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

May 15, 2015

The answer is : $\frac{7}{8} < \frac{22}{25} < \frac{8}{9}$.

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

$\frac{22}{25} = \frac{22 \cdot 9 \cdot 8}{25 \cdot 9 \cdot 8} = \frac{1584}{1800}$

$\frac{8}{9} = \frac{8 \cdot 25 \cdot 8}{9 \cdot 25 \cdot 8} = \frac{1600}{1800}$

$\frac{7}{8} = \frac{7 \cdot 25 \cdot 9}{8 \cdot 25 \cdot 9} = \frac{1575}{1800}$

Therefore, $\frac{7}{8} = \frac{1575}{1800} < \frac{22}{25} = \frac{1584}{1800} < \frac{8}{9} = \frac{1600}{1800}$.

May 16, 2015

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

$\frac{22}{25} , \frac{8}{9} , \frac{7}{8}$

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

First the easy pair $\frac{8}{9} , \frac{7}{8}$

The least common denominator is $9 \times 8$, I don't care what the number really is. It's the numerators I need to compare.

$\frac{8}{9} = \frac{8 \times 8}{9 \times 8} = \frac{64}{9 \times 8}$

$\frac{7}{8} = \frac{7 \times 9}{9 \times 8} = \frac{63}{9 \times 8}$

So $\frac{7}{8} < \frac{8}{9}$
(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 $\frac{7}{8}$ is $\frac{1}{8}$ less than $1$, while $\frac{8}{9}$ is $\frac{1}{9}$ less than $1$, so $\frac{7}{8} < \frac{8}{9}$)

$\frac{22}{25} , \frac{8}{9}$

The least common denominator is $9 \times 25$, Again, I don't care what the number really is. It's the numerators I need to compare.

$\frac{22}{25} = \frac{22 \times 9}{25 \times 9}$

$\frac{8}{9} = \frac{25 \times 8}{25 \times 9}$

The numerators are:

$22 \times 9$ $\textcolor{w h i t e}{\text{ssssssssssssssssssss}}$and $25 \times 8$, which we can rewrite as:

$22 \times \left(8 + 1\right) = 22 \times 8 + 22$ and $\left(22 + 3\right) 8 = 22 \times 8 + 24$

Whatever $22 \times 8$ is, adding 24 will give a bigger total than adding 22. The second numerator is greater. So

$\frac{22}{25} < \frac{8}{9}$

Third pair
$\frac{22}{25} , \frac{7}{8}$ Denominator $8 \times 25$,

Numerators:

$22 \times 8$ $\textcolor{w h i t e}{\text{ssssssssssssssssssss}}$and $25 \times 7$

$22 \times 8 = 22 \left(7 + 1\right) = 22 \left(7\right) + 22$ and $25 \times 7 = \left(22 + 3\right) 7 = 22 \left(7\right) + 21$

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

$\frac{7}{8} < \frac{22}{25}$

$\frac{7}{8} < \frac{22}{25} < \frac{8}{9}$