4th Grade · 5th Grade · 4.NF.C.7 · 5.NBT.A.3

Comparing decimals in 4th and 5th grade

Comparing decimals means deciding which of two decimal numbers is greater, such as 0.8 and 0.75, and writing the answer with >, < or =. In 4th grade, your child compares decimals to hundredths with pictures and reasoning. In 5th grade, the decimals go to thousandths, and reading and writing decimals in words and in expanded form becomes part of the work.

What the standards say

The official Common Core wording, quoted exactly. The plain version follows.

4.NF.C.7 Compare 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.

5.NBT.A.3 Read, write, and compare decimals to thousandths.
5.NBT.A.3.a Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × ( $\frac{1}{10}$ ) + 9 × ( $\frac{1}{100}$ ) + 2 × ( $\frac{1}{1000}$ ).
5.NBT.A.3.b Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

What it means, in plain words

4th grade: comparing tenths and hundredths 4.NF.C.7

A decimal is another way to write a fraction with 10 or 100 on the bottom. The first digit after the decimal point is the tenths place, and the second is the hundredths place. So 0.8 means 8 tenths, and 0.75 means 75 hundredths.

Comparing decimals in 4th grade means comparing the tenths first. 0.8 has 8 tenths and 0.75 has only 7, so 0.8 is greater, even though 75 looks bigger than 8. Children check this with a picture, such as a square cut into 100 small squares with part of it shaded, or with a number line. The symbol > means "is greater than" and < means "is less than".

Your child also learns that two decimals can only be compared when they are parts of the same whole. 0.3 of a large sheet cake is more cake than 0.3 of a cupcake, even though both numbers are 0.3.

Your child should be able to:

Read a decimal to hundredths as tenths and hundredths
Decide which of two decimals is greater by comparing the tenths first, then the hundredths
Record the result with >, < or =
Show the comparison with a shaded square or a number line

Example

Which is greater, 0.8 or 0.75?

Compare the tenths: 0.8 has 8 tenths and 0.75 has 7 tenths.
Another way: write 0.8 as 0.80. 80 hundredths is more than 75 hundredths.
$0.8 > 0.75$ , so 0.8 is greater.

5th grade: reading, writing and comparing to thousandths 5.NBT.A.3

In 5th grade, decimals reach the thousandths place, the third digit after the decimal point. Reading and writing decimals means moving between three forms of the same number. 4.326 is the number written in digits. "Four and three hundred twenty-six thousandths" is the number in words, where "and" stands for the decimal point. Expanded form shows what each digit is worth: $4 \times 1 + 3 \times \frac{1}{10} + 2 \times \frac{1}{100} + 6 \times \frac{1}{1000}$ .

Comparing decimals in 5th grade uses the same idea as in 4th grade, one place at a time from the left. Your child lines up the decimal points and finds the first place where the digits are different. Zeros at the end do not change the value, so 0.7, 0.70 and 0.700 are equal.

Your child should be able to:

Read and write decimals to thousandths in digits, in words and in expanded form
Line up the decimal points and compare one place at a time from the left
Explain why 0.7 and 0.700 are equal
Put several decimals in order from least to greatest

Example

Which is greater, 2.09 or 2.1?

Line up the decimal points and add a zero so both have the same number of places: 2.09 and 2.10.
The ones are the same (2). Compare the tenths: 0 against 1.
1 tenth is more than 0 tenths, so $2.1 > 2.09$ . 2.1 is greater.

Example

How do you write 6.054 in words and in expanded form?

In words: six and fifty-four thousandths.
In expanded form: $6 \times 1 + 0 \times \frac{1}{10} + 5 \times \frac{1}{100} + 4 \times \frac{1}{1000}$ .
The 0 holds the tenths place, so the 5 stays in the hundredths place.

Where children usually get stuck

Your child says 0.45 is greater than 0.6, because 45 is bigger than 6.
With whole numbers, a number with more digits is bigger. That rule does not work after the decimal point, where each place is a smaller piece.
What helps. Write both with the same number of digits: 0.45 and 0.60. Then compare 45 hundredths with 60 hundredths.
Your child says 0.3 is greater than 0.35, because tenths are bigger than hundredths.
The child has learned that tenths are bigger pieces than hundredths. Now the decimal with fewer digits looks like it always wins.
What helps. Put both on a number line from 0 to 1 marked in tenths. 0.35 sits halfway between 0.3 and 0.4, so it is past 0.3.
Your child writes "three and twelve thousandths" as 3.12.
The child writes the number that is heard, 12, right after the decimal point. Thousandths need three places, so a zero has to hold the tenths place.
What helps. Draw three boxes after the decimal point and label the last one thousandths. Fill them from the right: 12 takes the last two boxes and a 0 goes in the first, which gives 3.012.

Try four practice questions

Real 4th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Maya and Tom share a pizza. Maya eats 0.7 of the pizza, and Tom eats 0.3. Maya says she ate more because 0.7 is greater than 0.3. Is Maya's comparison valid?
A
Valid, because both decimals describe parts of the same whole.
B
Valid, because both decimals are written in tenths, so they can always be compared.
C
Not valid, because the decimals describe parts of different wholes.
D
Not valid, because we cannot compare decimals from two different people.
Show the answer and how to solve it
The comparison is valid because both 0.7 and 0.3 describe parts of one shared pizza — the same whole. When the wholes match, comparing the decimals tells you who ate more.
1. 1First, find out what each decimal describes. Maya's 0.7 is a part of the pizza, and Tom's 0.3 is a part of the same pizza.
2. 2To compare decimals in a word problem, both decimals must describe parts of the SAME whole. Read carefully: Maya and Tom share a pizza, and both decimals tell parts of that one pizza.
3. 3Since both 0.7 and 0.3 describe parts of one shared pizza, the wholes are equal. So 0.7 of the pizza really is more than 0.3 of the pizza, and Maya's comparison is valid.
Common mistake. A common mistake is thinking two decimals can be compared whenever they are written in the same place value (both tenths). Same place value alone is not enough — both decimals must describe parts of the SAME whole.
The idea underneath. Decimals only have meaning in a story when they are paired with a whole. Two decimals can be compared only when both describe parts of the same whole — context matters as much as the numbers.
2On level
Drag the decimal cards into order from least to greatest.
Drag a card to move it, or tap a card, then tap where it should go.
1
$\displaystyle 3.2$
2
three and one hundred five thousandths
3
$\displaystyle 3 + 0.1 + 0.08$
4
$\displaystyle 3.12$
Show the answer and how to solve it
From least to greatest, the cards are three and one hundred five thousandths, 3.12, $3 + 0.1 + 0.08$ , and 3.2 because their values are 3.105, 3.120, 3.180, and 3.200.
1. 1Rewrite each card as a decimal to thousandths so every value has the same number of places.
2. 2All four values share the ones digit 3. Compare tenths: 3.105 and 3.120 have tenths digit 1, while 3.180 and 3.200 also have tenths digits 1 and 2 respectively. Within the 3.1xx group, compare hundredths: 0 < 2 < 8, so 3.105 < 3.120 < 3.180. Then 3.200 is largest.
3. 3So the decimals go from least to greatest in this order.
Common mistake. A common mistake is treating 3.2 as smaller than 3.12 because 12 looks bigger than 2. Rewriting to thousandths as 3.200 and 3.120 makes clear that 3.200 is the greatest of the four values.
The idea underneath. Adding zeros to the right of a decimal does not change its value, so writing each number to the same place helps you compare digits place by place.

3On level

Decimal A is

0.31

and Decimal B is

0.38

. Use the place-value chart to compare A and B. Complete each step.

Decimal	Ones	Tenths	Hundredths
A	0	3	1
B	0	3	8

1. Compare the tenths digits.
2. Decide about the hundredths digits.
3. Decide which decimal is greater.

Show the answer and how to solve it

B is greater. Both decimals have $3$ tenths, so we compare the hundredths. B has $8$ hundredths and A has $1$ hundredth, so $0.38 > 0.31$ .

1Read the digits in each place from the place-value chart. Both A and B have $0$ in the ones place and $3$ in the tenths place. A has $1$ in the hundredths place, and B has $8$ in the hundredths place.
2Compare the tenths digits. Both A and B have $3$ tenths, so the tenths are equal. The tenths alone do not tell us which decimal is greater, so we need to look at the hundredths.
3Compare the hundredths digits. A has $1$ hundredth and B has $8$ hundredths. Since $8 > 1$ , B has more hundredths.
4Both decimals have the same number of tenths, but B has more hundredths. So $0.38 > 0.31$ , which means B is greater.

Common mistake. A common mistake is forgetting to compare the hundredths when the tenths are equal. Since both decimals have $3$ tenths, you must look at the hundredths. B has $8$ hundredths and A has $1$ hundredth, so B is greater.

The idea underneath. When comparing decimals, work from left to right by place value. If the larger places are equal, look at the next smaller place to find the difference.

4Stretch
Maya wants to compare 0.452 and four hundred fifty-two thousandths. Her work is shown below.
$\displaystyle \begin{array}{l} 0.452\; ? \;\text{four hundred fifty-two thousandths} \\ 0.452 \rightarrow 0.452 \\ \text{four hundred fifty-two thousandths} \rightarrow 0.52 \\ 0.452 < 0.52 \end{array}$
Select the statement that best describes Maya's work.
A
Maya's work shows that she read four hundred fifty-two thousandths as 0.52 instead of 0.452.
B
Maya's work shows that she compared digit strings and decided 452 is less than 52.
C
Maya's work shows that she ignored the zero placeholder in 0.452 and changed it to 0.45.
D
Maya's work is correct because four hundred fifty-two thousandths equals 0.52.
Show the answer and how to solve it
Maya read four hundred fifty-two thousandths as 0.52 instead of 0.452. The actual comparison is 0.452 = 0.452.
1. 1Inspect Maya's rewrite lines in the work shown.
2. 2Convert the word form correctly: four hundred fifty-two thousandths means 452 thousandths.
3. 3Compare the actual values by writing both to thousandths. Both sides equal 0.452, so the true relation is =.
4. 4Maya rewrote four hundred fifty-two thousandths as 0.52 instead of 0.452. She treated 452 thousandths as if it were 52 hundredths, dropping the first significant digit's positional value.
5. 5The correct description is that Maya read four hundred fifty-two thousandths as 0.52 instead of 0.452.
Common mistake. A common mistake is treating a three-digit thousandths word form as a two-digit hundredths value, so 452 thousandths becomes 0.52 rather than 0.452.
The idea underneath. When comparing decimals, align place values; zeros inside decimal places can change the value and cannot simply be dropped.