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.

• A. 7:6
• B. 15:13
• C. 6:7
• D. None of these

Answer 1

The Correct Option is B

1. The husband and wife's ages when they got married were in the ratio of 6:5. 2. 4 years and 6 years after their marriage, they had their 1st and 2nd children. This means that the current ages of the children are 4 and 6. 3. The sum of the present ages of the husband, wife, and children is the average age of the family multiplied by the number of family members: \(18.5 \times 4 = 74\). This sum includes the ages of the husband, wife, and two children. 4. The sum of the present ages of the husband and wife is 6.4 times the sum of the present ages of their children. This can be expressed as an equation: \[H + W = 6.4 \times (4 + 6)\] \[H + W = 64\] 5. 6 years after marriage, they had their 2nd child. At that time, the sum of their ages (husband and wife) was \(64 - 6 = 58\). Since the 2nd child was born 6 years after marriage, the sum of their ages when they got married was \(58 - 12 = 46\), and their ages were in the ratio 6:5. 6. Let the husband's age at marriage be \(6x\) and the wife's age at marriage be \(5x\), where \(x\) is a positive integer. 7. When the 2nd child was born, their ages had increased by 6 years each, so their ages were \(6x + 6\) and \(5x + 6\). 8. Their ages at marriage added up to \(6x + 5x = 11x\), which is equal to 46 (the sum of their ages when their 2nd child was born). Solving for \(x\), we get \(x = 4\). 9. Using \(x = 4\), we find their ages at the birth of the 2nd child were 30 (husband) and 26 (wife). 10. The ratio of their ages when the 2nd child was born is \(30 : 26\), which simplifies to \(15 : 13\). So, the required ratio of the ages of the husband and wife when their second child was born is \(15 : 13\).

Answer 2

Together, the four family members' ages number 74, with an average age of 18.5 years old. 
The husband and wife's total age is 64, whereas the children's total age is 10, given that their combined age is 6.4 times that of their offspring. 
The children are currently 4 and 6 years old, respectively, because one of them is two years older than the other. 
The husband and wife's combined age at the time of the second child's birth was 56 years old. 
The husband and wife's combined age at marriage was 44, with a 6:5 age ratio, meaning that they were 24 and 20 years old, respectively, because the second kid was born six years after the marriage.
 After the birth of the second kid, the husband and wife were now thirty and twenty-six years old, respectively, having aged by six years. As a result, their age difference is 15:13.