document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151006>Question 1151006</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The swimming team has competed in 45 races this season. They have won 30 races so far. How many races will the team need to win today for the team to have a 75% success rate?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #772601 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=772601\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "A = 30 = number of races won in the past<BR>\n" );
document.write( "B = 45 = number of races attempted in the past<BR>\n" );
document.write( "A/B = 30/45 = 0.667 = 66.7% of the races have been won so far (excluding today).\r<BR>\n" );
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document.write( "x = number of additional races won today<BR>\n" );
document.write( "y = number of additional races attempted today<BR>\n" );
document.write( "A+x = 30+x = new number of wins overall (old wins+today's wins)<BR>\n" );
document.write( "B+y = 45+y = new number of attempts (old total+today's total), consists of wins and losses\r<BR>\n" );
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document.write( "The fraction (30+x)/(45+y) represents the new proportion of wins out of total races done. We want this to be equal to 75/100 as this is one fractional form of 75%. Recall that \"percent\" means \"per one hundred\". In general, x% = x/100. So all we do is stick the 75 over 100.\r<BR>\n" );
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document.write( "The equation we have is<BR>\n" );
document.write( "(30+x)/(45+y) = 75/100\r<BR>\n" );
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document.write( "At this point we have a problem in that one equation is not enough to solve for two simultaneous variables. We need two equations to solve for two variables at the same time. Example: two linear equations are needed to solve for two unknowns.\r<BR>\n" );
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document.write( "At this point, you should ask your teacher to clarify as there isn't enough information. You should be given how many races the team attempts (the value of y) so you can solve for x.\r<BR>\n" );
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document.write( "You can choose to stop here, or you can continue onto the next section below. \r<BR>\n" );
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document.write( "-------------------------------------------------------\r<BR>\n" );
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document.write( "<font color=red>Disclaimer: This section entirely depends on the condition that the team wins all of its races today. It's certainly not impossible, but its fairly unlikely given their current record isn't 100%.</font>\r<BR>\n" );
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document.write( "If the team wins all of the races it has done today, then y = x.<BR>\n" );
document.write( "So for example, if the team attempted y = 10 races and won x = 10 races, then y = x.\r<BR>\n" );
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document.write( "We can update<BR>\n" );
document.write( "(30+x)/(45+y) = 75/100<BR>\n" );
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document.write( "(30+x)/(45+x) = 75/100<BR>\n" );
document.write( "after replacing y with x\r<BR>\n" );
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document.write( "Solve for x<BR>\n" );
document.write( "(30+x)/(45+x) = 75/100<BR>\n" );
document.write( "100(30+x) = 75(45+x) ..... cross multiply<BR>\n" );
document.write( "3000+100x = 3375+75x ..... distribution rule<BR>\n" );
document.write( "100x-75x = 3375-3000<BR>\n" );
document.write( "25x = 375<BR>\n" );
document.write( "x = 375/25 ..... divide both sides by 25<BR>\n" );
document.write( "x = 15\r<BR>\n" );
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document.write( "If the team attempts 15 races, and wins all 15 of them, then we have<BR>\n" );
document.write( "(30+x)/(45+x) = (30+15)/(45+15) = 45/60 = 0.75 = 75%<BR>\n" );
document.write( "Move the decimal point 2 spots to the right to go from decimal form to percent form.\r<BR>\n" );
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document.write( "So in short, (30+x)/(45+x) = 75% when x = 15\r<BR>\n" );
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document.write( "Again, this solution is only possible if the team wins all of its races today.<BR>\n" );
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