document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204445>Question 1204445</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangular room is completely tiled by 1-foot square tiles. All the adjacent to a door or wall are purple, and the rest of the tiles are white. If exactly 2/7 of the tiles are purple, then what is the smallest possible area of the room, in square feet?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840709 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13215)</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=840709\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let L and W be the length and width of the room.<br><BR>\n" );
document.write( "Then the total number of tiles is LW.<br><BR>\n" );
document.write( "And the number of purple tiles is 2L+2W-4.  The -4 is because 2L+2W together counts the 4 corner squares twice.<br><BR>\n" );
document.write( "The purple tiles are 2/7 of the total:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282L%2B2W-4%29%2F%28LW%29=2%2F7 ALIGN=MIDDLE  ALT=\"%282L%2B2W-4%29%2F%28LW%29=2%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28L%2BW-2%29%2F%28LW%29=1%2F7 ALIGN=MIDDLE  ALT=\"%28L%2BW-2%29%2F%28LW%29=1%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=LW=7L%2B7W-14 ALIGN=MIDDLE  ALT=\"LW=7L%2B7W-14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This is a Diophantine equation -- two variables with only one equation; but the number of solutions is finite because the variables are non-negative integers.<br><BR>\n" );
document.write( "Here is a solution to the problem using standard methods for solving linear Diophantine equations.<br><BR>\n" );
document.write( "Solve the equation for one variable in terms of the other.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=LW-7L=7W-14 ALIGN=MIDDLE  ALT=\"LW-7L=7W-14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28W-7%29=7W-14 ALIGN=MIDDLE  ALT=\"L%28W-7%29=7W-14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=%287W-14%29%2F%28W-7%29 ALIGN=MIDDLE  ALT=\"L=%287W-14%29%2F%28W-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Make it easier to find integer solutions by performing the division on the right and expressing the result as quotient and remainder.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=%287W-14%29%2F%28W-7%29=%287W-49%2B35%29%2F%28W-7%29=%287W-49%29%2F%28W-7%29%2B35%2F%28W-7%29=7%2B35%2F%28W-7%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In that form of the equation, 7 is an integer, and L has to be an integer; that means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=35%2F%28W-7%29 ALIGN=MIDDLE  ALT=\"35%2F%28W-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has to be an integer.  And that means (W-7) is a factor of 35.<br><BR>\n" );
document.write( "The factors of 35 are 1, 5, 7, and 35.  We are to find the minimum value of the product LW.<br><BR>\n" );
document.write( "To do that, we find the product LW for each of the possible values of (W-7).<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "  W-7  W   L   LW\r\n" );
document.write( " -----------------\r\n" );
document.write( "   1   8  42  336\r\n" );
document.write( "   5  12  14  168\r\n" );
document.write( "   7  14  12  168\r\n" );
document.write( "  35  42   8  336</PRE><BR>\n" );
document.write( "Note in the table that the dimensions and area of the room in the second half of the table repeat those in the first half.  That is because the original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=LW=7L%2B7W%2B14 ALIGN=MIDDLE  ALT=\"LW=7L%2B7W%2B14\"   DISABLED_style= \"max-width:90%; height:auto;\" > is symmetric in the two variables L and W.  If we note that early in our work, then we know we don't have to complete the whole table.<br><BR>\n" );
document.write( "ANSWER: The smallest possible area of the room in square feet is 168.<br><BR>\n" );
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