document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202892>Question 1202892</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Officials begin to release water from a full man-made lake at a rate that would empty the lake in 12<BR>\n" );
document.write( " weeks, but a river that can fill the lake in 20<BR>\n" );
document.write( " weeks is replenishing the lake at the same time. How many weeks does it take to empty the lake? Express your answer as a fraction reduced to lowest terms, if needed.<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 #838090 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=838090\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The response from the other tutor solves the problem by choosing a \"nice\" number for the number of gallons in the full lake.  That of course is a valid way of solving the problem.<br><BR>\n" );
document.write( "If you wanted a formal algebraic solution, it might look something like this:<br><BR>\n" );
document.write( "Let x be the number of gallons in the full lake.  Then, according to the given information,<br><BR>\n" );
document.write( "(1/12)x = gallons released in 1 week<BR>\n" );
document.write( "(1/20)x = gallons filled by the river in 1 week<br><BR>\n" );
document.write( "The number of gallons in the lake decreases by the difference between those two amounts.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F12%29x-%281%2F20%29x+=+%285%2F60%29x-%283%2F60%29x+=+%282%2F60%29x+=+%281%2F30%29x ALIGN=MIDDLE  ALT=\"%281%2F12%29x-%281%2F20%29x+=+%285%2F60%29x-%283%2F60%29x+=+%282%2F60%29x+=+%281%2F30%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since the capacity of the lake in gallons is x, and the number of gallons in the lake is decreasing by (1/30)x each week, the number of weeks needed to empty the lake is 30.<br><BR>\n" );
document.write( "ANSWER: 30<br><BR>\n" );
document.write( "And, if formal algebra is not required, here is an informal method for solving this kind of problem that avoids working with fractions.<br><BR>\n" );
document.write( "Consider the least common multiple of the two rates for filling and emptying the lake.  The least common multiple of 12 and 20 is 60.<br><BR>\n" );
document.write( "In 60 weeks, the lake could be emptied 60/12 = 5 times by releasing water; in those 60 weeks, the lake could be filled by the river 60/20 = 3 times.<br><BR>\n" );
document.write( "That means that if water is flowing into and out of the lake at the same time, in 60 weeks the lake could be emptied 5-3 = 2 times.<br><BR>\n" );
document.write( "Then, since the lake would be emptied twice in 60 weeks, the number of weeks needed to drain the lake (once) is 60/2 = 30.<br><BR>\n" );
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