Step by step
The question
Question: put 0.333…, 0.3, 1/4, 2/3 in order, from smallest to largest.
1 Write each as a decimal
To compare, we write every number the same way: as a decimal.
A decimal is tenths, hundredths or thousandths.
4 times 25 is 100.
So we multiply the top and the bottom by 25.
25 hundredths is 0.25.
So 1/4 is 0.25.
The bottom 3 does not go into 10, 100 or 1000.
So we divide: 2 divided by 3, one digit at a time.
20 divided by 3 is 6, with 2 left.
2 left again: the 6 repeats for ever.
So 2/3 is 0.666….
Careful!
0.333… is not 0.3.
Its 3s go on for ever, so it is a bit more than 0.3.
2 Same number of digits
Zeros at the end of a decimal do not change it: 0.3 is the same as 0.300.
A repeating decimal goes on for ever, so we can write as many of its digits as we need.
So we give every number 3 digits after the point.
3 Compare
Now they all have 3 digits after the point.
So we can forget the point and compare them like whole numbers: 250 < 300 < 333 < 666.
4 Write the answer
Now we write the numbers as they were given, in the same order: 1/4, 0.3, 0.333…, 2/3.
5 Check
Let's check on a number line.
Smaller numbers are further to the left.
We put a dot for each number.
From left to right: 1/4, 0.3, 0.333…, 2/3.
The same order.
So the answer is right!
Let's look back.
First: write each as a decimal.
Then: same number of digits.
Then: compare.
Then: write the answer.
Last: check.