document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=175252>Question 175252</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Barbara and Ramir raced each other.  Barbara averaged 9 mph for the entire course while Ramir averaged 8 mph for the fist half of the course and 10 mph for the second half. Who won? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #130349 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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=130349\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> it appears that barbara is the winner.<BR>\n" );
document.write( "here's why:<BR>\n" );
document.write( "---<BR>\n" );
document.write( "let t = time it took barbara.<BR>\n" );
document.write( "since rate * time = distance, then <BR>\n" );
document.write( "9*t = d<BR>\n" );
document.write( "which means that<BR>\n" );
document.write( "t = d/9<BR>\n" );
document.write( "barbara took d/9 hours to cover the distance.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "let t1 = time it took ramir to cover half the distance at 8 mph.<BR>\n" );
document.write( "since rate * time = distance, then<BR>\n" );
document.write( "8*t1 = d/2<BR>\n" );
document.write( "which means that<BR>\n" );
document.write( "t1 = d/16<BR>\n" );
document.write( "---<BR>\n" );
document.write( "let t2 = time it took ramir to cover half the distance at 10 mph.<BR>\n" );
document.write( "since rate * time = distance, then<BR>\n" );
document.write( "10*t2 = d/2<BR>\n" );
document.write( "which means that<BR>\n" );
document.write( "t2 = d/20<BR>\n" );
document.write( "---<BR>\n" );
document.write( "the total time it took ramir is equal to t1 + t2.<BR>\n" );
document.write( "since t1 = d/16 and t2 = d/20, then<BR>\n" );
document.write( "t1 + t2 = d/16 + d/20<BR>\n" );
document.write( "---<BR>\n" );
document.write( "d/16 + d/20 = (20d + 16d) / (16*20)<BR>\n" );
document.write( "which equals<BR>\n" );
document.write( "36d / 320<BR>\n" );
document.write( "which equals<BR>\n" );
document.write( "d/8.888888888..... hours<BR>\n" );
document.write( "---<BR>\n" );
document.write( "t1 + t2 = d/8.8888888.... hours which is the time it took ramir.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "since d/9 is a smaller number than d/8.8888888....., this means that barbara is the winner.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "this answer should be good at any distance.<BR>\n" );
document.write( "as an example, let d = 90<BR>\n" );
document.write( "it took barbara 10 hours<BR>\n" );
document.write( "it took ramir 10.125 hours.<BR>\n" );
document.write( "this is 45/8 + 45/10 since he covered half the distance at 8 mph and half the distance at 10 mph.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "let d = 1500 miles<BR>\n" );
document.write( "it took barbara 166.7 hours (rounded)<BR>\n" );
document.write( "it took ramir 168.75 hours.<BR>\n" );
document.write( "this is 750/8 + 750/10 since he covered half the distance at 8 mph and half the distance at 10 mph.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "answer checks out ok.<BR>\n" );
document.write( "barbara is the winner at any distance. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );