The sum of the ages of father and his son is 45 years. Five years ago, the product of their ages was 34. The age of father is:

39 years

The best way to find answer:

Check answer options and satisfy two equations:

a) F + S = 45 and b) (F - 5)(S - 5) = 34

Now,

     F    :   S          (F-5) × (S-5)

a) 40   :    5      ⇒    35   ×   1      ≠ 34

b) 15   :    30    ⇒    Not possible

c) 39   :    6        ⇒      34   ×   1        = 34

d) 38   :    7      ⇒    33  ×    3      ≠ 34

Hence, Father's age is = 39 years

Alternate way:

Father's age = F

Son's age = S

Five years ago father's age = F -5

Five years ago son's age = S -5

ATQ,

F + S = 45 years

(F - 5) × (S - 5) = 34

FS - 5F -5S + 25 = 34

FS -5 (F + S) = 9

FS - 5(45) = 9

FS = 234

F (45 - F) = 234  .......(i)

Put and check answer options:

From option (3)

put F = 39 years in equation (i)

39 (45 - 39) = 234

234 = 234

Hence, Father's age = 39 years.