Problem on ages

Question

The ratio of the ages of a husband and his wife when they got married was 6 : 5. 4 years and 6 years after their marriage they
had their 1st and 2nd children. The sum of the present ages of the husband and wife is 6.4 times the sum of the present ages of
their children. The average age of the family at present is 18.5 years. Find the ratio of the ages of the husband and wife when
their second child was born.

Answer 1

The sum of the present ages of the 4 members in the family is 18.5 × 4 = 74. Since the sum of the present ages of the husband and
wife is 6.4 times the sum of the present ages of their children, the sum of the ages of the husband and wife must be 64 while the
sum of the ages of the children must be 10. Since one child is 2 years older than the other one, the present ages of the children are
4 and 6. Since the 2nd child was born 4 years ago, the sum of the ages of the husband and wife then was 64 – 8 = 56. Since the
2nd child was born 6 years after marriage, the sum of the ages of the husband and wife when they got married was 56 – 12 = 44
and their ages were in the ratio 6 : 5 ⇒ their ages were 24 and 20. When the 2nd child was born, their ages had increased by 6
years each, i.e., their ages were 30 and 26. Thus the required ratio is 15 : 13

Answer 2

Let the age of the husband and wife be 6x and 5x when they got married.

Let present age of husband be H, wife be W, and 1st and 2nd child be a,b.

Average age of the family be 18.5 years. then,

H+W+a+b = 18.5*4 = 74 ----------  1

Also, The sum of the present ages of the husband and wife is 6.4 times the sum of the present ages of their children. Hence,

H+W = 6.4(a+b) , on substituting in equation no. 1, you get a+b = 10.

also, if a+b = 10 then H+W = 64 ------------------------------------------  2

the difference between first and second child is of 2 years therefore a = 6 , b = 4 years old.

It means b is younger and a is older, this implies their marriage must have happened before 10 years ago. why?

present age of older child(a or $1^{st}$) = 6 and he was born after 4 years from their marriage, or present age of younger child(b or $2^{nd}$) 4 years and he was born after 6 years from their marriage.

Hence,

H-10+W-10 = 6x + 5x (sum of the age 10 years before from current age = when they got married)

H+W - 20 = 11x

64-20 = 11x (From II)

x = 44/11 = 4

Hence when they married husband's age was 6x = 6*4 = 24 and wife age = 5x = 5*4=20.

They have asked ratio of their ages after second child was born i.e. after 6 years. Hence new ratio will be -

$\frac{H_{new}}{W_{new}} = \frac{24+6}{20+6} = \frac{30}{26} = \frac{15}{13}$ .