Why does 1:2 = 1/2?

Question

My natural instincts tells me 1:2 = 1/3, because the LHS is 1 part of 3.

For example, if something was in the ratio 1:1, I would expect the LHS to be 1/2 of the whole, so why is 1:1 = 1?

Answer 1

I think you are getting some basic facts confused:

• If $x$ and $y$ are in the ratio $1:2$, then $\frac{x}{y} = \frac{1}{2}$.

• Whereas if $z$ is split in the ratio $1:2$, then the '$1$' part is equivalent to $\frac{z}{3}$ while the '$2$' part is equivalent to $\frac{2z}{3}$. But again, here we have $\frac{z}{3} = \frac{1}{2} \times \frac{2z}{3}$, which is loosely what I said in bullet point 1.

Answer 2

Usually, the $:$ symbol is defined as "odds against" in the following way

$$ a:b \equiv \frac{b}{a+b} $$

Using this definition, $1:2 = \frac{2}{1+2} = \frac{2}{3}$.

---

Sometimes, the $:$ symbol is defined as "odds in favor" in the following way

$$ a:b \equiv \frac{a}{a+b} $$

Using this definition, $1:2 = \frac{1}{1+2} = \frac{1}{3}$.

---

Very rarely (in the US, see Arthur's comment), the $:$ symbols is defines as "ratio" in the following way

$$ a:b \equiv \frac{a}{b} $$

Using this definition. $1:2 = \frac{1}{2}$.

Answer 3

You can’t really say that they’re “equal.” The colon is a notation; the ratio is a number.

At any rate, if the ratio of A-to-B is $a:b$, then the total number of items is $a+b$, so the fraction of A is $a/(a+b)$ and vice versa for B.

Answer 4

$1:2$ will always mean "half", one way or another. It never means anything else. That being said, there are different answers as to what is half of what.

If you're following a recipe, and you're supposed to mix ingredients A and B in a $1:2$ ratio, then the amount of A is half the amount of B. But at the same time, the amount of A will be one third the full amount of mixture. Similarly, if you and your brother get an allowance from your parents, and you split it $1:2$, then you get half of what your brother gets, but you get one third of the total allowance.

On the other hand, we say that the odds of getting a heads when you flip a coin is $1:2$. In this case, the number of heads is half the total number of throws, so there are equally many heads and tails.

Basically, whether the ratio is given between two parts of a whole, or between one part and the whole differs from context to context. You are more or less expected to know which one is used in any given problem.