Dear Uncle Colin,

Why is dividing by a half the same as doubling?

- How Arithmetic Leverages Fractions

Hi, HALF, and thanks for your message!

I’m going to give a couple of reasons: first, the algebra of it, then the logic.

## Algebra

Let’s set this up as $\frac{x}{\br{\frac{1}{2}}} = y$ - you’re asking why $y=2x$.

The ugly thing here is the fraction on the bottom, so let’s multiply both sides by that:

$x = \frac{1}{2}y$

Now double both sides:

$2x = y$, as required. Magic!

## Logic

When you divide by a half, you’re asking “how many halves go evenly into this number?”

You know that 1 is made of two halves, 2 is made of four halves, and so on - your number $x$ is made up of $2x$ halves.

In general, if you ask how many $\frac{a}{b}$s go into $x$, you can reason that 1 is made up of $\frac{b}{a}$ of them, so $x$ is made up of $x$ lots of $\frac{b}{a}$.

Hope that helps!

- Uncle Colin