Add and Subtract Decimals by Place Value | Grade 5 Math | LevelKite

You already know how to add and subtract decimals with place-value models. In a model you always put like with like: hundredths with hundredths, tenths with tenths, ones with ones.

Now you will do the same job with digits on paper, much as you do with whole numbers such as $26$ or $475$ . You will find a sum (the total when you add) and a difference (what is left when you subtract).

Here is the key. Write one number under the other so that each column holds just one place value. A column is a stack of digits, one above the other. A place value is what a place in a number is worth, such as ones, tenths or hundredths.

The decimal point (the dot in a decimal) shows you how to do it. The ones place is always just to the left of the decimal point. The tenths place is always just to the right of it. So when the decimal points sit in one straight line, every digit lands in its correct column.

The chart below shows this for $4.75$ and $2.6$ .

The decimal points line up, so each column holds one place value.

Number	Ones	Decimal point	Tenths	Hundredths	In words
$4.75$	$4$	.	$7$	$5$	4 ones, 7 tenths and 5 hundredths
$2.6$	$2$	.	$6$	$0$	2 ones, 6 tenths and 0 hundredths

Key concept

Line up the decimal points

Write the numbers one under the other with the decimal points in one straight line. Then each column holds one place value, so you add or subtract hundredths with hundredths, tenths with tenths and ones with ones.

Work one column at a time, starting at the right. Sometimes you must regroup: trade $10$ of one place for $1$ of the next place to the left, or trade $1$ back for $10$ . In addition, regroup when a column adds up to $10$ or more. In subtraction, regroup when the top digit is smaller than the digit below it. This works just as it does with whole numbers, because $10$ hundredths make $1$ tenth and $10$ tenths make $1$ one.

Write the decimal point of the answer straight below the other decimal points.

Worked example

Add decimals with regrouping

Find $4.75 + 2.6$ .

Write 2.6 under 4.75 with the decimal points in one straight line. Now ones are under ones and tenths are under tenths. The hundredths column holds only the 5, because 2.6 has 0 hundredths.
$\begin{array}{r} 4.75 \\ +\,2.6\phantom{0} \end{array}$
Start at the right, with the hundredths, so that anything you regroup can be added in the next column. There are 5 hundredths and 0 hundredths, which make 5 hundredths. Write 5 in the hundredths column.
$5 \text{ hundredths} + 0 \text{ hundredths} = 5 \text{ hundredths}$
Add the tenths. Together, 7 tenths and 6 tenths make 13 tenths. A column holds only one digit, so regroup: 10 of the tenths make 1 one. Write the 3 tenths that are left in the tenths column, and move the 1 one over to the ones column.
$7 \text{ tenths} + 6 \text{ tenths} = 13 \text{ tenths} = 1 \text{ one} + 3 \text{ tenths}$
Add the ones. Together, 4 ones, 2 ones and the 1 regrouped one make 7 ones. Write 7 in the ones column.
$4 \text{ ones} + 2 \text{ ones} + 1 \text{ one} = 7 \text{ ones}$
Write the decimal point of the answer straight below the other decimal points. That keeps the 7 in the ones place, the 3 in the tenths place and the 5 in the hundredths place.
$\begin{array}{r} 4.75 \\ +\,2.6\phantom{0} \\ \hline 7.35 \end{array}$

Answer

The sum of $4.75$ and $2.6$ is $7.35$ .

Subtraction uses the same columns. Line up the decimal points, start at the right, and take away one place value at a time. When the top digit in a column is smaller than the digit below it, regroup: trade $1$ from the next place to the left for $10$ of the place you are working on. If that next place holds $0$ , first regroup into it from the place on its left, just as you do with whole numbers.

There is one new thing to watch. The two numbers may have different numbers of decimal places. A number's decimal places are the digits to the right of its decimal point. For example, $8.3$ has one decimal place and $2.57$ has two. When that happens, some column at the right end has a digit in one number and an empty space in the other. Write a $0$ in each of those empty spaces. Now every column has a top digit to subtract from and a bottom digit to take away.

Worked example

Subtract with a placeholder zero

Find $8.3 - 2.57$ .

Line up the decimal points. The hundredths column has a 7 below and an empty space above, so write a placeholder zero after the 3. The top number has not changed, because 3 tenths is the same amount as 30 hundredths.
$\begin{array}{r} 8.30 \\ -\,2.57 \end{array}$
Subtract the hundredths. The top has 0 hundredths, which is fewer than 7, so regroup: trade 1 tenth for 10 hundredths. The top number now has 2 tenths and 10 hundredths.
$10 \text{ hundredths} - 7 \text{ hundredths} = 3 \text{ hundredths}$
Subtract the tenths. The 2 tenths that are left are fewer than 5, so regroup again: trade 1 one for 10 tenths. The top number now has 7 ones and 12 tenths.
$12 \text{ tenths} - 5 \text{ tenths} = 7 \text{ tenths}$
Subtract the ones: 7 ones take away 2 ones leaves 5 ones. Write the decimal point straight below the others.
$\begin{array}{r} 8.30 \\ -\,2.57 \\ \hline 5.73 \end{array}$

Answer

The difference $8.3 - 2.57$ is $5.73$ .

The same subtraction in a place-value chart. The first three rows are all worth 8.30.

Step	Ones	Tenths	Hundredths
Start with $8.30$	$8$	$3$	$0$
Trade $1$ tenth for $10$ hundredths	$8$	$2$	$10$
Trade $1$ one for $10$ tenths	$7$	$12$	$10$
Take away $2.57$	$2$	$5$	$7$
What is left: $5.73$	$5$	$7$	$3$

Sometimes you add three or more decimals. Adding puts amounts together, and the total does not depend on which amounts you join first. Two properties of addition (facts that are always true when you add) say this. With each one, the sum stays the same.

Commutative property: you may change the order of the numbers you add. For example, $0.3 + 4.5 = 4.5 + 0.3$ .
Associative property: you may change the grouping. The grouping is your choice of which numbers to add first, and parentheses ( ) mark that choice. For example, $(4.5 + 0.3) + 0.7 = 4.5 + (0.3 + 0.7)$ .

Both groupings give $5.5$ , but the second is easier, because $0.3 + 0.7 = 1$ , a whole number. So look for partners: two numbers that make a whole number, or a sum you can find in your head. Use the properties to bring partners together, and add them first.

These properties are true for addition, so use them when every sign is a plus sign. They are not true for subtraction: $(9 - 4) - 2 = 3$ , but $9 - (4 - 2) = 7$ .

Worked example

Add four decimals by pairing partners

Find $2.75 + 1.2 + 0.25 + 2.3$ .

Look for partners. The numbers 2.75 and 0.25 make a whole number: 75 hundredths and 25 hundredths are 100 hundredths, which is 1 one, and with the 2 ones that makes 3. The other two numbers, 1.2 and 2.3, make an easy sum: 1 one and 2 ones are 3 ones, and 2 tenths and 3 tenths are 5 tenths.
$2.75 + 0.25 = 3 \qquad 1.2 + 2.3 = 3.5$
Use the commutative property to change the order. Swap 1.2 and 0.25 so that the partners sit side by side.
$2.75 + 1.2 + 0.25 + 2.3 = 2.75 + 0.25 + 1.2 + 2.3$
Use the associative property to group each pair of partners. The parentheses show which sums to find first.
$2.75 + 0.25 + 1.2 + 2.3 = (2.75 + 0.25) + (1.2 + 2.3)$
Add inside each pair of parentheses. Then add the two results: 3 ones and 3 ones are 6 ones, and the 5 tenths stay as they are.
$(2.75 + 0.25) + (1.2 + 2.3) = 3 + 3.5 = 6.5$

Answer

The sum $2.75 + 1.2 + 0.25 + 2.3$ is $6.5$ .

Here is the idea to keep. A digit's place tells you what it is worth, so you add or subtract only digits that share a place. Lining up the decimal points puts every place value in its own column. Then you work from right to left and regroup, just as you do with whole numbers. Placeholder zeros fill the empty places after the decimal point, so every column is ready. And when you add three or more decimals, you may change the order and the grouping to add partners first.

Next, you will use these skills to find a missing number in a decimal addition or subtraction sentence, to solve word problems, and to find and fix mistakes.