document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099298>Question 1099298</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are 100 seats for 100 people at the movie theater.  The total price for the 100 people is $1.00. Men will be charged $.05 each, women $.02 and children 10 for $.01.  How many men, women, and children will there be? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #713732 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=713732\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br>Let m be the number of men, w the number of women, and c be the number of children.  Then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Bw%2Bc+=+100 ALIGN=MIDDLE  ALT=\"m%2Bw%2Bc+=+100\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [the total number of people is 100]<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.05m%2B.02w%2B.001c+=+1 ALIGN=MIDDLE  ALT=\".05m%2B.02w%2B.001c+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [the total cost of the tickets is $1.00]<br><BR>\n" );
document.write( "This is a problem with Diophantine equations.  Diophantine equations are systems of equations in which there are fewer equations than variables, but in which it is possible to find the solution (or a family of solutions) using the requirement that all the variables must have values that are integers (usually, as in this case, positive integers).<br><BR>\n" );
document.write( "With two variables and three equations, the procedure is to use the two equations to eliminate one of the variables, leaving one equation with two variables.<BR>\n" );
document.write( "Then solve that one equation for one variable in terms of the other, and use the requirement that both variables have positive integer values to find solution(s).<br><BR>\n" );
document.write( "With the two equations in this problem, we can eliminate variable c by multiplying the second equation by 1000 and subtracting the two equations:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50m%2B20w%2Bc+=+1000 ALIGN=MIDDLE  ALT=\"50m%2B20w%2Bc+=+1000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Bw%2Bc+=+100 ALIGN=MIDDLE  ALT=\"m%2Bw%2Bc+=+100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49m%2B19w+=+900 ALIGN=MIDDLE  ALT=\"49m%2B19w+=+900\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19w+=+900-49m ALIGN=MIDDLE  ALT=\"19w+=+900-49m\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%28900-49m%29%2F19 ALIGN=MIDDLE  ALT=\"w+=+%28900-49m%29%2F19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "With the equation in this form, we know that, because w has to be an integer, (900-49m) must be divisible by 19.<br><BR>\n" );
document.write( "There are various ways to find what value(s) of m make this an integer.  With pencil and paper, the best way is to rewrite the numerator (900-49m) in two parts, the first of which is exactly divisible by 19.<br><BR>\n" );
document.write( "Proceeding with that idea...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%28912-57m%29%2F19+%2B+%288m-12%29%2F19+=+%2848-3m%29+%2B+4%282m-3%29%2F19 ALIGN=MIDDLE  ALT=\"w+=+%28912-57m%29%2F19+%2B+%288m-12%29%2F19+=+%2848-3m%29+%2B+4%282m-3%29%2F19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "With the equation in this form, we see that (2m-3) must be divisible by 19; and some quick calculations show that m must be 11.  Then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%2848-33%29+%2B+4%2819%2F19%29+=+15%2B4+=+19 ALIGN=MIDDLE  ALT=\"w+=+%2848-33%29+%2B+4%2819%2F19%29+=+15%2B4+=+19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And then since there are 11 men and 19 women, the number of children must be 70.<br><br><BR>\n" );
document.write( "Now here is a different approach to the problem which uses more standard algebraic techniques.<br><BR>\n" );
document.write( "Since the price for children is 10 for a penny, the number of children must be a multiple of 10.<BR>\n" );
document.write( "Furthermore, a bit of logical analysis tells us that, since that average cost of the tickets is a penny, most of the people must be children.<br><BR>\n" );
document.write( "So see if you can find a solution if the number of children is 90; if not, then try 80; then 70; and so on, until you find a solution.<br><BR>\n" );
document.write( "If c = 90...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Bw+=+10 ALIGN=MIDDLE  ALT=\"m%2Bw+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5m%2B2w+=+91 ALIGN=MIDDLE  ALT=\"5m%2B2w+=+91\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiply the first equation by 5 and then subtract the two equations:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5m%2B5w+=+50 ALIGN=MIDDLE  ALT=\"5m%2B5w+=+50\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3w+=+-41 ALIGN=MIDDLE  ALT=\"3w+=+-41\"   DISABLED_style= \"max-width:90%; height:auto;\" >  obviously no good...<br><BR>\n" );
document.write( "If c = 80...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Bw+=+20 ALIGN=MIDDLE  ALT=\"m%2Bw+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5m%2B2w+=+92 ALIGN=MIDDLE  ALT=\"5m%2B2w+=+92\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5m%2B5w+=+100 ALIGN=MIDDLE  ALT=\"5m%2B5w+=+100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3w+=+8 ALIGN=MIDDLE  ALT=\"3w+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >  no; 8 is not divisible by 3<br><BR>\n" );
document.write( "If c = 70...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%2Bw+=+30 ALIGN=MIDDLE  ALT=\"m%2Bw+=+30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5m%2B2w+=+93 ALIGN=MIDDLE  ALT=\"5m%2B2w+=+93\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5m%2B5w+=+150 ALIGN=MIDDLE  ALT=\"5m%2B5w+=+150\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3w+=+57 ALIGN=MIDDLE  ALT=\"3w+=+57\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+19 ALIGN=MIDDLE  ALT=\"w+=+19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This works. c = 70; w = 19; that makes m = 11.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );