document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1022114>Question 1022114</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Jenny and Penny can eat a whole gallon of ice cream together in 2 hours. Penny and Lenny can eat a whole gallon of ice cream together in 1 (2/3) hours. All three together can eat a whole gallon of ice cream together in 1 hour. How many hours would it take Jenny to eat a whole gallon of ice cream all by herself? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #637853 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39620)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=637853\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> * ADDED EASIER SOLUTION BELOW - SEE ASTERISK (*).\r<BR>\n" );
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document.write( "Rates of more than one eater at the same time are additive.\r<BR>\n" );
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document.write( "j, time for Jenny<BR>\n" );
document.write( "p, time for Penny<BR>\n" );
document.write( "y, time for Lenny<BR>\n" );
document.write( "To eat the one gallon on each's own\r<BR>\n" );
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document.write( "Notice that the numerators in all terms are 1, and only a variable is used in every denominator on the left hand members.<BR>\n" );
document.write( "An alternative is to not to focus on the times as variables, but to focus on the RATES directly as variables, although each rate will still be a fraction.\r<BR>\n" );
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document.write( "You might be more comfortable reassigning new variables and forming<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28J%2BP=1%2F2%2CP%2BY=1%2F%281%262%2F3%29%2CJ%2BP%2BY=1%29 ALIGN=MIDDLE  ALT=\"system%28J%2BP=1%2F2%2CP%2BY=1%2F%281%262%2F3%29%2CJ%2BP%2BY=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now you have a system of LINEAR equations instead of RATIONAL equations.  Your goal is to solve for J  or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2FJ ALIGN=MIDDLE  ALT=\"1%2FJ\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "I used matrix operations, not shown here, and found  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28P=1%2F10%2CY=1%2F2%29 ALIGN=MIDDLE  ALT=\"system%28P=1%2F10%2CY=1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so you can find J using  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=J%2B1%2F10%2B1%2F2=1 ALIGN=MIDDLE  ALT=\"J%2B1%2F10%2B1%2F2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Tell me if you really need to see the matrix row operation steps.<BR>\n" );
document.write( "Jenny takes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%282%261%2F2%29 ALIGN=MIDDLE  ALT=\"highlight%282%261%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > hours to eat one gallon of ice cream by herself.\r<BR>\n" );
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document.write( "*--------EVEN EASIER FOR THIS EXERCISE---------\r<BR>\n" );
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document.write( "Notice that two of the equations have an expression, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fj%2B1%2Fp ALIGN=MIDDLE  ALT=\"1%2Fj%2B1%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "First a small adjustment to be made to the system,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%281%2Fj%2B1%2Fp=1%2F2%2C1%2Fp%2B1%2Fy=3%2F5%2C1%2Fj%2B1%2Fp%2B1%2Fy=1%29 ALIGN=MIDDLE  ALT=\"system%281%2Fj%2B1%2Fp=1%2F2%2C1%2Fp%2B1%2Fy=3%2F5%2C1%2Fj%2B1%2Fp%2B1%2Fy=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "There is a very easy elimination you can perform.  Working with first and third equations to do this...<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fj%2B1%2Fp%2B1%2Fy-1%2Fj-1%2Fp=1%2F2 ALIGN=MIDDLE  ALT=\"1%2Fj%2B1%2Fp%2B1%2Fy-1%2Fj-1%2Fp=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281%2Fy=1%2F2%29 ALIGN=MIDDLE  ALT=\"highlight%281%2Fy=1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >------giving one of your answer parts, meaning <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=2%29 ALIGN=MIDDLE  ALT=\"highlight%28y=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, two hours for Lenny to eat one gallon on his own.\r<BR>\n" );
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document.write( "Now look at the second equation of the system.  You already found value for y.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fp=6%2F10-5%2F10 ALIGN=MIDDLE  ALT=\"1%2Fp=6%2F10-5%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fp=1%2F10 ALIGN=MIDDLE  ALT=\"1%2Fp=1%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28p=10%29 ALIGN=MIDDLE  ALT=\"highlight%28p=10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, meaning Penny needs 10 hours on her own to eat one gallon of ice cream.\r<BR>\n" );
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document.write( "Now return to the first equation of the system, only with j for Jenny being unknown.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fj=1%2F2-1%2F10 ALIGN=MIDDLE  ALT=\"1%2Fj=1%2F2-1%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fj=%285-1%29%2F10 ALIGN=MIDDLE  ALT=\"1%2Fj=%285-1%29%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fj=4%2F10 ALIGN=MIDDLE  ALT=\"1%2Fj=4%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28j=5%2F2%29 ALIGN=MIDDLE  ALT=\"highlight%28j=5%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >-----Meaning Jenny needs 2.5 hours to eat one gallon ice cream. </SPAN>\n" );
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