In a group of 150 people, $\frac{2}{5}$ are men , $\frac{1}{3}$ are women and the rest are children. The average of the women is $\frac{4}{5}$ of the average age of the men. The average age of the children is $\frac{1}{5}$ of the average age of the men. If the average age of the men is 50 years, then the average age of all the people in the group is:

36 years 

Number of man = \(\frac{2}{5}\) × 150 = 60

Number of Women = \(\frac{1}{3}\) × 150 = 50

Number of children = 150 - 60 - 50 = 40

Ratio of average age of women to men = 4 : 5

Ratio of average age of children to men = 1 : 5

Now , Women : Men : Children = 4 : 5 : 1

ATQ , Average age of men = 50

5R = 50

SO , Average age of women = 40 & average age of children = 10

Sum of age of 60 men = 60 × 50 = 3000

Sum of age of women = 50 ×  40 = 2000

Sum of age of children = 10 ×  40 = 400

Average of all = \(\frac{3000 + 2000 + 400}{5}\)

=  \(\frac{5400}{150}\)

= 36