Add, Subtract and Estimate Decimals | Grade 6 Math | LevelKite

You already add and subtract whole numbers by working with ones, tens and hundreds. Decimals work the same way once you know what each digit counts.

A decimal is a number written with a dot called the decimal point, such as $3.75$ or $2.6$ . Digits to the right of the decimal point count pieces smaller than one. A tenth is one of $10$ equal pieces of $1$ . A hundredth is one of $100$ equal pieces of $1$ . So $10$ hundredths make $1$ tenth, and $10$ tenths make $1$ one.

Adding joins two amounts into one total, called the sum. Subtracting takes one amount away from another and leaves the difference. Either way, you can only add or subtract digits that count the same size of piece: hundredths with hundredths, tenths with tenths, ones with ones.

What each digit counts

Number	Ones	Tenths	Hundredths
$3.75$	$3$	$7$	$5$
$2.6$	$2$	$6$	none
$2.60$	$2$	$6$	$0$

The position of a digit tells you what size of piece it counts. That size is the digit's place value. In the table, the $7$ in $3.75$ and the $6$ in $2.6$ both count tenths, so those two digits belong together.

Now compare the last two rows. Both numbers are $2$ ones and $6$ tenths, and $0$ hundredths adds nothing, so $2.6 = 2.60$ . A zero written after the last digit to the right of the decimal point never changes a number's value. A whole number has its decimal point just after the ones digit, so $8 = 8.0 = 8.00$ .

Key concept

Line up the decimal points

To add or subtract decimals, write one number under the other with the decimal points in one column. Then every column holds digits with the same place value. Fill any empty place to the right of a decimal point with a zero. Work one column at a time, starting from the right. When you need to, regroup: trade $10$ of one piece for $1$ of the next larger piece, or $1$ larger piece for $10$ of the next smaller piece, just as you do with whole numbers. Put the decimal point in the answer directly below the other decimal points.

Worked example

Add decimals with different numbers of digits

Find the sum $3.75 + 2.6$ .

Write the numbers one under the other with the decimal points in one column. Write 2.6 as 2.60 so both numbers have a hundredths digit. The value does not change, because 0 hundredths adds nothing.
$\begin{array}{r} 3.75 \\ +\,2.60 \end{array}$
Start at the right with the hundredths, because regrouping passes to the next column on the left. Add 5 hundredths and 0 hundredths.
$5 \text{ hundredths} + 0 \text{ hundredths} = 5 \text{ hundredths}$
Add the tenths: 7 tenths and 6 tenths make 13 tenths. Regroup, because 10 tenths make 1 one. So 13 tenths is 1 one and 3 tenths. Write 3 in the tenths column and carry the 1 to the ones column.
$7 \text{ tenths} + 6 \text{ tenths} = 13 \text{ tenths} = 1 \text{ one} + 3 \text{ tenths}$
Add the ones, and include the 1 you carried: 3 ones, 2 ones and 1 one make 6 ones.
$3 + 2 + 1 = 6$
Write the decimal point in the answer directly below the other decimal points, so each digit keeps its place value.
$\begin{array}{r} 3.75 \\ +\,2.60 \\ \hline 6.35 \end{array}$

Answer

The sum of $3.75$ and $2.6$ is $6.35$ .

Subtraction uses the same layout. In this lesson you always subtract a smaller number from a larger one, so the larger number goes on top.

Sometimes the top digit in a column is smaller than the digit below it. Then you regroup: trade $1$ larger piece for $10$ of the next smaller piece. For example, $1$ tenth becomes $10$ hundredths. The top number keeps its value. It is only written with different pieces.

Worked example

Subtract with regrouping

Find the difference $18.2 - 5.63$ .

Write 18.2 as 18.20, because 5.63 has a hundredths digit and the top number needs a hundredths place to subtract from. Line up the decimal points. The 5 goes under the 8, because both digits count ones.
$\begin{array}{r} 18.20 \\ -\,5.63 \end{array}$
Hundredths: you cannot take 3 from 0. Regroup 1 tenth as 10 hundredths. In place of 2 tenths, the top number now shows 1 tenth and 10 hundredths. That is the same amount, so its value has not changed. Subtract the hundredths.
$10 \text{ hundredths} - 3 \text{ hundredths} = 7 \text{ hundredths}$
Tenths: only 1 tenth is left on top, and you cannot take 6 from 1. Regroup 1 one as 10 tenths. In place of 8 ones and 1 tenth, the top number now shows 7 ones and 11 tenths. Subtract the tenths.
$11 \text{ tenths} - 6 \text{ tenths} = 5 \text{ tenths}$
Ones: 7 ones are left on top, and 7 ones minus 5 ones is 2 ones. Tens: nothing is taken from the 1 ten, so it stays. Write the decimal point in the answer directly below the other decimal points.
$\begin{array}{r} 18.20 \\ -\,5.63 \\ \hline 12.57 \end{array}$
Check with addition. Adding back the number you took away must return the number you started with, because addition undoes subtraction. The sum is 18.20, which equals 18.2, so the answer checks.
$12.57 + 5.63 = 18.20$

Answer

The difference $18.2 - 5.63$ is $12.57$ .

An estimate is a number close to the exact answer that you can find with quick arithmetic. An estimate helps when a rough amount is enough. It also helps you check an exact answer.

In this lesson you estimate a sum or a difference by first rounding each decimal to the nearest whole number. Rounding replaces a number with a nearby number that is easier to use. For a decimal greater than zero, look at the tenths digit:

If it is $5$ or more, round up to the next whole number. So $4.8$ rounds to $5$ .
If it is $4$ or less, keep the whole-number part and drop the digits after the decimal point. So $9.36$ rounds to $9$ .

Then add or subtract the rounded numbers. The symbol $\approx$ means is about, so you can write $4.8 \approx 5$ .

Worked example

Estimate a sum and a difference

Estimate $3.75 + 2.6$ and $18.2 - 5.63$ by rounding each number to the nearest whole number. Then compare each estimate with the exact answer from the earlier examples, $6.35$ and $12.57$ .

Round the two numbers in the sum. In 3.75 the tenths digit is 7, and in 2.6 it is 6. Both digits are 5 or more, so each number rounds up to the next whole number.
$3.75 \approx 4 \qquad 2.6 \approx 3$
Add the rounded numbers. Their sum is close to the exact sum, because each number moved only a little when you rounded it.
$3.75 + 2.6 \approx 4 + 3 = 7$
Round the two numbers in the difference. In 18.2 the tenths digit is 2, which is 4 or less, so keep 18. In 5.63 the tenths digit is 6, which is 5 or more, so round up to 6.
$18.2 \approx 18 \qquad 5.63 \approx 6$
Subtract the rounded numbers. Their difference is close to the exact difference for the same reason.
$18.2 - 5.63 \approx 18 - 6 = 12$
Compare with the exact answers. The exact sum 6.35 is more than 6 and less than 8, so it is within 1 of the estimate 7. The exact difference 12.57 is more than 11 and less than 13, so it is within 1 of the estimate 12.

Answer

The sum $3.75 + 2.6$ is about $7$ , and the difference $18.2 - 5.63$ is about $12$ . Each estimate is within $1$ of the exact answer.

Those estimates were close for a reason. Rounding to the nearest whole number moves a number by at most $0.5$ . So when you add or subtract two numbers rounded this way, the estimate is within $1$ of the exact answer. The exact answer is at least the estimate minus $1$ and at most the estimate plus $1$ .

That makes an estimate a useful check. Recall the mistaken sum $4.01$ for $3.75 + 2.6$ . The estimate is $7$ , so the exact sum must be at least $6$ and at most $8$ . Since $4.01$ is less than $6$ , it cannot be right. If an exact answer falls outside that range, look for a mistake in the exact work or in the rounding. An answer inside the range is reasonable, but a small error can still hide in it.

A number sentence is a statement such as $3.75 + 2.6 = 6.35$ that says two amounts are equal. Sometimes one number is missing and a blank stands in its place. In this lesson the letter $n$ stands for the missing number.

Think of the three numbers as two parts and one total. In $3.75 + 2.6 = 6.35$ , the parts are $3.75$ and $2.6$ , and the total is $6.35$ . Take one part away from the total and the other part is left: $6.35 - 2.6 = 3.75$ and $6.35 - 3.75 = 2.6$ . All three sentences describe the same relationship.

This is why addition and subtraction are inverse operations: each one undoes the other. You know the same idea from whole numbers. Because $6 \times 4 = 24$ , you also know $24 \div 4 = 6$ .

\text{part} + \text{part} = \text{total} \qquad \text{total} - \text{part} = \text{other part}

To complete a sentence, first decide whether $n$ is a part or the total. In an addition sentence, the total is the sum, the number that stands alone on one side of the equals sign. In a subtraction sentence, the total is the number written before the minus sign.

If $n$ is a part, subtract the known part from the total.
If $n$ is the total, add the two parts.

Each rule rewrites the same relationship in another form, so the three numbers still fit together. In this lesson every part is greater than zero, so the total is always the largest of the three numbers. Use that as a quick check.

Three kinds of sentence with a missing number n

Sentence	The missing number is	What you compute	Completed sentence
$2.4 + n = 5.9$	a part	total minus known part: $5.9 - 2.4 = 3.5$	$2.4 + 3.5 = 5.9$
$n - 1.2 = 3.6$	the total	part plus part: $3.6 + 1.2 = 4.8$	$4.8 - 1.2 = 3.6$
$7.5 - n = 4.1$	a part	total minus known part: $7.5 - 4.1 = 3.4$	$7.5 - 3.4 = 4.1$

Worked example

Complete a subtraction sentence

Find the number $n$ that completes the sentence $9.2 - n = 3.48$ .

Decide what the missing number is. In a subtraction sentence the total is the number before the minus sign, so 9.2 is the total. The missing number n and 3.48 are the two parts.
$9.2 - n = 3.48$
Rewrite the relationship. Taking either part away from the total leaves the other part, so n equals the total minus the known part. Do not add here: n is a part, so it must be smaller than the total.
$n = 9.2 - 3.48$
Estimate first so that you can check your work. Both tenths digits are 4 or less, so 9.2 rounds to 9 and 3.48 rounds to 3.
$n \approx 9 - 3 = 6$
Subtract exactly. Write 9.2 as 9.20 and line up the decimal points. Regroup 1 tenth as 10 hundredths: 10 minus 8 is 2 hundredths. That leaves 1 tenth, so regroup 1 one as 10 tenths: 11 minus 4 is 7 tenths. Last, 8 ones minus 3 ones is 5 ones.
$\begin{array}{r} 9.20 \\ -\,3.48 \\ \hline 5.72 \end{array}$
Check. The two parts must add up to the total, and they do. Also, 5.72 is within 1 of the estimate 6, and it is smaller than the total, 9.2.
$5.72 + 3.48 = 9.20$

Answer

The missing number is $5.72$ , so the completed sentence is $9.2 - 5.72 = 3.48$ .

Takeaway. Decimals add and subtract just like whole numbers once every digit sits in the column for its place value. Line up the decimal points, fill empty places with zeros, and regroup when you need to. Round each number to the nearest whole number to estimate, and use the estimate to check an exact answer. To find a missing number, decide whether it is a part or the total, then subtract or add. Next, you will use these skills to solve word problems and to spot mistakes in decimal addition and subtraction.