document.write( "Question 696107: A guy is her mom's age with digits reversed (e.g. 34 and 43). A year ago he was half her mom's age. How old are they now? \n" ); document.write( "

Algebra.Com's Answer #428820 by KMST(5328) $\"\"$ $\"About$

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THE ALGEBRA WAY TO SOLVE IT:
\n" ); document.write( "Setting up:
\n" ); document.write( " $\"a\"$ = first (tens) digit of that guy's age.
\n" ); document.write( " $\"b\"$ = second (ones) digit of that guy's age.
\n" ); document.write( "The guy's age is $\"10a%2Bb\"$ .
\n" ); document.write( "His mother's age is $\"10b%2Ba\"$ .
\n" ); document.write( "Last year, the guy was $\"10a%2Bb-1\"$
\n" ); document.write( "and his mother was $\"10b%2Ba-1\"$ ,
\n" ); document.write( "which was twice the guy's age, so
\n" ); document.write( " $\"10b%2Ba-1=2%2810a%2Bb-1%29\"$
\n" ); document.write( "
\n" ); document.write( "Solving:
\n" ); document.write( " $\"10b%2Ba-1=2%2810a%2Bb-1%29\"$ --> $\"10b%2Ba-1=20a%2B2b-2\"$ --> $\"10b%2Ba-1%2B2-a-2b=20a%2B2b-2%2B2-a-2b\"$ --> $\"highlight%288b%2B1=19a%29\"$
\n" ); document.write( "At this point we could start trying values for $\"a\"$ ,
\n" ); document.write( "knowing that it is a small odd number.
\n" ); document.write( "It must be odd to make $\"19a\"$ equal to the obviously odd $\"8b%2B1\"$ .
\n" ); document.write( "It is small because it is the first digit of the son's age,
\n" ); document.write( "and the mother's is more than twice that, but no more than 99.
\n" ); document.write( "
\n" ); document.write( "Otherwise, $\"8b%2B1=19a\"$ <--> $\"8b%2B1=16a%2B3a\"$
\n" ); document.write( "We see that $\"3a\"$ must be a multiple of 8 plus 1.
\n" ); document.write( " $\"3a=8n%2B1\"$
\n" ); document.write( "At this point, we should see $\"n=1\"$ with $\"highlight%28a=3%29\"$ as an obvious answer.
\n" ); document.write( "
\n" ); document.write( "(Otherwise, we could try $\"n\"$ values starting with $\"n=1\"$ to find that only $\"n=1\"$ works,
\n" ); document.write( "because $\"n=2\"$ and $\"n=3\"$ would yield fractional values for $\"a\"$ ,
\n" ); document.write( "and $\"n%3E=4\"$ would yield $\"a%3E=11\"$ ).
\n" ); document.write( "
\n" ); document.write( "Substituting into $\"highlight%288b%2B1=19a%29\"$ , we get
\n" ); document.write( " $\"8b%2B1=19%2A3\"$ --> $\"8b%2B1=57\"$ --> $\"8b%2B1-1=57-1\"$ --> $\"8b=56\"$ --> $\"8b%2Fb=56%2Fb\"$ --> $\"highlight%28b=7%29\"$
\n" ); document.write( "The guy is $\"highlight%2837%29\"$ and his mom is $\"highlight%2873%29\"$ . \n" ); document.write( "

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