Write Decimals in Expanded Form | Grade 5 Math | LevelKite

You already know the places in a decimal, from the ones down to the thousandths. Now you will use those places to take a number apart.

Look at $3.47$ . In words, it is 3 ones, 4 tenths, and 7 hundredths. Each digit has a value. The value is how much the digit is worth because of the place it sits in. The 3 is in the ones place, so its value is $3$ . The 4 is in the tenths place, so its value is $0.4$ . The 7 is in the hundredths place, so its value is $0.07$ .

Expanded form is a way to write a number as a sum of the values of its digits. You write the values out with plus signs between them, so you can see what every digit is worth.

The table takes $3.47$ apart one digit at a time. Put plus signs between the numbers in the last column and you have its expanded form.

Taking 3.47 apart, one digit at a time

Digit	Place	Value in words	Value as a number
$3$	ones	3 ones	$3$
$4$	tenths	4 tenths	$0.4$
$7$	hundredths	7 hundredths	$0.07$

3.47 = 3 + 0.4 + 0.07

Worked example

From a decimal to a sum

Write $26.185$ in expanded form as a sum of the values of its digits.

Start with the digits to the left of the decimal point. A digit's worth depends on its place, so name each place. The 2 is in the tens place, so it is worth 20. The 6 is in the ones place, so it is worth 6.
$2 \text{ tens} = 20, \quad 6 \text{ ones} = 6$
Do the same for the digits to the right of the decimal point. The 1 is in the tenths place, the 8 is in the hundredths place, and the 5 is in the thousandths place.
$1 \text{ tenth} = 0.1, \quad 8 \text{ hundredths} = 0.08, \quad 5 \text{ thousandths} = 0.005$
Join the five values with plus signs, because the number is all of these parts added together.
$26.185 = 20 + 6 + 0.1 + 0.08 + 0.005$
Check by putting the parts back. 20 and 6 make 26. Then 1 tenth, 8 hundredths, and 5 thousandths fill the three places after the decimal point. You get 26.185 again.
$20 + 6 + 0.1 + 0.08 + 0.005 = 26.185$

Answer

In expanded form, $26.185 = 20 + 6 + 0.1 + 0.08 + 0.005$ .

There is a second way to write expanded form. It shows two things for every digit: the digit itself and the size of its place.

Think about $0.4$ . It means 4 tenths. One tenth is the fraction $\frac{1}{10}$ , and 4 tenths is 4 of them. So $0.4$ is the same amount as $4 \times \frac{1}{10}$ .

In the same way, $0.07$ is 7 hundredths, which is $7 \times \frac{1}{100}$ . A digit to the left of the decimal point works the same way: 3 ones is $3 \times 1$ .

The number you multiply the digit by is the place value. It is how much a 1 in that place is worth. The table lists the place value of each place.

Each place, its place value, and a digit in that place

Place	Place value	A digit in that place	Digit times place value
tens	$10$	2 tens is $20$	$2 \times 10$
ones	$1$	3 ones is $3$	$3 \times 1$
tenths	$\frac{1}{10}$	4 tenths is $0.4$	$4 \times \frac{1}{10}$
hundredths	$\frac{1}{100}$	7 hundredths is $0.07$	$7 \times \frac{1}{100}$
thousandths	$\frac{1}{1000}$	5 thousandths is $0.005$	$5 \times \frac{1}{1000}$

Key concept

Expanded form shows what each digit is worth

Expanded form writes a number as a sum of the values of its digits. The value of a digit is the digit times the place value of its place. You can write that value as a decimal, such as $0.07$ , or as the digit times its place value, such as $7 \times \frac{1}{100}$ . Both name the same amount, so both sums equal the same number.

3.47 = 3 + 0.4 + 0.07 = 3 \times 1 + 4 \times \frac{1}{10} + 7 \times \frac{1}{100}

Worked example

Expanded form using fractions

Write $8.406$ in expanded form using fractions.

Say what each digit stands for, because each digit becomes one part of the sum. The number has 8 ones, 4 tenths, 0 hundredths, and 6 thousandths.
Write each digit times the place value of its place. A 1 in the ones place is worth 1. A 1 in the tenths place is worth one tenth, in the hundredths place one hundredth, and in the thousandths place one thousandth.
$8 \times 1, \quad 4 \times \frac{1}{10}, \quad 0 \times \frac{1}{100}, \quad 6 \times \frac{1}{1000}$
Look at the hundredths part. Zero hundredths is nothing, and adding 0 does not change a sum. So you may leave this part out. Keeping it is also correct.
$0 \times \frac{1}{100} = 0$
Join the three parts that are left with plus signs. Together they make the number.
$8.406 = 8 \times 1 + 4 \times \frac{1}{10} + 6 \times \frac{1}{1000}$

Answer

In expanded form using fractions, $8.406 = 8 \times 1 + 4 \times \frac{1}{10} + 6 \times \frac{1}{1000}$ .

You can also go the other way. Start with an expanded form and write the decimal.

Each part tells you one digit and its place. The parts $0.07$ and $7 \times \frac{1}{100}$ say the same thing: put a 7 in the hundredths place. So read each part and put its digit in its place.

Sometimes no part names a place. You saw why in the last example: a digit of 0 adds nothing, so its part can be left out. When you write the decimal, you must put that 0 back whenever the empty place comes between two of your digits. The 0 holds the place open, so the digits after it stay in their own places.

Try $2 + 0.07 + 0.005$ . The parts give you 2 ones, 7 hundredths, and 5 thousandths. No part has tenths, and the tenths place comes between the 2 and the 7. So write 0 in the tenths place. The decimal is $2.075$ . The table shows each digit going to its place.

Each part of 2 + 0.07 + 0.005 goes to its own place

Place	Ones	Tenths	Hundredths	Thousandths
Part that names it	$2$	no part	$0.07$	$0.005$
Digit to write	$2$	$0$	$7$	$5$

Worked example

From an expanded form back to a decimal

Write $6 \times 10 + 3 \times \frac{1}{10} + 5 \times \frac{1}{1000}$ as a decimal.

Read each part as a digit and a place. The number a digit is multiplied by is its place value, so it tells you the place. The parts are 6 tens, 3 tenths, and 5 thousandths.
Name the places in order from the tens to the thousandths and look for any that no part names. The ones and the hundredths are missing. Each comes between two of your digits, so each gets a 0.
$6 \text{ tens}, \quad 0 \text{ ones}, \quad 3 \text{ tenths}, \quad 0 \text{ hundredths}, \quad 5 \text{ thousandths}$
Write the digits in that order. Put the decimal point between the ones and the tenths, because it marks where the ones end and the tenths begin.
$6 \times 10 + 3 \times \frac{1}{10} + 5 \times \frac{1}{1000} = 60.305$
Check by writing your answer in expanded form as a sum. The parts 60, 0.3, and 0.005 are 6 tens, 3 tenths, and 5 thousandths. They match the parts you started with.
$60.305 = 60 + 0.3 + 0.005$

Answer

Written as a decimal, $6 \times 10 + 3 \times \frac{1}{10} + 5 \times \frac{1}{1000}$ is $60.305$ .

Here is the idea to keep. Every digit in a decimal has a value, and the number is the sum of those values. Expanded form writes that sum out. You can write each part as a decimal, like $0.07$ , or as a digit times its place value, like $7 \times \frac{1}{100}$ . To go back to a decimal, put each digit in its place and write 0 in any empty place that comes between two digits.

Next, you will use place value to compare decimals one place at a time from the left, and to put decimals in order.