Dividing by 2 numbers at once, what is the answer?

Solution 1:

In the USA and Canada, perhaps other places too, we use the BEDMAS system for interpreting expressions. First, we calculate the result of bracketed sub-expressions. Then exponentiation. Then from left to right, resolving divisions or multiplications as they show up. Then from left to right, resolving additions and subtractions as they show up.

So in the BEDMAS convention, we would deal with your 4/1/5 by doing the divisions left-to-right. 4/5.

Edit:

As the comments point out, some use PEDMAS instead. P for parenthesis instead of brackets ;)

Solution 2:

The double fraction $a/b/c$ should not be allowed because it does not make sense : which operation should we do first ? To make sense, you must add parenthesis (this way, you specify which operation is done first). There are two ways to do it :

$$\frac{\Big( \displaystyle \frac{a}{b} \Big)}{c} = \frac{a}{bc} = (a/b)/c $$

and

$$\frac{a}{\Big( \displaystyle \frac{b}{c} \Big)} = \frac{ac}{b} = a/(b/c). $$

Therefore, $\displaystyle \frac{\Big( \displaystyle \frac{a}{b} \Big)}{c} \neq \frac{a}{\Big( \displaystyle \frac{b}{c} \Big)}$ in general. Be careful to the way you put parenthesis. In general :

$$\frac{\Big( \displaystyle \frac{a}{b} \Big)}{\Big( \displaystyle \color{red}{\frac{c}{d}} \Big)} = \frac{a}{b} \times \color{\red}{\frac{d}{c}} = \frac{ad}{bc}.$$