document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1138693>Question 1138693</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An ice-cream parlor has one size of ice-cream cone for 7 cents and another for a cents. One afternoon  when the smaller cones outsold the larger ones two to one, the sale totaled 6.21 pesos. How many cones of each size were sold? </SPAN>\n" );
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document.write( "\"... a cents...\"???<br><BR>\n" );
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document.write( "After seeing the response from the other tutor, I see that perhaps you really do mean a cents each for the other cones, and we are supposed to find all solutions for which the total sales is 6.21 pesos = 621 cents.<br><BR>\n" );
document.write( "So let x be the number of cones sold at a cents each; then the number sold at 7 cents each is 2x.  Then if the total sales was 621 cents,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%287%29%2Bx%28a%29+=+621 ALIGN=MIDDLE  ALT=\"2x%287%29%2Bx%28a%29+=+621\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14x%2Bax+=+621 ALIGN=MIDDLE  ALT=\"14x%2Bax+=+621\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2814%2Ba%29x+=+621 ALIGN=MIDDLE  ALT=\"%2814%2Ba%29x+=+621\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To find solutions in integers, find the prime factorization of the total sales number.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2814%2Ba%29x+=+621+=+3%2A3%2A3%2A23 ALIGN=MIDDLE  ALT=\"%2814%2Ba%29x+=+621+=+3%2A3%2A3%2A23\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So (14+a) has to be in integer that is a divisor of 621; with this prime factorization of 621, there are two possibilities:<BR>\n" );
document.write( "(14+a) = 23  -->  a = 9; or<BR>\n" );
document.write( "(14+a) = 3*3*3 = 27  -->  a = 13<br><BR>\n" );
document.write( "Solution #1:  If (14+a) = 23, making a = 9, then x = 3*3*3 = 27.  This gives us x=27 cones at 9 cents each and 2x=54 cones at 7 cents each.<br><BR>\n" );
document.write( "CHECK: 27(9)+54(7) = 243+378 = 621<br><BR>\n" );
document.write( "Solution #2:  If (14+a) = 3*3*3 = 27, making a = 13, then x = 23.  This gives us x=23 cones at 13 cents each and 2x=46 cones at 7 cents each.<br><BR>\n" );
document.write( "CHECK: 23(13)+46(7) = 299+322 = 621<br><BR>\n" );
document.write( "  </SPAN>\n" );
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