Comparing and Ordering Decimals

0.7 or 0.35? Most children pick the wrong one, and they pick it for a reason that has been correct every other time they have used it, which is what makes the habit so hard to shift.

Decimals and percentages · topic 2 of 7Grades 4 to 6Number7 min read7 practice questions

Why children get this wrong

With whole numbers, more digits has always meant a bigger number, and 350 beats 7 without anybody having to think about it. Children spend the first several years of school building that rule out of thousands of examples, none of which contradicted it, and then decimals arrive and break it without warning.

0.7 is bigger than 0.35. Not by a little either. It is twice as big.

The digits after the point are not an ordinary number that happens to sit there. Each one is an amount in a named column, and the columns shrink as you move right, so the tail of a decimal has to be read column by column rather than as a lump.

Compare from the left

The rule

Start with the whole numbers. If they differ, you are already finished.

If they match, compare the tenths. Then the hundredths. Stop at the first column where they differ.

Take 0.7 against 0.35. The whole numbers are both 0, so move right to the tenths, where it is 7 against 3 and the question is already settled. Nothing further along can overturn that, because even nine hundredths is less than one whole tenth, so a gap in the tenths column can never be closed by anything to its right.

ones3.tenths4hundredths530.40.05each digit is worth this much
Compare the columns that match each other. Tenths against tenths, hundredths against hundredths. Never tenths against hundredths.

The zero padding trick

Decimals of different lengths feel harder to compare than they are. Pad the shorter one with zeros on the right and the difficulty disappears.

0.7 becomes 0.70, and comparing 0.70 with 0.35 is now just 70 against 35, which no child has ever got wrong.

Why adding zeros is safe

A zero on the RIGHT-hand end of a decimal changes nothing. 0.7 and 0.70 and 0.700 are all the same amount, the same way 7 tenths and 70 hundredths are the same amount. A zero anywhere else does change the number, so only ever pad the right-hand end.

Zooming in on the line

When two decimals really are close together, zoom in on the line. Between 0.3 and 0.4 there are ten more steps waiting, each one a hundredth, and between any two of those there are ten more again.

0.300.310.320.330.340.350.360.370.380.390.40
0.35 sits exactly halfway between 0.3 and 0.4. Every gap on a number line can be cut into ten smaller gaps, forever.

Worked examples

1

Different whole numbers

Which is larger, 2.9 or 3.05?
  1. Look at the whole numbers first: 2 against 3.
  2. They are different, so the comparison is already over.
  3. 3 is more than 2, so 3.05 is larger.
  4. The 9 in 2.9 is tempting, but nine tenths is still less than one whole.
3.05 is larger
2

Same start, different lengths

Which is larger, 0.4 or 0.38?
  1. Whole numbers match, both 0. Move on.
  2. Pad to the same length: 0.4 becomes 0.40.
  3. Now compare tenths: 4 against 3. That already decides it.
  4. Or read them as hundredths: 40 against 38.
0.4 is larger, by two hundredths
3

Putting four in order

Order these smallest first: 0.5, 0.45, 0.09, 0.54
  1. Pad everything to two decimal places: 0.50, 0.45, 0.09, 0.54.
  2. Now they are all hundredths, so read them as 50, 45, 9 and 54.
  3. Order those: 9, then 45, then 50, then 54.
  4. Write the answers back in their original form.
0.09, then 0.45, then 0.5, then 0.54

Common mistakes

1. Longer means bigger

0.35 gets picked over 0.7 because it has more digits. This is not carelessness and it should not be treated as such: it is a rule that worked flawlessly for years and has only just stopped working, and the child has had no reason to suspect it. Say the replacement out loud until it sticks. After the point, more digits means finer pieces, not more stuff.

2. Padding on the wrong side

0.4 gets padded into 0.04 to match the length of 0.38, which does make the two the same length and also makes the first one ten times smaller. Zeros are only free on the right-hand end. Put one immediately after the point and every digit slides into a smaller column.

3. Carrying on after the decision is made

3.05 and 2.9 get compared, the whole numbers settle it immediately, and then the child keeps going and gets stuck on the 9. Once a column differs, the comparison is over. No collection of tenths under ten ever adds up to a whole one, so nothing further right can change the verdict.

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For parents

Do not simply tell your child that the longer-means-bigger rule is wrong, because it is not wrong, it is out of range. Tell them it holds for whole numbers and fails after a decimal point, then ask them to work out why. That last step matters: the old rule was built out of years of evidence, and it will not be dislodged by being contradicted once. It gets replaced when the child can explain the difference themselves.

Keep going

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