document.write( "Question 295272: Exactly 100 people live in a village. The oldest person in the village was born in 1900 and everybody in the village was born in a different year but all on January 1st. In 1999, the sum of the digits in Julie's birth year was equal to her age. How old was she?
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #212887 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
Exactly 100 people live in a village. The oldest person in the village was born in 1900 and everybody in the village was born in a different year but all on January 1st. In 1999, the sum of the digits in Julie's birth year was equal to her age. How old was she?
\n" ); document.write( "

\n" );
document.write( "Suppose Julie's was born in the year \"19tu\", then the last
\n" );
document.write( "two digits of her birth year is 10t+u, and her age in 
\n" );
document.write( "1999 was therefore \"99-%2810t%2Bu%29\". 
\n" );
document.write( "The sum of the digits of her birth year is \"1%2B9%2Bt%2Bu\"
\n" );
document.write( "Since her age = the sum of the digits of her birth year, then
\n" );
document.write( "\"99-%2810t%2Bu%29=1%2B9%2Bt%2Bu\"
\n" );
document.write( "\"99-10t-u=10%2Bt%2Bu\"
\n" );
document.write( "\"89=11t%2B2u\"
\n" );
document.write( "The least coefficient in absolute value is 2, so
\n" );
document.write( "we write all other integers in that equation in terms of
\n" );
document.write( "their nearest multiple of 2:
\n" );
document.write( "\"88%2B1=%2810%2B1%29t%2B2u\"
\n" );
document.write( "\"88%2B1=10t%2Bt%2B2u\"
\n" );
document.write( "We divide through by 2:
\n" );
document.write( "\"88%2F2%2B1%2F2=%2810t%29%2F2%2Bt%2F2%2B%282u%29%2F2\"
\n" );
document.write( "\"44%2B1%2F2=5t%2Bt%2F2%2Bu\"
\n" );
document.write( "Isolate the fractions on the right:
\n" );
document.write( "\"44-5t-u=t%2F2-1%2F2\"
\n" );
document.write( "Since the left side is an integer, the right side is too.
\n" );
document.write( "Let that integer be A. So both sides = A:
\n" );
document.write( "\"system%28t%2F2-1%2F2=A%2C44-5t-u=A%29\".
\n" );
document.write( "Clear the first equation of fractions:
\n" );
document.write( "\"t-1=2A\"
\n" );
document.write( "\"t=2A%2B1\"
\n" );
document.write( "Substituting in
\n" );
document.write( "\"44-5t-u=A\"
\n" );
document.write( "\"44-5%282A%2B1%29-u=A\"
\n" );
document.write( "\"44-10A-5-u=A\"
\n" );
document.write( "\"39-11A=u\"
\n" );
document.write( "\"u=39-11A\"
\n" );
document.write( "u is a digit, so
\n" );
document.write( "\"0%3C=u%3C=9\"
\n" );
document.write( "Substituting \"39-11A\" for u
\n" );
document.write( "\"0%3C=39-11A%3C=9\"
\n" );
document.write( "\"-39%3C=-11A%3C=-30\"
\n" );
document.write( "\"%28-39%29%2F%28-11%29%3E=A%3E=%28-30%29%2F%28-11%29\"
\n" );
document.write( "\"3%266%2F11%3E=A%3E=2%268%2F11\"
\n" );
document.write( "Since A is an integer it can only be 3,
\n" );
document.write( "for 3 is the only integer in that interval.
\n" );
document.write( "So \"A=3\"
\n" );
document.write( "\"t=2A%2B1\"
\n" );
document.write( "\"t=2%283%29%2B1\"
\n" );
document.write( "\"t=6%2B1\"
\n" );
document.write( "\"t=7\"
\n" );
document.write( "\"u=39-11A\"
\n" );
document.write( "\"u=39-11%283%29\"
\n" );
document.write( "\"u=39-33\"
\n" );
document.write( "\"u=6\"
\n" );
document.write( "Therefore she was born in 1976 and therefore in 1999, she was 23,
\n" );
document.write( "and indeed the sum of the digits of her birth year is 1+9+7+6=23,
\n" );
document.write( "which is her age.
\n" );
document.write( "Edwin
\n" );
document.write( "    \n" );
document.write( "
\n" );