# Algorithms, continued

Frank Noschese (whose TEDxNYED talk is totally worth your time, by the way) gave me +K for algorithms on Klout the other day. This was presumably in return for the +K he received from me for textbooks a few days prior.

To celebrate this achievement (and to demonstrate that it is well deserved), I want to share the common numerator division algorithm.

Here’s how it goes.

$\frac{a}{b}\div\frac{a}{c}=\frac{c}{b}$

In words, if two fractions have a common numerator, then their quotient is a fraction whose numerator is the denominator of the divisor and whose denominator is the denominator of the dividend.

As an example, $\frac{2}{3}\div\frac{2}{6}=\frac{6}{3}=2$

Two thirds compared to two sixths

Or, if you start with different numerators, $\frac{2}{3}\div\frac{1}{4}=\frac{2}{3}\div\frac{2}{8}=\frac{8}{3}$

Two thirds compared to one fourth

Two thirds compared to two eighths

Now, unlike the common numerator algorithm for adding fractions, I can actually describe what’s going on here.

Think about a fraction as a part-whole relationship. When we have two fractions with common numerators but different denominators (and assuming each is a fraction of the same-sized whole), then each fraction represents the same number of pieces, but these pieces are different sizes.

The quotient, then, is the ratio of the sizes of these pieces. In the first example, thirds are twice as big as sixths. In the second example, thirds are two-and-two-thirds times as big as eighths.

The common denominator algorithm for dividing fractions emphasizes a measurement interpretation of division (the quotient answers the question, “How many of this are in that?”)

The invert-and-multiply algorithm emphasizes formal numerical/algebraic relationships (the quotient is obtained by using the properties of multiplicative inverses).

The common numerator algorithm, though? It emphasizes that fractions and quotients are ratios. The quotient is a multiplicative comparison of the sizes of the pieces.