The lucky cancellation

Statement

Trying to simplify $\dfrac{19}{95}$, a student cancels the $9$ from numerator and denominator, obtaining $\dfrac{1}{5}$, which by luck is the correct answer. Trying the same trick on another fraction, they get $\dfrac{1}{4}$, again correct. Which digit did they cancel this time?

Solution

We need a fraction of two two-digit numbers that, after cancelling the repeated digit, gives $\dfrac14$. It is $\frac{16}{64} = \frac{1}{4},$ and cancelling the $6$ leaves exactly $\dfrac{1}{4}$. The cancelled digit is $\boxed{6}$.