document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1109307>Question 1109307</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have to spend $100 to buy 100 animal dog cost $10 cat cost $3 mice cost $.50 I have to use at least one of each. How much animal of each I will need to equal $100 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #724303 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=724303\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc+=+100 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc+=+100\"   DISABLED_style= \"max-width:90%; height:auto;\" >  the total number of dogs, cats, and mice is 100<BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10a%2B3b%2B.5c+=+100 ALIGN=MIDDLE  ALT=\"10a%2B3b%2B.5c+=+100\"   DISABLED_style= \"max-width:90%; height:auto;\" >  the total cost of the dogs ($10 each), cats ($3 each), and mice ($.50 each) is $100<br><BR>\n" );
document.write( "How to solve a problem like this:<BR>\n" );
document.write( ".. eliminate one variable to give you one equation in two variables.<BR>\n" );
document.write( ".. solve that equation for either variable in terms of the other.<BR>\n" );
document.write( ".. use the fact that the variables must have whole number values to find the solution(s)<br><BR>\n" );
document.write( "The way this problem is stated, there should only be one answer.  In similar problems, there might be several solutions.<br><BR>\n" );
document.write( "Multiply equation (2) by 2 and subtract equation (1) from the result, eliminating variable c:<BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20a%2B6b%2Bc+=+200 ALIGN=MIDDLE  ALT=\"20a%2B6b%2Bc+=+200\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc+=+100 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc+=+100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(3) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19a%2B5b+=+100 ALIGN=MIDDLE  ALT=\"19a%2B5b+=+100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A little logical analysis at this point can find the answer.  So let's do that; but then I will show you the formal algebraic way to finish the problem -- so you can apply the method the next time you see a problem like this.<br><BR>\n" );
document.write( "In equation 3, \"5b\" is a multiple of 5, because b is a whole number; and 100 is also a multiple of 5.  That means \"19a\" must be a multiple of 5, with a being a whole number.  Since 19 is a prime number, that means a must be a multiple of 5.  And with the two equations we started with, it is clear that a must be 5.<br><BR>\n" );
document.write( "So a=5; equation (3) then tells us that b is 1; and then equation (1) tells us c is 94.<br><BR>\n" );
document.write( "Answer: 5 dogs, 1 cat, 94 mice.<br><BR>\n" );
document.write( "And here is the formal algebraic method for finishing the problem, starting with equation (3).<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19a%2B5b+=+100 ALIGN=MIDDLE  ALT=\"19a%2B5b+=+100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5b+=+100-19a ALIGN=MIDDLE  ALT=\"5b+=+100-19a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5b+=+%28100-20a%29%2Ba ALIGN=MIDDLE  ALT=\"5b+=+%28100-20a%29%2Ba\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [do this so that (100-20a) is evenly divisible by 5]<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+%2820-4a%29%2B%28a%2F5%29 ALIGN=MIDDLE  ALT=\"b+=+%2820-4a%29%2B%28a%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In this form of the equation, b and a are whole numbers, so (20-4a) is a whole number; that means (a/5) must be a whole number.<br><BR>\n" );
document.write( "So, as with the logical reasoning we used above, we see that a must be a multiple of 5.  And, as before, the conditions of the problem only allow one choice for the value of a: a=5.<br><BR>\n" );
document.write( "And once again (of course), the solution is 5 dogs, 1 cat, and 94 mice. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );