document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=947653>Question 947653</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the digits of a three digit natural number are in A.p and their sum is 18.the number obtained by reversing the digits is 594 less than the original number.find the original number. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #578639 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=578639\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> WITHOUT ALGEBRA:<BR>\n" );
document.write( "The average of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > consecutive terms (or any odd number of terms) of an A.P. (arithmetic progression, or arithmetic sequence, depending on where you live) is equal to the middle term.<BR>\n" );
document.write( "So, the middle digit is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2F3=6 ALIGN=MIDDLE  ALT=\"18%2F3=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since reversing the digits makes a lesser number (594 less than the original number),<BR>\n" );
document.write( "the digits decrease from left to right.<BR>\n" );
document.write( "(They do not stay the same or increase).<BR>\n" );
document.write( "The first digit, being greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "could be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "If the first digit were <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=6%2B3 ALIGN=MIDDLE  ALT=\"9=6%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the last digit would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3=6-3 ALIGN=MIDDLE  ALT=\"3=6-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the 3-digit number would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=963 ALIGN=MIDDLE  ALT=\"963\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then, reversing the digits we would get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=369 ALIGN=MIDDLE  ALT=\"369\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the difference would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=963-369=594 ALIGN=MIDDLE  ALT=\"963-369=594\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "If the first digit were <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8=6%2B2 ALIGN=MIDDLE  ALT=\"8=6%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the last digit would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4=6-2 ALIGN=MIDDLE  ALT=\"4=6-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If the first digit were <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7=6%2B1 ALIGN=MIDDLE  ALT=\"7=6%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the last digit would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=6-1 ALIGN=MIDDLE  ALT=\"5=6-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In either case the difference between the number obtained by reversing the digits and the original number would be less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=594 ALIGN=MIDDLE  ALT=\"594\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
document.write( "\n" );
document.write( "Som the original number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28963%29 ALIGN=MIDDLE  ALT=\"highlight%28963%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . There is no other choice.<BR>\n" );
document.write( " <BR>\n" );
document.write( "USING ALGEBRA TO MAKE IT MORE COMPLICATED:<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > be the first digit and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > be the common difference.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bd ALIGN=MIDDLE  ALT=\"a%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the second digit, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B2d ALIGN=MIDDLE  ALT=\"a%2B2d\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the third digit.<BR>\n" );
document.write( "so, \"their sum is 18\" translates as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B%28a%2Bd%29%2B%28a%2B2d%29=18 ALIGN=MIDDLE  ALT=\"a%2B%28a%2Bd%29%2B%28a%2B2d%29=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a%2B3d=18 ALIGN=MIDDLE  ALT=\"3a%2B3d=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28a%2Bd%29=18 ALIGN=MIDDLE  ALT=\"3%28a%2Bd%29=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bd=18%2F3%29%29%29+---%3E+%7B%7B%7Ba%2Bd=6%29%29%29%0D%0AThe+value+of+the+original+number+is%0D%0A%7B%7B%7B100a%2B10%28a%2Bd%29%2B%28a%2B2d%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The value of the number obtained by reversing the digits is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%28a%2B2d%29%2B10%28a%2Bd%29%2Ba ALIGN=MIDDLE  ALT=\"100%28a%2B2d%29%2B10%28a%2Bd%29%2Ba\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since the value of the number obtained by reversing the digits is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=594 ALIGN=MIDDLE  ALT=\"594\"   DISABLED_style= \"max-width:90%; height:auto;\" > less than the value of the original number,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%28a%2B2d%29%2B10%28a%2Bd%29%2Ba=100a%2B10%28a%2Bd%29%2B%28a%2B2d%29-594 ALIGN=MIDDLE  ALT=\"100%28a%2B2d%29%2B10%28a%2Bd%29%2Ba=100a%2B10%28a%2Bd%29%2B%28a%2B2d%29-594\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100a%2B200d%2B10a%2B10d%2Ba=100a%2B10a%2B10d%2Ba%2B2d-594 ALIGN=MIDDLE  ALT=\"100a%2B200d%2B10a%2B10d%2Ba=100a%2B10a%2B10d%2Ba%2B2d-594\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28100%2B10%2B1%29a%2B%28200%2B10%29d=%28100%2B10%2B1%29a%2B%2810%2B2%29d-594 ALIGN=MIDDLE  ALT=\"%28100%2B10%2B1%29a%2B%28200%2B10%29d=%28100%2B10%2B1%29a%2B%2810%2B2%29d-594\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=210d=12d-594 ALIGN=MIDDLE  ALT=\"210d=12d-594\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=210d-12d=-594 ALIGN=MIDDLE  ALT=\"210d-12d=-594\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=198d=-594 ALIGN=MIDDLE  ALT=\"198d=-594\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-594%2F198 ALIGN=MIDDLE  ALT=\"d=-594%2F198\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-3 ALIGN=MIDDLE  ALT=\"d=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting that value into the other equation, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bd=6 ALIGN=MIDDLE  ALT=\"a%2Bd=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B%28-3%29=6 ALIGN=MIDDLE  ALT=\"a%2B%28-3%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-3=6 ALIGN=MIDDLE  ALT=\"a-3=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=6%2B3 ALIGN=MIDDLE  ALT=\"a=6%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=9 ALIGN=MIDDLE  ALT=\"a=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and having the second digit, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bd=6 ALIGN=MIDDLE  ALT=\"a%2Bd=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "the first digit, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=9 ALIGN=MIDDLE  ALT=\"a=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the common difference <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-3 ALIGN=MIDDLE  ALT=\"d=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we can find the third digit,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B2d=9%2B2%28-3%29=9-6=3 ALIGN=MIDDLE  ALT=\"a%2B2d=9%2B2%28-3%29=9-6=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and then we know that the original number was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28963%29 ALIGN=MIDDLE  ALT=\"highlight%28963%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );