document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1128413>Question 1128413</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A park has two sprinklers that are used to fill a fountain. One sprinkler can fill the fountain in 4 h, whereas the second sprinkler can fill the fountain in 12 h. How long will it take to fill the fountain with both sprinklers operating? <BR>\n" );
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document.write( "The classic way to solve this kind of problem is to think of the fractions of the job each sprinkler does in one hour.<br><BR>\n" );
document.write( "In this problem, one sprinkler fills 1/4 of the fountain in 1 hour and the other fills 1/12 of it in one hour.  Together, then, the fraction of the fountain they fill is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4%2B1%2F12+=+3%2F12%2B1%2F12+=+4%2F12+=+1%2F3 ALIGN=MIDDLE  ALT=\"1%2F4%2B1%2F12+=+3%2F12%2B1%2F12+=+4%2F12+=+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then, since together they fill 1/3 of the fountain in one hour, it takes them 3 hours to fill to fountain.<br><BR>\n" );
document.write( "This is a very common problem that I see frequently on high school math competition tests.  An experienced student will just write down the answer without having to do much work at all; the answer is just \"product divided by sum\" -- in this problem, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812%2A4%29%2F%2812%2B4%29+=+48%2F16+=+3 ALIGN=MIDDLE  ALT=\"%2812%2A4%29%2F%2812%2B4%29+=+48%2F16+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Here is how that works in general....<br><BR>\n" );
document.write( "Suppose the two times for the two workers are x and y.  Then the fractions they do in one unit of time are 1/x and 1/y.  The fraction they do together in one unit of time is 1/x+1/y = (x+y)/(xy); then the amount of time they take to do the job together is the reciprocal of that number, which is (xy)/(x+y).<br><BR>\n" );
document.write( "There is also another easy alternative method for solving this type of problem that many students like.<br><BR>\n" );
document.write( "Consider the least common multiple of the two times for the two sprinklers; the LCM of 4 and 12 is 12.<br><BR>\n" );
document.write( "Now imagine each sprinkler working for 12 hours.  The one that can fill the fountain in 4 hours could fill the fountain 3 times in 12 hours; the one that can fill it in 12 hours could fill the fountain 1 time in 12 hours.<br><BR>\n" );
document.write( "Together, then, they could fill the fountain 4 times in 12 hours; that means they could fill it once in 3 hours. </SPAN>\n" );
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