document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=126664>Question 126664</A>: <I><SPAN STYLE=\"word-break: break-all;\">         I have a 2 questions, they are comfusing me, Laura and Brent paddled a canoe 6 miles upstream in four hours. The return trip took 3 hours. Find the rate at which Laura and Brent paddled the canoe in still water......that is one, and the other is, The sum of the digits of a 2-digit # is 11. If the digits are reversed, the new number is 45 more than the original numer. Find the number </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #92795 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=92795\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You know that distance = rate X time or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=rt ALIGN=MIDDLE  ALT=\"d=rt\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or, since we are trying to determine a rate: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=d%2Ft ALIGN=MIDDLE  ALT=\"r=d%2Ft\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "For both the upstream and downstream trips we know the distance is 6 miles and the times, 4 hours and 3 hours respectively.\r<BR>\n" );
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document.write( "For the upstream trip, the true rate of speed of the boat is the boat's rate of speed in still water MINUS the rate of the current.  For the downstream trip, the true rate is the still water rate PLUS the rate of the current.\r<BR>\n" );
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document.write( "Upstream equation:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bs%5D-r%5Bc%5D=6%2F4=3%2F2 ALIGN=MIDDLE  ALT=\"r%5Bs%5D-r%5Bc%5D=6%2F4=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bs%5D ALIGN=MIDDLE  ALT=\"r%5Bs%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the rate in still water and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bc%5D ALIGN=MIDDLE  ALT=\"r%5Bc%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the rate of the current.\r<BR>\n" );
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document.write( "Downstream equation:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bs%5D%2Br%5Bc%5D=6%2F3=2 ALIGN=MIDDLE  ALT=\"r%5Bs%5D%2Br%5Bc%5D=6%2F3=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bs%5D-r%5Bc%5D=3%2F2 ALIGN=MIDDLE  ALT=\"r%5Bs%5D-r%5Bc%5D=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bs%5D%2Br%5Bc%5D=2 ALIGN=MIDDLE  ALT=\"r%5Bs%5D%2Br%5Bc%5D=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now add the equations, term-by-term:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2r%5Bs%5D%2B0r%5Bc%5D=7%2F2 ALIGN=MIDDLE  ALT=\"2r%5Bs%5D%2B0r%5Bc%5D=7%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And solve:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bs%5D=7%2F4 ALIGN=MIDDLE  ALT=\"r%5Bs%5D=7%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the rate in still water is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F4 ALIGN=MIDDLE  ALT=\"7%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > mph.\r<BR>\n" );
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document.write( "Check:<BR>\n" );
document.write( "To go 6 miles in 4 hours, the true rate must have been <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > mph.  To go 6 miles in 3 hours, the true rate must have been <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > mph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F4-r%5Bc%5D=3%2F2 ALIGN=MIDDLE  ALT=\"7%2F4-r%5Bc%5D=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5Bc%5D=1%2F4 ALIGN=MIDDLE  ALT=\"r%5Bc%5D=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F4%2B1%2F4=8%2F4=2 ALIGN=MIDDLE  ALT=\"7%2F4%2B1%2F4=8%2F4=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, answer checks.\r<BR>\n" );
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document.write( "Let the 10s digit be x and the 1s digit be y.  The sum of the digits is 11, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=11 ALIGN=MIDDLE  ALT=\"x%2By=11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And the original number is n, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=10x+%2B+y ALIGN=MIDDLE  ALT=\"n=10x+%2B+y\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If you reverse the digits, the new number which is 45 more than the old number would be given by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2B45=10y+%2B+x ALIGN=MIDDLE  ALT=\"n%2B45=10y+%2B+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but this can be written:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=10y%2Bx-45 ALIGN=MIDDLE  ALT=\"n=10y%2Bx-45\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now we have two expressions that are equal to n, so we can set these two expressions equal to each other:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%2By=10y%2Bx-45 ALIGN=MIDDLE  ALT=\"10x%2By=10y%2Bx-45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x-x-10y%2By=-45 ALIGN=MIDDLE  ALT=\"10x-x-10y%2By=-45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x-9y=-45 ALIGN=MIDDLE  ALT=\"9x-9y=-45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y=-5 ALIGN=MIDDLE  ALT=\"x-y=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From our first equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=11 ALIGN=MIDDLE  ALT=\"x%2By=11\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=11-x ALIGN=MIDDLE  ALT=\"y=11-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we can substitute this expression for y into the previous result:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-%2811-x%29=-5 ALIGN=MIDDLE  ALT=\"x-%2811-x%29=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-11=-5 ALIGN=MIDDLE  ALT=\"2x-11=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x=-5%2B11 ALIGN=MIDDLE  ALT=\"2x=-5%2B11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x=6 ALIGN=MIDDLE  ALT=\"2x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now substituting this value for x into the first equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2By=11 ALIGN=MIDDLE  ALT=\"3%2By=11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=8 ALIGN=MIDDLE  ALT=\"y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the original number is 38.\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "3 + 8 = 11.\r<BR>\n" );
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document.write( "83 - 38 = 45.  Answer checks. </SPAN>\n" );
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