Question

Average age of 6 sons of a family is 8 years. Average age of sons together with their parents is $22$ years. If the father is older than the mother by 8 years. The age of the mother (in years) is
$A.44$
$B.52$
$C.60$
$D.68$

Answer 1

Hint: Take the mother age as $x$. Then the father is $8$ years older than the mother. Then the average age of $6$ sons of a family is $8$ years. Average age of sons together with their parents is $22$ years. We need to find the sum of sons and sons with their parents. Subtract these two we get the age of parents. Then add the mother age and father age and equal the age of the parent we get the mother age.

Complete step-by-step answer:
Take the mother's age as $x$.
Therefore,
Mother age $=x$ years
Then the father is $8$ years older than the mother.
Therefore add $8$ years to mother age we get father age so,
Father age$=(x+8)$years
 Then we know the average age of $6$ sons of a family is $8$ years. We need to find the sum so that we need to multiply the number of sons and average age of sons. So we get,
$\Rightarrow $ Sum of $6$ sons $=8\times 6$
$\Rightarrow $ Sum of $6$ sons $=48$ years
Average age of sons together with their parents is $22$ years. We need to find the sum so that we need to multiply the average age of sons and average age of sons together with their parents . So we get,
$\Rightarrow $ Sum of $6$ sons and parents $=22\times 8$
$\Rightarrow $ Sum of $6$ sons and parents $=176$years
Then subtract these two years to get sum of the age of parents
$\Rightarrow $ Sum of age of parents $=176-48$
$\Rightarrow $ Sum of age of parents $=128$ years
Therefore we get,
$\Rightarrow x+x+8=128$
$\Rightarrow 2x+8=128$
Further we get,
$\Rightarrow 2x=128-8$
$\Rightarrow 2x=120$
Take this $2$RHS we get,
$\Rightarrow x=\dfrac{120}{2}$
$\Rightarrow x=60$
Therefore the mother age is $60$ years

Note: If the sum of the sons is given we need to find the average of that in this case we need to divide the sum by the number of sons. Sum is a preferred measure when we just need total value or total items. Average is more like a statistical measure and that is the mean mostly commonly used form of average.