document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=97502>Question 97502</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/1592009425/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Heres the question:<BR>\n" );
document.write( "A farmer wants to buy 100 animals for $100.00 (must buy at least one of them)\r<BR>\n" );
document.write( "\n" );
document.write( "Horses-$5.00<BR>\n" );
document.write( "Cows-$3.00<BR>\n" );
document.write( "Chicks-.50 \r<BR>\n" );
document.write( "\n" );
document.write( "How many horses,cows and chicks does the farmer need to have 100 and 100.00? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #70951 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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=70951\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem involves diophantine equations, where you have more unknowns than equations.<BR>\n" );
document.write( "The solutions, however, are limited by the fact that the unknowns must be integers, i.e. whole cows, whole horses, etc.<BR>\n" );
document.write( "We can set up two equations as follows: H = number of horses, C = number of cows, and ch = number of chicks.\r<BR>\n" );
document.write( "\n" );
document.write( "From the problem description, we can write:<BR>\n" );
document.write( "H+C+ch = 100 The total number of animals is 100.<BR>\n" );
document.write( "($5)H+($3)C+($0.50)ch = $100 The total amount the farmer spends.<BR>\n" );
document.write( "The two equations are:\r<BR>\n" );
document.write( "\n" );
document.write( "1) H+C+ch = 100<BR>\n" );
document.write( "2) 5H+3C+0.5ch = 100 Multiply this equation by 2 and subtract equation 1) from the result.\r<BR>\n" );
document.write( "\n" );
document.write( "1) H+C+ch = 100<BR>\n" );
document.write( "2a) 10H+6C+ch = 200<BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "3) 9H+5C = 100 Subtract 9H from both sides.<BR>\n" );
document.write( "3a) 5C = 100-9H  Now divide  both sides by 5.<BR>\n" );
document.write( "3b) C = 20-(9/5)H Now C can only be an integer if H is a multiple of 5.  Let's try H = 5.<BR>\n" );
document.write( "3c) C = 20-(9/5)(5)<BR>\n" );
document.write( "C = 20 - 9<BR>\n" );
document.write( "C = 11 This is one possibility. Now let's try H = 10<BR>\n" );
document.write( "3d) C = 20 - (9/5)(10)<BR>\n" );
document.write( "C = 20 - 18<BR>\n" );
document.write( "C = 2 This is another possibility. <BR>\n" );
document.write( "H cannot be greater than 10 otherwise you would end up with C being a non-integer or a negative number.\r<BR>\n" );
document.write( "\n" );
document.write( "So, if C = 11 when H = 5, then ch = 100-11-5 = 84 - the number of chicks.<BR>\n" );
document.write( "Let's check this solution. \r<BR>\n" );
document.write( "\n" );
document.write( "H+C+ch = 100<BR>\n" );
document.write( "5+11+84 = 100  and...<BR>\n" );
document.write( "5($5) + 11($3) + 84($0.50) = $25 + $33 + $42 = $100 This is a valid solution!\r<BR>\n" );
document.write( "\n" );
document.write( "Now if C = 2 when H = 10, then ch = 100-2-10 = 88 - the number of chicks.<BR>\n" );
document.write( "Let's check this solution.<BR>\n" );
document.write( "H+C+ch = 100<BR>\n" );
document.write( "10+2+88 = 100 and...<BR>\n" );
document.write( "10($5)+2($3)+88($0.50) = $50+$6+$44 = $100 This solution is also valid!\r<BR>\n" );
document.write( "\n" );
document.write( "The two solutions are:\r<BR>\n" );
document.write( "\n" );
document.write( "1) 5 horses, 11 cows, and 84 chicks.\r<BR>\n" );
document.write( "\n" );
document.write( "2) 10 horses, 2 cows, and 88 chicks. <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );