversal method).\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "Start with what we want to prove<BR>\n" );
document.write( "a/(a+b) = c/(c+d)<BR>\n" );
document.write( "the goal is to get to<BR>\n" );
document.write( "a/b = c/d\r<BR>\n" );
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document.write( "a/(a+b) = c/(c+d)<BR>\n" );
document.write( "(a+b)/a = (c+d)/c<BR>\n" );
document.write( "a/a + b/a = c/c + d/c<BR>\n" );
document.write( "1 + b/a = 1 + d/c<BR>\n" );
document.write( "b/a = d/c<BR>\n" );
document.write( "a/b = c/d\r<BR>\n" );
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document.write( "Now reverse the order to get...<BR>\n" );
document.write( "a/b = c/d<BR>\n" );
document.write( "b/a = d/c<BR>\n" );
document.write( "1 + b/a = 1 + d/c<BR>\n" );
document.write( "a/a + b/a = c/c + d/c<BR>\n" );
document.write( "(a+b)/a = (c+d)/c<BR>\n" );
document.write( "a/(a+b) = c/(c+d)\r<BR>\n" );
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document.write( "-------------------------------------------------------<BR>\n" );
document.write( "Part (d)\r<BR>\n" );
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document.write( "The goal is to show that<BR>\n" );
document.write( "(a+b)/(c+d) = b/d<BR>\n" );
document.write( "leads to <BR>\n" );
document.write( "a/b = c/d<BR>\n" );
document.write( "so we can reverse it later.\r<BR>\n" );
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document.write( "(a+b)/(c+d) = b/d<BR>\n" );
document.write( "d(a+b) = b(c+d)<BR>\n" );
document.write( "ad+bd = bc+bd<BR>\n" );
document.write( "ad = bc<BR>\n" );
document.write( "a/b = c/d\r<BR>\n" );
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document.write( "Now reverse those steps<BR>\n" );
document.write( "a/b = c/d<BR>\n" );
document.write( "ad = bc<BR>\n" );
document.write( "ad+bd = bc+bd<BR>\n" );
document.write( "d(a+b) = b(c+d)<BR>\n" );
document.write( "(a+b)/(c+d) = b/d<BR>\n" );
document.write( "Let me know if you have any questions.\r<BR>\n" );
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document.write( "I recommend trying out numeric examples that the tutor @Theo has mentioned, or coming up with your own numbers to try out. <BR>\n" );
document.write( "Keep in mind that the numeric examples do not constitute a full formal proof (since you'd need to check infinitely many numeric values). <BR>\n" );
document.write( "The proof is when we did all those algebraic steps. <BR>\n" );
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