UPSC NDA Math Model Paper 9

Let's say that there are two sets with number of elements n1 and n2 and averages X1 and X2.
The sum of the values of both the sets will be n1X1+n2X2 and the number of elements combined will be n1+n2.
Since X1<X2, we can say that:
(n1+n2)X1<n1X1+n2X2< (n1+n2)X2

⇒

(n1+n2)X1

n1+n2

<

n1X1+n2X2

n1+n2

<

(n1+n2)X2

n1+n2

X1<X<X2
∴ The combined average of two sets always lies between the individual averages, i.e.
X1<X<X2