How do you order the rational numbers from least to greatest: -4 3/5 -3 2/5, -4.65, -4.09?

When all numbers are negative, the number having greatest numerical value is least and number having least numerical value is greatest.

But for that it is always preferable to write all the numbers up to a reasonable number of decimal places for comparing them, let us say three here.

Then `-4 3/5=-4.600`
`-3 2/5=-3.400`
`-4.65=-4.650` and
`-4.09=-4.090`

As `-4.65` has the greatest numerical value, it is the least.
Then comes `-4 3/5`, followed by `-4.09` and finally `-3 2/5`, which is greatest.

Hence order from least to greatest is `{-4.65,-4 3/5,-4.09,-3 2/5}`

Notice that the fractions are in 5ths and that the smallest decimal part in 100ths. It is relatively straightforward to change these all into the same units for direct comparison.

`color(blue)("Converting the decimals and fractions into 100ths")`

`color(brown)(-4 3/5)" " ->" " -4 6/10" " ->" " -4 60/100" "color(green)(larr" value 1")`

`color(brown)(-3 2/5)" "->" "-3 4/10" "->" "-3 40/100" "color(green)(larr" value 2")`

`color(brown)(-4.65) " ...................................."-4 65/100" "color(green)(larr" value 3")`

`color(brown)(-4.09) "......................................."-4 9/100" "color(green)(larr" value 4")`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Ordering the 'value numbers' by inspection")`

`color(green)(" "2"; "4"; "1"; "3)" "` giving:

`color(brown)(-3 2/5"; "-4.09"; "-4 3/5"; "-4.65)`