First, separate negatives from positives, because negatives are smaller (less positive, more negative) than positives. Or think of the numbers on a number line—the smaller numbers are further to the left. That means you can go on to work out the order of negatives separately from the order of positives.
So the negatives are −⅓ and -0.3. And the positives are ⅚, 0.8 and ⁵⁄₇.
Multiplying numbers by a positive number doesn’t change their comparative size, so if we carefully choose the multiplier it’s possible to see at a glance their relative sizes. Starting with the negatives, choose 3 as the multiplier: -1 and -0.9. -1 is more negative than -0.9 so the order is -1, -0.9.
For the positives we can get rid of the fractions by multiplying by 42 (6×7 is the lowest common denominator). 35, 33.6, 30. We can ignore the decimal: 35, 33, 30 can easily be put in order: that is: 30, 33.6, 35.
Now we switch back to the originals: -⅓, -0.3, ⁵⁄₇, 0.8, ⅚.