Simplifying Fractions

$\frac{12}{30}$ has a simpler name. 12 and 30 are both divisible by 6, their greatest common factor. Divide the top and the bottom by 6: 12 divided by 6 is 2, and 30 divided by 6 is 5. So $\frac{12}{30}$ is $\frac{2}{5}$.

Why is that allowed? It is the last lesson run backwards: multiply the top and the bottom of $\frac{2}{5}$ by 6 and you are back at $\frac{12}{30}$. Dividing both by the same number makes the pieces bigger and fewer, and the amount stays exactly the same.

A fraction is in lowest terms when its top and bottom share no factor except 1. 2 and 5 share none, so $\frac{2}{5}$ is finished. Dividing by the GCF gets there in one step, because it removes every shared factor at once. The board accepts a line like $\frac{6}{15}$, since its value is right, but it will not call the problem finished until the fraction is in lowest terms.