document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1006572>Question 1006572</A>: <I><SPAN STYLE=\"word-break: break-all;\"> My hippo's bathtub takes 6 minutes to fill to the top. On the other hand, it empties somewhat slower: it takes 9 minutes to empty the full tub. One day my hippo was very busy and didn't notice that the stopper wasn't in when he turned the water on. How long did it take to fill the tub this time? How much water was wasted? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #622680 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=622680\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Filling the tub:\r<BR>\n" );
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document.write( "it takes 6 minutes to fill the tub, so we have this ratio\r<BR>\n" );
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document.write( "6 minutes: 1 tub\r<BR>\n" );
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document.write( "divide both parts by 6 to get\r<BR>\n" );
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document.write( "1 minute: 1/6 tub\r<BR>\n" );
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document.write( "this says \"in one minute, 1/6 of the tub is full\"\r<BR>\n" );
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document.write( "The fill rate is essentially 1/6 of a tub per minute\r<BR>\n" );
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document.write( "Emptying the tub:\r<BR>\n" );
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document.write( "Using similar logic as shown above, if it takes 9 minutes to empty the tub, then the drain rate is 1/9 of a tub per minute\r<BR>\n" );
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document.write( "Combining the two rates\r<BR>\n" );
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document.write( "The fill rate is 1/6 of a tub per min<BR>\n" );
document.write( "The drain rate is 1/9 of a tub per min\r<BR>\n" );
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document.write( "Think of the fill rate as a positive quantity. The drain rate is the opposite: a negative quantity\r<BR>\n" );
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document.write( "so we have 1/6 and -1/9 and they combine to...\r<BR>\n" );
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document.write( "1/6 + (-1/9) = 1/6 - 1/9<BR>\n" );
document.write( "1/6 + (-1/9) = 3/18 - 2/18<BR>\n" );
document.write( "1/6 + (-1/9) = (3-2)/18<BR>\n" );
document.write( "1/6 + (-1/9) = 1/18\r<BR>\n" );
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document.write( "The combined rate is 1/18 of a tub per min. Recall that we assigned positive rates to mean that the tub is filling. So overall, the tub is indeed filling up. The rate at which it fills up is 1/18 of a tub per min\r<BR>\n" );
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document.write( "Put another way, it takes 18 minutes to fill up the entire bath tub. \r<BR>\n" );
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document.write( "Now onto the second part of the question: how much water was wasted?\r<BR>\n" );
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document.write( "We know that it takes 18 minutes for the tub to fill IF the stopper isn't in. Let's see how much water drains out in 18 min\r<BR>\n" );
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document.write( "(-1/9 of a tub per min)*(t minutes) = (-1/9)*(18) = -2\r<BR>\n" );
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document.write( "so 2 full tubs of water are drained in the span of 18 minutes<BR>\n" );
document.write( "Note: 1/6*18 = 3 full tubs of water are poured from the faucet. Notice how 3+(-2) = +1\r<BR>\n" );
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document.write( "We lost 2 tubs of water when we drained it out of the stopper. We used 3 tubs of water to get the tub full again. In total, we used 2+3 = 5 tubs of water. In the most perfect world, we'd use only 1 tub of water to fill the entire tub. Since the tub, by definition, has a volume of \"1 tub\".\r<BR>\n" );
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document.write( "So we wasted 5-1 = 4 tubs of water.\r<BR>\n" );
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document.write( "Note: \"tubs\" aren't an official volume liquid measurement, but no other units like gallons, liters, etc were given.  </SPAN>\n" );
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