Explanation:

Here is the breakdown of the problem:

- Present Age of Mother: 42 years.

- After 12 Years, mother's age will be 54 years (42 + 12).

- Father's Age After 12 Years: Let it be \( F + 12 \).

- Given Ratio after 12 years: (Father's Age) : (Mother's Age) = 10 : 9.

Thus, $$(F + 12) / 54 = 10 / 9$$. Solving gives:

$$F + 12 = 60$$. Therefore, \( F = 48 \).

- Father's Present Age: 48 years.

- Father's Age Relation: Father’s age is four times daughter's age.

Hence, daughter's present age is \( 48 / 4 = 12 \).

- Son and Daughter's Present Age Ratio: 5 : 6.

$$(Son's Age) / 12 = 5 / 6$$. Solving gives:

Son's age = 10 years.

- Sum of Son and Daughter's Ages after 12 Years:

Son's age + 12 = 22 years,

Daughter's age + 12 = 24 years,

Sum = 22 + 24 = 46 years.

Therefore, the sum of the ages of the son and daughter after 12 years is 46 years.

Option: 2, 46 years

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