Jonathan Allan have answered correctly.
To get better understanding why the answer is 8,8,4,2
I give my version
1st : Sum of their multiplicative inverse is 1
All age possibility are
2 , 3 , 7 , 42 sums 54
2 , 3 , 8 , 24 sums 37
2 , 3 , 9 , 18 sums 32
2 , 3 , 10 , 15 sums 30
2 , 3 , 12 , 12 sums 29
2 , 4 , 5 , 20 sums 31
2 , 4 , 6 , 12 sums 24
2 , 4 , 8 , 8 sums 22
2 , 5 , 5 , 10 sums 22
2 , 6 , 6 , 6 sums 20
3 , 3 , 4 , 12 sums 22
3 , 3 , 6 , 6 sums 18
3 , 4 , 4 , 6 sums 17
4 , 4 , 4 , 4 sums 16
2nd : Sum of their age is your sister age
Because A said that knowing the total (from the age of her sister) did not help, we know that knowing the sum of the ages does not give a definitive answer; thus, there must be more than one solution with the same total.
Only three sets of possible ages add up to the same totals:
2 , 4 , 8 , 8 sums 22
2 , 5 , 5 , 10 sums 22
3 , 3 , 4 , 12 sums 22
3rd clue : 2 oldest are twin
A concludes that the correct solution is 8,8,4,2.