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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?

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    $\begingroup$ From a Student: Well, why not use the Euclidean Algorithim to find $\gcd(a,b)$? $\endgroup$ – S.C.B. Apr 5 '16 at 13:27
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If you have the fraction $\displaystyle\frac ab$ then just find the $\gcd(a,b)$ and then cancel.

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    $\begingroup$ This seems like a good idea. I did not realize the Euclidean algorithm was actually an algorithm; I thought it was just a theorem. $\endgroup$ – goblin Apr 5 '16 at 13:36
  • $\begingroup$ @goblin Sir,in which class are the students you teach? $\endgroup$ – tatan Apr 5 '16 at 13:39
  • $\begingroup$ They're usually in Years 10 to 12, but some of the weaker ones need a brush up on fractions and basic arithmetic. Its remarkable how many Year 11 students can't add fractions, for example. $\endgroup$ – goblin Apr 5 '16 at 13:40
  • $\begingroup$ @goblin In Indian education system (as in my case) students are usually taught gcf and lcm around $9-10$ years.... $\endgroup$ – tatan Apr 5 '16 at 13:43
  • $\begingroup$ Its possible that they're taught to 9-10 year olds here in Australia too, but that I just wasn't listening at the time. Perhaps here it is one of those things that is usually taught, but never tested. $\endgroup$ – goblin Apr 5 '16 at 13:48

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