Is there a better way of simplifying fractions?

Question

The way I teach my students to simplify fractions is to first write the numerator and denominator as a product of prime numbers, and then cancel.

For instance: $$\frac{15}{20} = \frac{3 \times 5}{2 \times 2 \times 5} = \frac{3}{2 \times 2} = \frac{3}{4}$$

This ensures that there aren't any "hidden" cancellations remaining that they've missed.

But sometimes, this is really inefficient. For instance:

$$\frac{42}{2} = \frac{2 \times 3 \times 7}{2} = 3 \times 7 = 21$$

The part where we factorized $21$ into $3 \times 7$ was obviously a complete waste of time: they should just have written

$$\frac{42}{2} = \frac{2 \times 21}{2} = 21$$

I'd like to find a better way of doing this, so I can be a better math tutor.

Question. Is there a better way of simplifying fractions, that still ensures that the final result is in simplest form?

Answer 1

If you have the fraction $\displaystyle\frac ab$ then just find the $\gcd(a,b)$ and then cancel.