A fraction that simplifies by deleting a digit

Text

Something curious happens with the fraction $\frac{19}{95}$: “deleting” the $9$ in the numerator and denominator gives $\frac{1}{5}$, which really is the reduced value of the fraction. Check that $\frac{19}{95}=\frac{1}{5}$ and explain why here the deletion accidentally gives the right answer.

Solution

Just factor the denominator: $95 = 5\cdot 19$, so $\frac{19}{95} = \frac{19}{5\cdot 19} = \frac{1}{5} = \boxed{0.2}.$ The “digit deletion” is only a numerical coincidence: it works for very few special fractions (for example $\frac{16}{64}$ or $\frac{26}{65}$) and in general is a wrong procedure.