Puzzles/Arithmetical puzzles/Three Daughters/Solution

The answer is 2, 2 and 9.

The product of their ages is 36, leaving the possibilities

{1,1,36}, {1,2,18}, {1,3,12}, {1,4,9}, {1,6,6}, {2,2,9}, {2,3,6} and {3,3,4}.

Knowing the sum of their ages is not sufficient to determine their ages, eliminating {1,1,36} (38), {1,2,18} (21), {1,3,12} (16), {1,4,9} (14), {2,3,6} (11) and {3,3,4} (10), leaving only

{1,6,6} and {2,2,9}.

Knowing that there exists an eldest daughter eliminates {1,6,6}, leaving only {2,2,9}.

How does knowing that there exists an eldest daughter eliminate {1,6,6}? One daughter can be older than another but both have the same age in years.

Hey, questioner, we're going by the same number of years. We are taking that both daughters in the thrown out {1, 6, 6} choice are EXACTLY 6-years-old.