Question

How do you order these fractions smallest to largest: 22/25, 8/9, 7/8?

Answer 1

The answer is : `7/8 < 22/25 < 8/9`.

In order to see which one is larger than another, you need to put them all to the same denominators :

`22/25 = (22*9*8)/(25*9*8) = 1584/1800`

`8/9=(8*25*8)/(9*25*8) = 1600/1800`

`7/8=(7*25*9)/(8*25*9)=1575/1800`

Therefore, `7/8 = 1575/1800 < 22/25 = 1584/1800< 8/9=1600/1800`.

Answer 2

Same idea, but less actual arithmetic. (Is it easier? Probably not, but the arithmetic is easier, because we don't finish much of it.)

`22/25, 8/9, 7/8`

We can order them, pairwise (two at a time).

First the easy pair `8/9, 7/8`

The least common denominator is `9xx8`, I don't care what the number really is. It's the numerators I need to compare.

`8/9 = (8xx8)/(9xx8) = 64/(9xx8)`

`7/8 = (7xx9)/(9xx8) = 63/(9xx8)`

So `7/8 < 8/9`
(At the end, we won't need this as a separate step, but it's not difficult to do.)

Second Pair
(Note: it is even quicker to observe that `7/8` is `1/8` less than `1`, while `8/9` is `1/9` less than `1`, so `7/8 <8/9`)

`22/25 , 8/9`

The least common denominator is `9xx25`, Again, I don't care what the number really is. It's the numerators I need to compare.

`22/25 = (22xx9)/(25xx9)`

`8/9 = (25xx8)/(25xx9)`

The numerators are:

`22xx9` `color(white)"ssssssssssssssssssss"`and `25xx8`, which we can rewrite as:

`22xx(8+1)=22xx8 + 22` and `(22+3)8 = 22xx8 +24`

Whatever `22xx8` is, adding 24 will give a bigger total than adding 22. The second numerator is greater. So

`22/25 < 8/9`

Third pair
`22/25, 7/8` Denominator `8xx25`,

Numerators:

`22xx8` `color(white)"ssssssssssssssssssss"`and `25xx7`

`22xx8 = 22(7+1)=22(7)+22` and `25xx7=(22+3)7 = 22(7)+21`

Adding `22` will give a greater total than adding `21`, so the first number is greater:

`7/8 < 22/25`

Final Answer

`7/8 < 22/25 < 8/9`