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 Question from COCO, a student: Hello. I am having lots of trouble with these types of problems. Here is one of them:1/2 1/18 3/6. And I need to order them from greatest to least. Please also explain how to do it.

Firstly, two of these are equal, so trying to give them a definite order will be hard!

There are various ways to order positive fractions. Here are some, in roughly increasing order of difficulty:

(1) a/b > c/b if a>c: so 3/7 > 2/7

(2) a/b < a/c if b>c: so 3/7 < 3/5

Subtracting results in (2) from 1 we get

(3) (b-a)/b > (c-a)/c if b>c; so 4/7 > 2/5

Renaming: ka/kb = a/b, so ka/kb > c/d if a/b > c/d
(so 3/6 = 1/2 > 1/3)

Transitivity:

if a/b > c/d and c/d > e/f then a/b > e/f:
so 3/7 > 2/7 > 2/11, 3/7 > 2/11

A special case of this is "benchmarking" when the middle term is something obvious:

6/11 > 1/2 > 3/7

The "big hammer" that always works but needs some arithmetic:

a/b > c/d if and only if ad > bc

Good Hunting!

RD

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