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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?

user47860
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    Very often in probability and statistics, we will refer to the probability of an event occurring and we will refer to the odds of an event occurring. Probability and odds are related but are not the same. The probability of a fair coin being flipped heads is $\frac{1}{2}$. The odds that a fair coin is flipped as heads is $1:1$ (both for and against). They both convey the same information, but are written in different ways. Outside of probability and statistics, the uses of $:$ can and will vary, but generally a definition will be provided. – JMoravitz Apr 09 '18 at 22:47

4 Answers4

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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. $1:2 = \frac{1}{2}$.

Alex Jones
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    In several parts of the world, ":" is the conventional division symbol taught in school, rather than "$\div$" or "$/$". So your last case might not be as rare as you seem to imply. It's what I teach to my students in Norway. – Arthur Apr 09 '18 at 22:46
  • Thanks for the info, I've updated the answer to reflect this. – Alex Jones Apr 09 '18 at 22:51
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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.

PhysicsMathsLove
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    But again, here we have $\frac{z}{3} = \frac{1}{2} \times \frac{2z}{3}$, which is what I said in bullet point 1. This needs to be more accurate as it's not what the first bullet point says. – Shaun Apr 09 '18 at 22:49
  • @Shaun Well it is, since if $x = \frac{z}{3}$ and $y=\frac{2z}{3}$ are in the ratio $1:2$, then $\frac{x}{y} = \frac{\frac{z}{3}}{\frac{2z}{3}} = \frac{1}{2}$. I have just re-arranged that and left it to the OP. – PhysicsMathsLove Apr 09 '18 at 22:54
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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.

gen-ℤ ready to perish
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$1:2$ 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.

Arthur
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