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My first impression of this is that the numbers must be very small. So I try the smallest possible: Carl is 2 years old and Ginger, half his age, is then 1. But - “if Carl were 2 years younger” would make him 0 years old, so I abandon this.
\n" ); document.write( "Then I try the next possibility, working only with whole numbers. I set Carl’s age at 4, making Ginger 2. Then “if Carl were 2 years younger” - would make him 2. And if “Ginger were 3 years older” - would make Ginger 5. Voila! The difference is then 3 between their ages.
\n" ); document.write( "Answer: Carl is 4, Ginger is 2.\r
\n" ); document.write( "\n" ); document.write( "One sometimes shrinks from solving problems in this “dumb” way, i e by trial and error. However, if this produces a correct answer, why not?\r
\n" ); document.write( "\n" ); document.write( "A more “mathematical” way of solving this, calling Carl’s age c and Ginger’s age g, gives the two small equations below.
\n" ); document.write( " 1: c = 2g
\n" ); document.write( " 2: (g+3) - (c-2) = 3
\n" ); document.write( "1 and 2: (g+3) - (2g-2) = 3
\n" ); document.write( "=> -g + 5 = 3
\n" ); document.write( "=> g = 2
\n" ); document.write( "1: c = 2*2 = 4\r
\n" ); document.write( "\n" ); document.write( "Now, the acute observer will notice that I have “forgotten” another possibility for equation number 2: (c-2) - (g+3) = 3. Plugging in the value for c I get:
\n" ); document.write( "(2g-2) - (g+3) = 3
\n" ); document.write( "=> g - 5 = 3
\n" ); document.write( "=> g = 8
\n" ); document.write( "1: c = 2*8 = 16 (!)\r
\n" ); document.write( "\n" ); document.write( "This shows that the more “mathematical” way of solving this is better, since it reveals that there are two solutions to the problem. \n" ); document.write( "