In a room, there are n persons whose average height is 160 cm. If m more persons, whose average height is 172 cm, enter the room, then the average height of all persons in the room becomes 164 cm. Then m : n is

Initially, there are $$n$$ persons whose average height is 160 cm.

So, sum of all their heights =$$160n$$

Also, $$m$$ more persons enter the room with average height 172 cm.

Now, sum of their heights =$$172m$$

So, total height of all members =$$160n+172m$$

And total no. of persons are $$m+n$$

The final average height is given as 164

So, $$\dfrac{160n+172m}{m+n}=164$$

or, $$160n+172m=164\left(m+n\right)$$

or, $$160n+172m=164m+164n$$

or, $$8m=4n$$

or, $$m:n=1:2$$