document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169861>Question 1169861</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A number of two digits exceeds twice the sum of its digits by 26. If the<BR>\n" );
document.write( "digits are interchanged, the resulting number exceeds the given number<BR>\n" );
document.write( "by 18. Find the number. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #794693 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13216)</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=794693\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "A solution using formal algebra....<br><BR>\n" );
document.write( "x = tens digit<BR>\n" );
document.write( "y = units digit<BR>\n" );
document.write( "10x+y = original number<BR>\n" );
document.write( "10y+x = original number with digits reversed<br><BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%2By+=+2%28x%2By%29%2B26 ALIGN=MIDDLE  ALT=\"10x%2By+=+2%28x%2By%29%2B26\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [the number exceeds twice the sum of its digits by 26]<br><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810y%2Bx%29-%2810x%2By%29+=+18 ALIGN=MIDDLE  ALT=\"%2810y%2Bx%29-%2810x%2By%29+=+18\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [the number with the digits reversed exceeds the given number by 18]<br><BR>\n" );
document.write( "Use (2) to find y in terms of x:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810y%2Bx%29-%2810x%2By%29+=+18 ALIGN=MIDDLE  ALT=\"%2810y%2Bx%29-%2810x%2By%29+=+18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9y-9x+=+18 ALIGN=MIDDLE  ALT=\"9y-9x+=+18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%28y-x%29+=+18 ALIGN=MIDDLE  ALT=\"9%28y-x%29+=+18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-x+=+2 ALIGN=MIDDLE  ALT=\"y-x+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%2B2 ALIGN=MIDDLE  ALT=\"y+=+x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Substitute \"x+2\" for \"y\" in (1):<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%2By+=+2%28x%2By%29%2B26 ALIGN=MIDDLE  ALT=\"10x%2By+=+2%28x%2By%29%2B26\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%2Bx%2B2+=+2%28x%2Bx%2B2%29%2B26 ALIGN=MIDDLE  ALT=\"10x%2Bx%2B2+=+2%28x%2Bx%2B2%29%2B26\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11x%2B2+=+4x%2B30 ALIGN=MIDDLE  ALT=\"11x%2B2+=+4x%2B30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x+=+28 ALIGN=MIDDLE  ALT=\"7x+=+28\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4 ALIGN=MIDDLE  ALT=\"x+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%2B2+=+6 ALIGN=MIDDLE  ALT=\"y+=+x%2B2+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: the number is 10x+y = 46<br><BR>\n" );
document.write( "The first calculation above shows that, when a 2-digit number is subtracted from the same number with the digits reversed, the result is always a multiple of 9.  If a formal algebraic solution is not required (for example, in a timed math contest where a quick solution is wanted), you can use this fact along with trial and error to quickly find the answer.<br><BR>\n" );
document.write( "Since the difference in this case is 18, we know the difference between the digits is 2; and since the number with the digits reversed is greater than the given number, we know the units digit is 2 more than the tens digit.<br><BR>\n" );
document.write( "So the possible 2-digit numbers are 13, 24, 35, 46, 57, 68, and 79.<br><BR>\n" );
document.write( "Then simply find which of those satisfies the condition that the number exceeds twice the sum of its digits by 26.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );