document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192305>Question 1192305</A>: <I><SPAN STYLE=\"word-break: break-all;\"> On her birthday in 2007, Rachel's age is equal to twice the sum of the digits of the<BR>\n" );
document.write( "year in which she was born. How many possible years are there in which she could<BR>\n" );
document.write( "have been born? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849037 by </B><A HREF=/tutors/aboutme.mpl?userid=htmentor><B>htmentor(1343)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=htmentor><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=htmentor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849037\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, let's consider the possibility that Rachel was born in the 2000's. Then we can express the year she was born as 2000 + x. Thus, her age A = 2007 - (2000 + x)<BR>\n" );
document.write( "If twice the sum of the digits = her age, we can write A = 2(2 + x).<BR>\n" );
document.write( "Thus, we have two equations A = 4 + 2x and A = 7 - x.<BR>\n" );
document.write( "Multiplying the 2nd equation by 2, and adding (1) and (2) we get:<BR>\n" );
document.write( "3A = 18 -> A = 6.<BR>\n" );
document.write( "Now, we consider if Rachel is born in the year 19xy.<BR>\n" );
document.write( "Then we can express the year she was born as 1900 + 10x + y, and her age would be:<BR>\n" );
document.write( "A = 2007 - (1900 + 10x + y)<BR>\n" );
document.write( "And the equation for twice the sum of the digits is:<BR>\n" );
document.write( "2(1 + 9 + x + y) = A<BR>\n" );
document.write( "Simplifying we have 107 - 10x - y = A and 20 + 2x + 2y = A<BR>\n" );
document.write( "The expression for x in terms of A, which we get by multiplying (1) by 2 and adding, is: A = 78 - 6x<BR>\n" );
document.write( "Since x can take on the values [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], we have these possible ages: [78, 72, 66, 60, 54, 48, 42, 36, 30, 24].<BR>\n" );
document.write( "This gives possible birth years [1929, 1935, 1941, 1947, 1953, 1959, 1965, 1971, 1977, 1983].<BR>\n" );
document.write( "Of these, only 1959, 1965 and 1971 have the correct x value.<BR>\n" );
document.write( "So, the answer is 1959, 1965, 1971 and 2001. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );