document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=926219>Question 926219</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Gauss and Euler run a 12-mile race. Euler starts 20 minutes after Gauss and finishes 40 minutes before Gauss. If Gauss's rate is 1 mph more than 50% of Euler's rate, how many hours does it take Euler to finish the race? List all possible answers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #562098 by </B><A HREF=/tutors/aboutme.mpl?userid=ptaylor><B>ptaylor(2198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ptaylor><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=ptaylor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=562098\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Distance(d) equals Rate(r) times Time(t) or d=rt;  r=d/t and t=d/r<BR>\n" );
document.write( "Let r=Euler's rate<BR>\n" );
document.write( "50% of Euler's rate is 0.5r and 1 mph more is (0.5r+1)<BR>\n" );
document.write( "(I prefer to deal in fractions rather than decimals.  It eliminates round off errors and generally is not as messy).  Sooo<BR>\n" );
document.write( "(0.5r+1)=((r/2)+1)=(r+2)/2)<BR>\n" );
document.write( "So, Gauss's rate=(r+2)/2  mph<BR>\n" );
document.write( "(20 min=20/60=1/3 hr)<BR>\n" );
document.write( "When Euler starts, Gauss has already ran a distance of (1/3)(r+2)/2)=(r+2)/6  miles leaving (12-(r+2)/6) =(72-r-2)/6=(70-r)/6 miles to go<BR>\n" );
document.write( "Time it takes Euler to run 12 miles=12/r hours<BR>\n" );
document.write( "Time it takes Gauss to finish the 12 miles after Euler starts:<BR>\n" );
document.write( "((70-r)/6)/((r+2)/2)=2(70-r)/6(r+2)=(70-r)/3(r+2)<BR>\n" );
document.write( "Now  if we add 40 min=(2/3) hr to Euler’s time, we get Gauss’s time, sooo:<BR>\n" );
document.write( "(12/r)+2/3=(70-r)/3(r+2)  multiply each term by 3r(r+2)<BR>\n" );
document.write( "36(r+2)+2r(r+2)=r(70-r)<BR>\n" );
document.write( "36r+72+2r^2+4r=70r-r^2<BR>\n" );
document.write( "2r^2+r^2+36r+4r-70r+72=0<BR>\n" );
document.write( "3r^2+-30r+72=0<BR>\n" );
document.write( "r^2-10r+24=0   ---quadratic  in standard form and it can be factored<BR>\n" );
document.write( "(r-6)(r-4)=0<BR>\n" );
document.write( "So Euler’s rate is either <BR>\n" );
document.write( "r=6 mph or<BR>\n" );
document.write( "r=4 mph<BR>\n" );
document.write( "If Eulers rate is 6 mph, then Gauss’s rate is (r+2)/2=8/2= 4mph<BR>\n" );
document.write( "If Euler’s rate is 4 mph, then Gauss’s rate is ( 4+2)/2=3 mph\r<BR>\n" );
document.write( "\n" );
document.write( "If r=6, then time it takes Gauss to finish the race is <BR>\n" );
document.write( "(70-r)/(3(r+2)=64/24=8/3  hr= 2 and 2/3 hr=160 min; Euler would have taken 12/6=2 hr=120 minutes---40 min faster<BR>\n" );
document.write( "If r=4, then time it takes Gauss to finish the race is 66/18 hr=<BR>\n" );
document.write( "3 and 2/3 hr=220 min; Euler would have taken 12/4=3 hr=180 min----40 min faster\r<BR>\n" );
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document.write( "Hope this helps---ptaylor<BR>\n" );
document.write( " </SPAN>\n" );
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