Anomalous Cancellation

Number Theory Level 5

$\large\dfrac{1\not{5}}{\not{5}7}\ne\dfrac{1}{7}$

When teaching children how to reduce fractions, many of them will make mistakes like the one shown above. However, there are a few situations in which making this error does yield the correct answer, that is, $\frac{\overline{AB}}{\overline{BC}} = \frac{A}{C}$ . For how many fractions would this method of simplification work?

Note: $\overline{AB}$ simply denotes the decimal expansion of a two-digit number with $A$ and $B$ as the digits.