Step by step
The question
Question: put 1/9, 0.1, 1/8, −0.12 in order, from smallest to largest.
1 Write each as a decimal
To compare, we write every number the same way: as a decimal.
The bottom 9 does not go into 10, 100 or 1000.
So we divide: 1 divided by 9, one digit at a time.
10 divided by 9 is 1, with 1 left.
1 left again: the 1 repeats for ever.
So 1/9 is 0.111….
8 times 125 is 1000.
So we multiply the top and the bottom by 125.
125 thousandths is 0.125.
So 1/8 is 0.125.
Careful!
1/9 is not 0.1.
Its 1s go on for ever, so it is a bit more than 0.1.
2 Same number of digits
Zeros at the end of a decimal do not change it: 0.1 is the same as 0.100.
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: 100 < 111 < 125.
Negative numbers are smaller than every positive number.
So they come first.
4 Write the answer
Now we write the numbers as they were given, in the same order: −0.12, 0.1, 1/9, 1/8.
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: −0.12, 0.1, 1/9, 1/8.
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.