document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=127819>Question 127819</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Daddy Warbucks always carries a specific number of $100 and $500 bills for impulse purchases and $1 bills for tips. If he has 500 bills in his briefcase and they total $50000, how many of each denomination does he carry?\r<BR>\n" );
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document.write( "Please help me, I do not know how to set this up to solve. Thankyou<BR>\n" );
document.write( "My question is not from a textbook, but it is from my teacher in my Math 102, Intermediate Algebra class. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #93616 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=93616\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I used h for hundreds,  f for five hundreds, and s for singles.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2Bf%2Bs=500 ALIGN=MIDDLE  ALT=\"h%2Bf%2Bs=500\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Hundreds are worth $100, so the value of the h hundreds is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100h ALIGN=MIDDLE  ALT=\"100h\"   DISABLED_style= \"max-width:90%; height:auto;\" >, likewise the value of the f five hundreds is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500f ALIGN=MIDDLE  ALT=\"500f\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the value of the s singles is just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s ALIGN=MIDDLE  ALT=\"s\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and we know that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100h%2B500f%2Bs=50000 ALIGN=MIDDLE  ALT=\"100h%2B500f%2Bs=50000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "But now what?  We have three unknowns but only two equations.\r<BR>\n" );
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document.write( "The secret lies in realizing that the answers must be positive integers.\r<BR>\n" );
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document.write( "Let's rearrange the equations a little:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2Bs=500-f ALIGN=MIDDLE  ALT=\"h%2Bs=500-f\"   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=100h%2Bs=50000-500f ALIGN=MIDDLE  ALT=\"100h%2Bs=50000-500f\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now multiply the first equation by -1:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-h-s=-500%2Bf ALIGN=MIDDLE  ALT=\"-h-s=-500%2Bf\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And add it term-by-term to the second equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=99h=49500-499f ALIGN=MIDDLE  ALT=\"99h=49500-499f\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then divide by 99:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=%2849500-499f%29%2F99 ALIGN=MIDDLE  ALT=\"h=%2849500-499f%29%2F99\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we know that the smallest f can be is 0 and the largest it could be is 100, because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2A100=50000 ALIGN=MIDDLE  ALT=\"500%2A100=50000\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  We can actually exclude 100 because that gets us to 50,000 with only 100 bills.\r<BR>\n" );
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document.write( "The problem now becomes finding a whole number value for f so that the expression for h above comes out to be another whole number.  You could make a list of all the results of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2849500-499f%29%2F99 ALIGN=MIDDLE  ALT=\"%2849500-499f%29%2F99\"   DISABLED_style= \"max-width:90%; height:auto;\" > for each of the 100 possibilities for f, but that is a lot of calculator work.  If you have access to MS Excel, you can solve it the way I did.  I made a list of the numbers from 1 to 100, made a formula that calculated <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2849500-499f%29%2F99 ALIGN=MIDDLE  ALT=\"%2849500-499f%29%2F99\"   DISABLED_style= \"max-width:90%; height:auto;\" > for each of those numbers, and examined the list for an integer result.\r<BR>\n" );
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document.write( "There is an analysis method, but it is a bit convoluted.  First thing to notice is that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49500%2F99=500 ALIGN=MIDDLE  ALT=\"49500%2F99=500\"   DISABLED_style= \"max-width:90%; height:auto;\" >, an integer, therefore we only need to find f so that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=499f%2F99 ALIGN=MIDDLE  ALT=\"499f%2F99\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer.  Well, right from the start we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=0 ALIGN=MIDDLE  ALT=\"f=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a possibility, but I think we can exclude that result because the wording of the problem at least implies that he has at least one of each kind of bill.\r<BR>\n" );
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document.write( "Now notice that, in the case of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=1 ALIGN=MIDDLE  ALT=\"f=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the remainder using integer division of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=499%2F99 ALIGN=MIDDLE  ALT=\"499%2F99\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 4.  That means the remainder when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=2 ALIGN=MIDDLE  ALT=\"f=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be 8, and so on until you get to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=24 ALIGN=MIDDLE  ALT=\"f=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the remainder is 96.  The next one, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=25 ALIGN=MIDDLE  ALT=\"f=25\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the remainder reverts to 1 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=96%2B4-99=1 ALIGN=MIDDLE  ALT=\"96%2B4-99=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the 'by 4' count starts over.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=49 ALIGN=MIDDLE  ALT=\"f=49\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a remainder of 97, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=50 ALIGN=MIDDLE  ALT=\"f=50\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=2 ALIGN=MIDDLE  ALT=\"r=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=74 ALIGN=MIDDLE  ALT=\"f=74\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=98 ALIGN=MIDDLE  ALT=\"r=98\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=75 ALIGN=MIDDLE  ALT=\"f=75\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3 ALIGN=MIDDLE  ALT=\"r=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=98 ALIGN=MIDDLE  ALT=\"f=98\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=95 ALIGN=MIDDLE  ALT=\"r=95\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=99 ALIGN=MIDDLE  ALT=\"f=99\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=0 ALIGN=MIDDLE  ALT=\"r=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >!!!!  And we have found the answer.\r<BR>\n" );
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document.write( "Turns out there is only one integer result, excluding the trivial result of h = 500, f = 0, and s = 0, and that is:  h = 1, f = 99, and s = 400. </SPAN>\n" );
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