# What number is between 1/2 and 8/9?

My answer to a question on Quora: What number is between 1/2 and 8/9?

John K Williamsson gave a good answer: for what he called the “dirty sum” of $$\frac{1}{2}$$ and $$\frac{8}{9}$$ (the mathematical term for that is mediant):

$$\displaystyle{\frac{1}{2} < \frac{1+8}{2+9} < \frac{8}{9}}$$

He suggests to use algebra to prove the mediant inequality: if $$a,b,c,d$$ are positive numbers and

$$\displaystyle{\frac{a}{b} < \frac{c}{d}}$$

then

$$\displaystyle{\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}}.$$

I wish to add that, at the primary school level, the mediant inequality does not need an algebraic proof, it is sufficiently self-evident.

Indeed consider fractions $$\frac{1}{2}$$ and $$\frac{8}{9}$$ as descriptions of real-life situations:

$$\frac{1}{2}$$ : $$2$$ children have $$1$$ bag of fruits.

$$\frac{8}{9}$$ : $$9$$ children have $$8$$ bags of fruits.

They come together and share equally: $$1 + 8$$ bags of fruit between $$2+9 = 11$$ kids, that is, they form the mediant:

$$\displaystyle{\frac{1+8}{2+9}}$$

In this sharing, which group of kids looses and and which one gains? Of course $$2$$ children with $$1$$ bag gain: they have $$\frac{1}{2}$$ bags per head, the other group comes with bigger share per head: $$\frac{8}{9}$$. For the same reason, kids in the second group lose.

I use an example with kids and bags of sweets in my lectures; here I replaced sweets by more politically correct fruits — perhaps I have to go further and use green vegetables in place of fruits. The original idea belonged to the great Israel Gelfand, and was stated in a more colorful language:

You can explain mathematics to everyone, even to drunkards. If you ask some people drinking vodka on a park bench, what is is bigger, $$\frac{2}{3}$$ or $$\frac{3}{4}$$, they will respond with expletives. But if you ask them, what is better, $$2$$ bottles of vodka for $$3$$ people or $$3$$ bottles of vodka for $$4$$ people, they will instantly give you the right answer: of course, $$3$$ bottles for $$4$$ people.

And this instant conclusion comes from an argument which is the reversal of the informal proof of the mediant inequality: how to get from the situation “$$2$$ bottles for $$3$$ people” to the situation “$$3$$ bottles for $$4$$ people”? Of course, it means that a fourth man comes and brings with him a whole bottle — can you imagine, a whole bottle of vodka! In the mediant inequality,

$$\displaystyle{\frac{2}{3} < \frac{2+1}{3+1} < \frac{1}{1}},$$

or

$$\displaystyle{ \frac{2}{3} < \frac{3}{4} < 1.}$$

I have seen some papers which confirm that this is a typical pattern of arithmetical thinking, as done by “normal” people in real life situations (for example, I have seen a claim that it is used by hospital nurses for comparing doses of medication, which one is bigger and which one is smaller).