A man age is (3p+2q) years and his only son’s age is ‘p’ years while his only daughter’s age is ‘q’ years. Man’s son is three years elder than her sister. If man’s wife is five years younger than her husband who is 34 years old, then find average age of the family (in years).

The correct option is 19.0.

Let us break down the problem step-by-step to find the average age of the family.

Step 1: Understand the given ages in terms of variables

Man's age = $3p+2q$ years

Son's age = $p$ years

Daughter's age = $q$ years

Step 2: Formulate equations from the given conditions

1. The man's son is three years elder than his daughter (sister):

$p-q=3$     --- (Equation 1)

2. The man is 34 years old:

$3p+2q=34$     --- (Equation 2)

Step 3: Solve for variables $p$ and $q$

From Equation 1, we can express $p$ in terms of $q$:

$p=q+3$

Substitute $p=q+3$ into Equation 2:

$3(q+3)+2q=34$

$3q+9+2q=34$

$5q+9=34$

$5q=34-9$

$5q=25$

$q=5$

Now find $p$:

$p=5+3=8$

Step 4: Determine the age of each family member

• Man's age = 34 years

• Son's age = $p=8$ years

• Daughter's age = $q=5$ years

• Wife's age = five years younger than her husband = $34-5=29$ years

Step 5: Calculate the average age of the family

The family consists of 4 members: Man, Wife, Son, and Daughter.

Total age of the family = $34+29+8+5=76$ years

Average age = $\frac{76}{4}=19.0$ years

Thus, the average age of the family is 19.0 years.