Adding and Subtracting Decimals | Grade 7 Math | LevelKite

You already know how to compare decimals, which are numbers written with a decimal point. The decimal point is the dot that separates the whole part of a number, on its left, from the part smaller than one, on its right. In this lesson you will add and subtract decimals.

Each digit (one of the symbols $0$ to $9$ ) tells how many of one unit there are, and the place value of a digit is the size of that unit. In $4.6$ , the $4$ counts ones and the $6$ counts tenths. In $2.38$ , the $8$ counts hundredths. Each unit is $10$ times the size of the unit one place to its right, so $10$ hundredths make $1$ tenth and $10$ tenths make $1$ one.

Adding means joining amounts, and the result is the sum. Subtracting means taking an amount away, and the result is the difference. You can only join or take away units of the same size directly: tenths with tenths, hundredths with hundredths. That one idea explains every step in this lesson.

Each problem here uses positive numbers (numbers greater than zero), and in each subtraction you take the smaller number from the larger one, so every answer is positive too.

Adding 4.6 and 2.38 one place value at a time

Number	Ones	Tenths	Hundredths
$4.6$	$4$	$6$	$0$
$2.38$	$2$	$3$	$8$
Sum: $6.98$	$4 + 2 = 6$	$6 + 3 = 9$	$0 + 8 = 8$

Read the table one column at a time. The number $4.6$ has no hundredths, so its hundredths column holds a $0$ . Writing that zero gives $4.60$ . The value stays the same because $6$ tenths equal $60$ hundredths, so $4.6 = 4.60$ .

Now each column joins units of one size, so you can add the digits in it. The sum is $6$ ones, $9$ tenths and $8$ hundredths, which is $6.98$ .

Written in columns, with the decimal points lined up, the same addition looks like this:

\begin{array}{r} 4.60 \\ +\,2.38 \\ \hline 6.98 \end{array}

Key concept

Add and subtract units of the same size

To add or subtract decimals, write them in columns with the decimal points lined up, so each column holds one place value: ones under ones, tenths under tenths, hundredths under hundredths, and so on. Fill any empty place to the right of a decimal point with a zero, then work column by column from right to left. The answer's decimal point goes directly below the others. For example, $4.6 + 2.38 = 4.60 + 2.38 = 6.98$ .

When a column needs it, regroup: trade $10$ of one unit for $1$ of the next larger unit, or $1$ unit for $10$ of the next smaller unit. The value does not change, because $10$ hundredths equal $1$ tenth and $10$ tenths equal $1$ one.

This lesson uses positive numbers and takes the smaller number from the larger one.

Worked example

Adding with regrouping

Find $3.75 + 8.6$ .

Write 8.6 as 8.60 so that both numbers show hundredths. The value is the same because 6 tenths equal 60 hundredths. Line up the decimal points so that each column holds one place value.
$3.75 + 8.6 = 3.75 + 8.60$
Start with the smallest unit, so that any regrouped unit can join the next column before you add it. Both digits in this column count hundredths, so you can add them: 5 hundredths and 0 hundredths make 5 hundredths.
$5 + 0 = 5$
Add the tenths: 7 tenths and 6 tenths make 13 tenths. Regroup 10 of them as 1 one, because 10 tenths equal 1 one. Write 3 in the tenths place and move the 1 to the ones column.
$7 + 6 = 13 = 10 + 3$
Add the ones, including the regrouped 1: 3 ones, 8 ones and 1 one make 12 ones, which is 1 ten and 2 ones. Every digit stayed in its column, so the decimal point goes directly below the others, between the ones and the tenths.
$3.75 + 8.60 = 12.35$
Check with a comparison. The number 3.75 is between 3 and 4, and 8.6 is between 8 and 9, so the sum must be between 3 + 8 = 11 and 4 + 9 = 13. The result 12.35 is. A decimal point one place off would give 1.235 or 123.5, which are not. This check shows the answer is reasonable; it does not prove every digit.
$3 + 8 < 3.75 + 8.6 < 4 + 9$

Answer

The sum of $3.75$ and $8.6$ is $12.35$ .

Subtraction uses the same idea. With the number you start with on top, take away units of the same size: hundredths from hundredths, tenths from tenths, ones from ones.

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

Worked example

Subtracting with regrouping

Find $12.4 - 7.86$ .

Write 12.4 as 12.40 so that both numbers show hundredths. The value is the same because 4 tenths equal 40 hundredths. Line up the decimal points with 12.40 on top, because it is the number you start with.
$12.4 - 7.86 = 12.40 - 7.86$
Hundredths: the top has 0 hundredths, not enough to take away 6. Regroup 1 of the 4 tenths as 10 hundredths, because 1 tenth equals 10 hundredths. The top now has 3 tenths and 10 hundredths.
$10 - 6 = 4$
Tenths: the top has 3 tenths, not enough to take away 8. Regroup 1 of the 2 ones as 10 tenths, because 1 one equals 10 tenths. The top now has 1 one and 13 tenths.
$13 - 8 = 5$
Ones: the top has 1 one, not enough to take away 7. Regroup the 1 ten as 10 ones, because 1 ten equals 10 ones. The top now has 11 ones and no tens.
$11 - 7 = 4$
Each digit stays in the column where you found it, so write 4 ones, 5 tenths and 4 hundredths with the decimal point directly below the others.
$12.40 - 7.86 = 4.54$
Check by adding the difference to the number you took away. Hundredths: 4 + 6 = 10, so write 0 and regroup 1 tenth. Tenths: 5 + 8 + 1 = 14, so write 4 and regroup 1 one. Ones: 4 + 7 + 1 = 12. The total matches the number you started with.
$4.54 + 7.86 = 12.40$

Answer

The difference $12.4 - 7.86$ is $4.54$ .

Worked example

A two-step problem with a whole number

A ribbon is $5$ meters long. You cut off one piece that is $1.35$ meters long and another piece that is $2.8$ meters long. How many meters of ribbon are left?

Let r be the length of ribbon left, in meters. The ribbon left equals the starting length minus the total length you cut off, so first find that total.
$r = 5 - (1.35 + 2.8)$
Add the two pieces to find the total you cut off. Write 2.8 as 2.80 so both lengths show hundredths. Hundredths: 5 + 0 = 5. Tenths: 3 + 8 = 11, so write 1 and regroup 10 tenths as 1 one. Ones: 1 + 2 + 1 = 4. You cut off 4.15 meters in all.
$1.35 + 2.80 = 4.15$
Write 5 as 5.00 on top. The zeros come after the decimal point, so the value is the same. The top has 0 hundredths, not enough to take away 5, and no tenths to trade, so regroup twice: 1 of the 5 ones becomes 10 tenths, then 1 of those tenths becomes 10 hundredths. That leaves 4 ones, 9 tenths and 10 hundredths.
$5.00 = 4 + 0.9 + 0.10$
Now every column has enough on top, so subtract each place value. Hundredths: 10 - 5 = 5. Tenths: 9 - 1 = 8. Ones: 4 - 4 = 0. Write the 0 in the ones place so the decimal point is easy to see.
$r = 5.00 - 4.15 = 0.85$
Check by adding back: the ribbon left plus the total you cut off should make the whole ribbon. Hundredths: 5 + 5 = 10. Tenths, with the regrouped 1: 8 + 1 + 1 = 10. Ones, with the regrouped 1: 0 + 4 + 1 = 5.
$0.85 + 4.15 = 5.00$

Answer

There are $0.85$ meters of ribbon left.

Adding and subtracting decimals comes down to one idea: join or take away units of the same size. Lining up the decimal points puts each place value in its own column. Zeros at the right-hand end, after the decimal point, fill empty places without changing a value. Regrouping trades $10$ of one unit for $1$ of the next larger unit, or the reverse. A quick comparison or an add-back check then tells you whether your answer makes sense.

Next, you will multiply and divide decimals.