The average weight of $35$ apples is $50$ grams. If some apples of weight $40$ grams each were removed, then the average weight becomes $57.5$ grams. How many apples of weight $40$ grams were removed?

Total initial weight of $35$ apples $=35\times 50=1750$ grams.Let the number of apples removed be $n$.The weight of the removed apples $=40n$ grams.Remaining number of apples $=35-n$.Total weight of remaining apples $=1750-40n$.Given that the new average weight is $57.5$ grams:
$$\frac{1750-40n}{35-n}=57.5$$
$$1750-40n=57.5(35-n)$$
$$1750-40n=2012.5-57.5n$$
$$57.5n-40n=2012.5-1750$$
$$17.5n=262.5$$
$$n=\frac{262.5}{17.5}=15$$
Thus, $15$ apples were removed.