Compare and Order Decimals to Thousandths | Grade 5 Math | LevelKite

Which is greater, $0.4$ or $0.38$ ? It is tempting to pick $0.38$ , because $38$ looks bigger than $4$ . But that pick is wrong, and this lesson shows you why.

To compare two numbers means to decide which one is greater, which one is less, or whether they are equal. You record the result with a symbol:

$>$ means is greater than
$<$ means is less than
$=$ means is equal to

When you write $>$ or $<$ , point the open side at the greater number.

You already know what each place in a decimal is worth. That is the tool you need here. A digit gets its value from its place, so you can compare decimals by looking at their places.

0.4 and 0.38 in a place-value chart

Number	Ones	Tenths	Hundredths	In words
$0.4$	$0$	$4$	no digit	4 tenths
$0.38$	$0$	$3$	$8$	3 tenths and 8 hundredths

Read the chart from the left. The places on the left are worth the most, so they matter most.

Ones. Both numbers have $0$ ones. This place cannot tell them apart, so move one place to the right.

Tenths. $0.4$ has $4$ tenths and $0.38$ has $3$ tenths. Four tenths is more than three tenths.

Can the $8$ hundredths in $0.38$ make up for that? No. You know that $1$ tenth is the same as $10$ hundredths, so $8$ hundredths is less than $1$ tenth. That puts $0.38$ above $3$ tenths but still below $4$ tenths.

So the tenths place decides. In words: four tenths is greater than thirty-eight hundredths. In symbols: $0.4 > 0.38$ .

\underbrace{0.4}_{4\text{ tenths}} = \underbrace{0.40}_{40\text{ hundredths}} = \underbrace{0.400}_{400\text{ thousandths}}

Key concept

Compare place by place, from the left

A digit gets its value from its place. To compare two decimals:

Line up the places: ones with ones, tenths with tenths, and so on. Where a number has no digit, think of a $0$ .
Start at the left, with the place worth the most, and compare the two digits.
If they match, move one place to the right and compare again.
Stop at the first place where the digits are different. The number with the greater digit there is the greater number.

If the digits match in every place, the decimals are equal.

Why it works: in decimals to thousandths, the digits to the right of a place are always worth less than $1$ of that place. For example, the hundredths and thousandths together are less than $1$ tenth. So they can never make up the difference.

Worked example

Compare two decimals

Compare $3.47$ and $3.452$ . Write $>$ , $<$ or $=$ between them.

Line up the places. The number 3.47 stops at the hundredths place, and 3.452 goes on to the thousandths place. Write a zero on the end of 3.47 so both numbers show thousandths. The value stays the same, because the zero comes after the last digit to the right of the decimal point.
$3.47 = 3.470$
Start at the left. Both numbers have 3 ones, so the ones place cannot decide. Move one place to the right. Both numbers have 4 tenths, so the tenths place cannot decide either.
$\text{ones: } 3 = 3 \qquad \text{tenths: } 4 = 4$
Move right again, to the hundredths place. The number 3.470 has 7 hundredths and 3.452 has 5 hundredths. This is the first place where the digits are different, so it decides. Seven hundredths is more than five hundredths.
$\text{hundredths: } 7 > 5$
Stop here. You do not need the thousandths place. The 2 thousandths in 3.452 are worth less than 1 hundredth, so they cannot make up the difference. Write the result with the numbers as they were given.
$3.47 > 3.452$

Answer

$3.47$ is greater than $3.452$ , so you write $3.47 > 3.452$ .

To order a group of numbers means to put them in a row by size. You can do it two ways:

Least to greatest. The least number, the smallest one, comes first. The greatest number, the biggest one, comes last.
Greatest to least. The greatest number comes first and the least number comes last.

Ordering is comparing, done more than once. It helps to write zeros on the end first, so every number has the same number of digits after the decimal point. Then compare place by place from the left. Find the least number, then the next one, and keep going until every number has its spot.

Both orders use the same list. Once you have the numbers from least to greatest, read the list backward to get greatest to least.

Worked example

Order four decimals

Put $0.62$ , $0.6$ , $0.602$ and $0.26$ in order from least to greatest. Then write them from greatest to least.

Write every number to thousandths so that all four show the same places. Zeros on the end, after the decimal point, do not change the values. The number 0.602 already shows thousandths.
$0.62 = 0.620 \qquad 0.6 = 0.600 \qquad 0.26 = 0.260$
Compare from the left. Every number has 0 ones, so look at the tenths place. Three numbers have 6 tenths, and only 0.260 has 2 tenths. Two tenths is less than six tenths, so 0.260 is the least.
$2 \text{ tenths} < 6 \text{ tenths}$
The other three numbers, 0.620, 0.600 and 0.602, all have 6 tenths, so the tenths place cannot sort them. Move right to the hundredths place. Their hundredths digits are 2, 0 and 0. The number with 2 hundredths, 0.620, is the greatest.
$2 \text{ hundredths} > 0 \text{ hundredths}$
Two numbers are left, 0.600 and 0.602. They match in the ones, tenths and hundredths places. In the thousandths place, 0 is less than 2, so 0.600 is less than 0.602. Now every number has its spot. Write them from least to greatest, the way they were given.
$0.26 < 0.6 < 0.602 < 0.62$
For greatest to least, read the same list backward. The greatest number now comes first, so every symbol turns around.
$0.62 > 0.602 > 0.6 > 0.26$

Answer

From least to greatest, the order is $0.26$ , $0.6$ , $0.602$ , $0.62$ . From greatest to least, it is $0.62$ , $0.602$ , $0.6$ , $0.26$ .

A word problem wraps the numbers in a story. Before you compare, decide what the question asks for: the least number or the greatest number. Ask yourself, What do these numbers measure?

Who jumped the farthest? The numbers are distances, so the farthest jump has the greatest number. Say two jumps are $2.9$ meters and $2.85$ meters. Both have $2$ ones, and $9$ tenths is more than $8$ tenths, so the farthest jump is $2.9$ meters.
Which puppy is the lightest? The numbers are weights, so the lightest puppy has the least number.
Who won the race? The numbers are times. The winner takes the least time, so look for the least number, not the greatest.

The greatest number does not always win. Check that all the numbers use the same unit, such as seconds. Then decide, and then compare.

Worked example

Who wins the race?

Three friends run the same race. Maya finishes in $15.4$ seconds, Leo in $15.38$ seconds and Sam in $15.409$ seconds. Who wins? List the runners from first place to last place.

Decide what answers the question before you compare. Each number is a time in seconds, and the winner is the runner who takes the least time. So first place goes to the least number, and first place to last place means least to greatest.
Write each time to thousandths so the places line up. Zeros on the end, after the decimal point, do not change the times. Sam's time already shows thousandths.
$15.4 = 15.400 \qquad 15.38 = 15.380$
Compare from the left. All three times have 1 ten and 5 ones. In the tenths place, Leo's time has 3 tenths and the other two times have 4 tenths. Three tenths is less than four tenths, so Leo's time is the least.
$3 \text{ tenths} < 4 \text{ tenths}$
Compare the two times that are left, 15.400 and 15.409. Every place matches until the thousandths place, where 0 is less than 9. So Maya's time is less than Sam's time.
$15.400 < 15.409$
Write the times from least to greatest, the way they were given. Then match each time to its runner: Leo, then Maya, then Sam.
$15.38 < 15.4 < 15.409$

Answer

Leo wins the race, because his time of $15.38$ seconds is the least. From first place to last place, the runners are Leo, Maya and Sam.

A mistake to watch for. A student thinks: $125$ is more than $5$ , so $0.125$ must be more than $0.5$ . The student read the digits after the decimal point like whole numbers.

This fails because the two numbers count pieces of different sizes. $0.125$ is $125$ thousandths. $0.5$ is $5$ tenths, and each tenth is the same as $100$ thousandths. So $0.5$ is $500$ thousandths, which is more than $125$ thousandths.

To fix it, compare by place. In the tenths place, $1$ is less than $5$ , so $0.125 < 0.5$ .

Counting digits fails too, because extra digits after the decimal point sit in smaller and smaller places. Sometimes the decimal with more digits is greater, as in $0.602 > 0.6$ . Sometimes it is less, as in $0.125 < 0.5$ .

Find the mistake, then fix it. Try each one before you read across.

A student's mistake	What went wrong	The fix
$0.07 > 0.6$ , because $7$ is more than $6$	The digits sit in different places. The $7$ is in the hundredths place and the $6$ is in the tenths place.	The ones match. In the tenths place, $0$ is less than $6$ , so $0.07 < 0.6$ .
$2.305 > 2.35$ , because $2.305$ has more digits	The number of digits does not set the size. The digits must be compared place by place.	The ones and tenths match. In the hundredths place, $0$ is less than $5$ , so $2.305 < 2.35$ .
$0.80 > 0.8$ , because $80$ is more than $8$	The digits after the decimal point were read as whole numbers. But $80$ hundredths is the same as $8$ tenths.	The zero on the end adds nothing. Every place matches, so $0.80 = 0.8$ .
Least to greatest: $0.9$ , $0.45$ , $0.123$	The numbers were put in order by how many digits they have, not by the value of their places.	The tenths digits are $9$ , $4$ and $1$ . Least to greatest: $0.123$ , $0.45$ , $0.9$ .

Here is the idea to keep: a decimal's size comes from the value of its places, not from how many digits it has.

To compare, line up the places and start from the left. The first place where the digits are different decides.
Zeros on the end, after the decimal point, do not change a decimal. Write them when they help you line up the places.
To order, keep comparing until every number has its spot.
In a word problem, decide first whether the least or the greatest number answers the question.
To check a comparison, ask which place each digit sits in.