Arrange in ascending order: 0.3, 0.33, 0.303, 0.033

Question

Arrange the following decimals in ascending order (smallest to largest):

0.3, \quad 0.33, \quad 0.303, \quad 0.033

Solution — Step by Step

The easiest way to compare decimals is to make them all have the same number of decimal places by adding zeros at the end (this doesn’t change their value).

Find the maximum decimal places among all numbers: 0.303 and 0.033 have 3 decimal places. So convert all to 3 decimal places:

$0.3 = 0.300$
$0.33 = 0.330$
$0.303 = 0.303$ (already 3 places)
$0.033 = 0.033$ (already 3 places)

All four numbers are between 0 and 1 (all have 0 before the decimal point). So we look at digits after the decimal.

Now compare: $0.300$ , $0.330$ , $0.303$ , $0.033$

Look at the tenths place (first decimal digit):

$0.033$ → tenths digit = 0 (smallest!)
$0.300$ → tenths digit = 3
$0.330$ → tenths digit = 3
$0.303$ → tenths digit = 3

So $0.033$ is the smallest. The remaining three all have 3 in the tenths place.

For $0.300$ , $0.330$ , $0.303$ (all have 3 in tenths place):

Look at the hundredths digit (second decimal digit):

$0.300$ → hundredths digit = 0 (smallest)
$0.303$ → hundredths digit = 0 (also 0!)
$0.330$ → hundredths digit = 3 (largest)

So $0.300$ and $0.303$ are tied on hundredths. $0.330$ is larger than both.

Both $0.300$ and $0.303$ have the same tenths (3) and same hundredths (0). Compare thousandths:

$0.300$ → thousandths digit = 0
$0.303$ → thousandths digit = 3

So $0.300 < 0.303$ .

Combining all comparisons:

0.033 < 0.300 < 0.303 < 0.330

Ascending order: $0.033 < 0.3 < 0.303 < 0.33$

Why This Works

Adding trailing zeros after the last decimal digit does not change a number’s value: $0.3 = 0.30 = 0.300$ . This is because the extra zeros are in the thousandths, ten-thousandths etc. places and represent zero of those units — contributing nothing to the value.

Once all numbers have the same number of decimal places, comparing them is the same as comparing integers: just compare digit by digit from left to right.

Alternative Method

Convert to fractions with a common denominator:

$0.033 = \dfrac{33}{1000}$
$0.300 = \dfrac{300}{1000}$
$0.303 = \dfrac{303}{1000}$
$0.330 = \dfrac{330}{1000}$

Now it’s trivial: $\dfrac{33}{1000} < \dfrac{300}{1000} < \dfrac{303}{1000} < \dfrac{330}{1000}$

Same answer. The equal-decimal-places method is faster for this type of question.

Common Mistake

The most common error is comparing 0.3 and 0.33 and concluding that 0.33 is smaller because “it has more digits after the decimal.” This is backwards. More decimal places does not mean smaller. Compare $0.3 = 0.30$ and $0.33 = 0.33$ — the second digit after the decimal is 0 vs 3, so $0.30 < 0.33$ . More digits don’t make a decimal smaller.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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