document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=322395>Question 322395</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A boat goes up stream 30 miles and down stream 44 miles in 10 hours. Again it goes up stream 40 miles and down stream<BR>\n" );
document.write( "55 miles in 13 hours. Find the rates of the stream and of the boat. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #230842 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=230842\">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 r1=rate of stream<BR>\n" );
document.write( "And r2=rate of the boat<BR>\n" );
document.write( "r2-r1=rate upstream<BR>\n" );
document.write( "r2+r1=rate downstream<BR>\n" );
document.write( "On the first run:<BR>\n" );
document.write( "time upstream=30/(r2-r1)<BR>\n" );
document.write( "time downstream=44/(r2+r1)<BR>\n" );
document.write( "and we are told that the above times equals 10 hrs, so:<BR>\n" );
document.write( "30/(r2-r1) + 44/(r2+r1)=10-----------------------eq1<BR>\n" );
document.write( "On the second run:<BR>\n" );
document.write( "time upstream=40/(r2-r1)<BR>\n" );
document.write( "time downstream=55/(r2+r1)<BR>\n" );
document.write( "And these times equals 13 hours, so:<BR>\n" );
document.write( "40/(r2-r1)+55/(r2+r1)=13----------------------eq2<BR>\n" );
document.write( "In eq1, multiply each term by (r2-r1)(r2+r1) and we get:<BR>\n" );
document.write( "30(r2+r1)+44(r2-r1)=10(r2^2-r1^2) simplify<BR>\n" );
document.write( "30r2+30r1+44r2-44r1=10r2^2-10r1^2<BR>\n" );
document.write( "74r2-14r1=10r2^2-10r1^2-----------------eq1a<BR>\n" );
document.write( "Let try to simplify eq2 in the same manner<BR>\n" );
document.write( "40(r2+r1)+55(r2-r1)=13(r2^2-r1^2)<BR>\n" );
document.write( "40r2+40r1+55r2-55r1=13r2^2-13r1^2<BR>\n" );
document.write( "95r2-15r1=13r2^2-13r1^2------------eq2a\r<BR>\n" );
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document.write( "From eq1a, (r2^2-r1^2)=(74r2-14r1)/10.  Substitute this into eq2a:<BR>\n" );
document.write( "95r2-15r1=13(74r2-14r1)/10) multiply each side by 10<BR>\n" );
document.write( "950r2-150r1=962r2-182r1 simplify<BR>\n" );
document.write( "-150r1+182r1=962r2-950r2<BR>\n" );
document.write( "32r1=12r2 or<BR>\n" );
document.write( "8r1=3r2<BR>\n" );
document.write( "r1=(3/8)r2 and r1^2=(9/64)r2^2; substitute these into eq1a<BR>\n" );
document.write( "74r2-14(3/8)r2=10r2^2-10(9/64)r2^2 multiply each term by 64<BR>\n" );
document.write( "4736r2-336r2=640r2^2-90r2^2 simplify<BR>\n" );
document.write( "4400r2=550r2^2 divide each side by 10r2<BR>\n" );
document.write( "440=55r2<BR>\n" );
document.write( "r2=8 mph-----------------------------------------speed of boat<BR>\n" );
document.write( "r1=(3/8)r2=(3/8)*8=3 mph ------------------------speed of stream\r<BR>\n" );
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document.write( "CK<BR>\n" );
document.write( "30/(8-3)=6hr<BR>\n" );
document.write( "44/(8+3)=4hr<BR>\n" );
document.write( "10=10<BR>\n" );
document.write( "40/5=8 hr<BR>\n" );
document.write( "55/11=5 hrs<BR>\n" );
document.write( "13=13\r<BR>\n" );
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document.write( "You can look it over.  There may be an easier way.<BR>\n" );
document.write( "Hope this helps---ptaylor\r<BR>\n" );
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