document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186985>Question 1186985</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi<BR>\n" );
document.write( "Mary bought some shoes and dresses for $296.00. The dress cost $24  the shoes $20 . How many of each did she buy.\r<BR>\n" );
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document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #817954 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</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=817954\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Here first is a refinement, using logical analysis and simple mental arithmetic, of the informal solution shown by the other tutor...<br><BR>\n" );
document.write( "The total cost of all the items is $296, and the total cost of the shoes, at $20 per pair, is a multiple of $10.<BR>\n" );
document.write( "That means the total cost of the dresses, at $24 each, must be a number with units digit 6.<BR>\n" );
document.write( "That means the number of dresses must be 4, 9, 14, ...<BR>\n" );
document.write( "Since 14 dresses at $24 each is more than $296, there are exactly two solutions -- with either 4 or 9 dresses.<br><BR>\n" );
document.write( "<hr><BR>\n" );
document.write( "And here is how to find the solution(s) using formal mathematics....<br><BR>\n" );
document.write( "x = # of dresses at $24 each<BR>\n" );
document.write( "y = # of (pairs of) shoes at $20 each<br><BR>\n" );
document.write( "The total cost was $296:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24x%2B20y=296 ALIGN=MIDDLE  ALT=\"24x%2B20y=296\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Simplify...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%2B5y=74 ALIGN=MIDDLE  ALT=\"6x%2B5y=74\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Solve the equation for one variable<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y=-6x%2B74 ALIGN=MIDDLE  ALT=\"5y=-6x%2B74\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-6x%2B74%29%2F5 ALIGN=MIDDLE  ALT=\"y=%28-6x%2B74%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Perform the indicated division as quotient plus remainder, using integers -- instead of fractions<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%28-5x%2B70%29%2B%28-x%2B4%29%29%2F5+=+%28-5x%2B70%29%2F5%2B%28-x%2B4%29%2F5 ALIGN=MIDDLE  ALT=\"y=%28%28-5x%2B70%29%2B%28-x%2B4%29%29%2F5+=+%28-5x%2B70%29%2F5%2B%28-x%2B4%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-x%2B14%29%2B%28-x%2B4%29%2F5 ALIGN=MIDDLE  ALT=\"y=%28-x%2B14%29%2B%28-x%2B4%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In that form of the equation, x and y are non-negative integers, and 14 is an integer, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-x%2B4%29%2F5 ALIGN=MIDDLE  ALT=\"%28-x%2B4%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > has to be an integer.<br><BR>\n" );
document.write( "Inspection shows that x must be 4, or 4 plus or minus some multiple of 5.<br><BR>\n" );
document.write( "Use that and the knowledge that x and y are both non-negative integers to find the solution(s).<br><BR>\n" );
document.write( "(1) x=4<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-4%2B14%29%2B%28-4%2B4%29%2F5+=+10%2B0+=+10 ALIGN=MIDDLE  ALT=\"y=%28-4%2B14%29%2B%28-4%2B4%29%2F5+=+10%2B0+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: x=4 dresses and y=10 pairs of shoes<br><BR>\n" );
document.write( "CHECK: 4($24)+10($20) = $96+$200 = $296<br><BR>\n" );
document.write( "(2) x=9<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-9%2B14%29%2B%28-9%2B4%29%2F5+=+5-1=4 ALIGN=MIDDLE  ALT=\"y=%28-9%2B14%29%2B%28-9%2B4%29%2F5+=+5-1=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: x=9 dresses and y=4 pairs of shoes<br><BR>\n" );
document.write( "CHECK: 9($24)+4($20) = $216+$80 = $296<br><BR>\n" );
document.write( "(3) x=14<br><BR>\n" );
document.write( "14($24) is greater than $296...<br><BR>\n" );
document.write( "There are exactly two answers:<BR>\n" );
document.write( "(1) 4 dresses and 10 pairs of shoes<BR>\n" );
document.write( "(2) 9 dresses and 4 pairs of shoes<br><BR>\n" );
document.write( " </SPAN>\n" );
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