Decimals · Grades 4, 5
Comparing and Ordering Decimals
Quick answer
To compare two decimals, line up the decimal points and work left to right, stopping at the first place where the digits differ. 0.5 beats 0.45 because five tenths beats four tenths, whatever follows. Padding the shorter decimal with trailing zeros makes the comparison easier and changes nothing, since 0.5 and 0.50 are the same number.
What you'll learn
- Compare two decimals place by place
- Explain why a longer decimal is not automatically larger
- Order a list of decimals from smallest to largest
Left to right, one place at a time
Comparing decimals is comparing place values in order of importance, starting with the most important.
Compare and :
| Place | Decided? | ||
|---|---|---|---|
| tenths | yes — | ||
| hundredths | never reached |
The tenths place settled it. Nothing in the hundredths could overturn that, so the comparison stops there.
Why length says nothing
This is the mistake worth killing early: has more digits than and is still smaller.
With whole numbers, more digits does mean bigger — beats . That rule does not survive the decimal point, because places to the right get smaller rather than larger.
Why the first difference ends it
One unit in any place outweighs everything to its right put together.
Nine hundredths plus nine thousandths plus nine ten-thousandths plus nine hundred-thousandths still falls short of one tenth. It always will, because each place is only a tenth of the one before, and a run of nines can approach a whole tenth without ever reaching it.
So once a place differs, the larger digit wins and the rest of the number is irrelevant.
Padding with zeros
The comparison gets easier if both numbers have the same number of places. Adding trailing zeros does that and changes nothing:
Now compare against — the same answer, read as whole numbers.
| Original | Padded | Read as |
|---|---|---|
Once every number has the same number of decimal places, the decimal point stops mattering and ordinary whole-number ordering takes over. That is the whole trick for sorting a list.
Worked examples
Common mistakes
Practice problems
-
Which is bigger, or ?
Answer
Full solution
Tenths: against , so is bigger. Padding makes it plain: against .
-
Which is bigger, or ?
Answer
Full solution
Tenths: against . One tenth beats zero tenths, so .
-
Which is bigger, or ?
Answer
Full solution
Ones match. Tenths: against , so is bigger.
-
Are and equal?
Answer
Yes.
Full solution
The trailing zero adds nothing: .
-
Which is bigger, or ?
Hint
Do not be distracted by the extra digits.
Answer
Full solution
Tenths: against . The that follows cannot make up a whole tenth.
-
Order , and from smallest to largest.
Answer
Full solution
Pad to three places: , , .
Read as whole numbers: .
-
Which is bigger, or ?
Answer
Full solution
Ten wholes beats nine wholes, so the decimal parts never matter.
-
Fill the blank with or : .
Answer
Full solution
Pad to three places: against . Hundredths: against , so .
-
Three parcels weigh kg, kg and kg. Which is heaviest?
Answer
The kg parcel.
Full solution
Pad to three places: , , .
Read as whole numbers: , so kg is heaviest.
-
Ben orders , and as , then says he sorted them by counting digits. Is his order right, and is his reasoning?
Hint
A right answer can come from wrong reasoning.
Answer
The order is right; the reasoning is not.
Full solution
Padding gives , , , so the order is correct.
But digit count had nothing to do with it. came first because it has seven tenths against eight and nine — the tenths place decided all three comparisons.
His method would fail immediately on and , where the longer number is the smaller one.
Frequently asked questions
How do I compare two decimals?
Line up the decimal points and work left to right. The first place where the digits differ decides it, and nothing after that place matters.
Is a longer decimal always bigger?
No. 0.5 is bigger than 0.45, even though 0.45 has more digits. Places further right are worth less, so length says nothing about size.
Can I add zeros to the end?
Yes, and it often helps. Writing 0.5 as 0.50 gives both numbers the same number of places without changing either value, and then you can compare them like whole numbers.
Why does the first differing place decide everything?
Because one unit in any place outweighs everything to its right put together. Nine hundredths, nine thousandths and so on can never add up to a whole tenth.
How do I order a list of decimals?
Pad them all to the same number of decimal places, then sort them as if the points were not there. Reading the padded numbers as whole numbers gives the right order every time.
Key terms in this lesson
- Decimal
- A decimal writes a number using places smaller than one, separated from the whole part by a decimal point. 3.47 is three wholes plus four tenths plus seven hundredths.
- Place value
- Place value is the idea that a digit's worth depends on where it sits. The 4 in 40 is worth forty; the 4 in 0.4 is worth four tenths.
Standards alignment
This lesson covers the following Common Core State Standards for Mathematics.
- CCSS.MATH.CONTENT.5.NBT.A.3bNumber and Operations in Base TenCompare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
- CCSS.MATH.CONTENT.4.NF.C.7Number and Operations—FractionsCompare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.