12.40
+03.75
The decimal points line up, so every digit is paired with the same place value.
One large addition, split by place value
Before this page: know what each digit is worth in integers and decimals, and be comfortable adding two single digits without writing anything down — the KS2 page on addition and subtraction covers the second.
Held in the head, 4,687 + 2,759 is a blur of digits with no obvious place to begin, and any answer that emerges has to be taken on trust. The difficulty is not the arithmetic; it is trying to do all of it at once.
Place value offers the way out. The digits in a number are not interchangeable symbols: each one counts a different size of thing, and quantities of the same size can be added entirely on their own. Seven ones and nine ones make sixteen ones whatever stands to the left of them. One hard addition is therefore several easy ones in disguise.
Column addition is the arrangement that separates them and keeps them apart. Equal place values are written in a single column, each column is added by itself, and the answer is assembled from the results. The only thing left to manage is a column whose total is too large to fit in one digit. The same arrangement then serves whole numbers, decimals, money and measures alike, because all of them are written in the same place-value system.
100s
10s
1s
3
4
2
+
5
2
6
8
6
8
The columns are added in this direction
- Place-value columnsThe headings name what every digit in that column is worth. Ones sit beneath ones, tens beneath tens, hundreds beneath hundreds.
- One column at a timeThe ones are added first, then the tens, then the hundreds. Each is a small addition on its own.
- Answer lineEvery column total is written beneath the line, in the column it came from.
Why the columns must align
The alignment is not a matter of neatness. Each column contains quantities of one size only: a digit in the tens column counts groups of ten, a digit in the hundredths column counts groups of one hundredth. Digits taken from different columns count different things, and a total that mixes them measures nothing.
12.4
+3.75
Aligning the final digits places hundredths beneath tenths and changes the calculation.
Regrouping a column
A column has room for one digit, so a total of ten or more will not fit. Nothing is lost: the excess is simply worth more than the column can express, and it moves to the column where it can be expressed. Split the total into ones and tens, write the ones in the current column, and carry the tens into the column immediately to the left. Regrouping renames a quantity; it never changes it.
10 ones
1 ten
The same split happens inside a calculation. If the tens column holds 8 and 6 with a regrouped 1 above it, the column totals 15:
Tens column
18615
5is written in the tens column
1is regrouped into the hundreds column, where it joins the next addition
Add one place-value column at a time
- 1
Align equal place values.
- 2
Begin with the furthest column to the right.
- 3
Write the result digit and regroup any tens.
- 4
Continue left, including every regrouped amount.
Adding whole numbers
Larger numbers need nothing new, only more columns. Keep the digits in clear columns and write each regrouped amount above the column that will use it, so that it is waiting there when that column is reached rather than being remembered.
Example: calculate 4,687 + 2,759
4,687 + 2,759
Align equal place values
Begin with equal place values directly beneath one another. The ones sit under the ones, the tens under the tens, and so on.
4,687 + 2,759 = 7,446
Adding decimals
Decimals need no new method either. The place-value columns continue across the decimal point, which marks the boundary between ones and tenths. Align that boundary in both numbers, write the answer's point directly beneath it, then add from the smallest occupied place towards the left, exactly as before.
Example: calculate 18.75 + 6.9
18.75 + 6.9
Align equal place values
Begin with the decimal points directly beneath one another. Write 6.9 as 6.90 so every occupied place is visible.
18.75 + 6.90 = 25.65
Adding in context
The context decides what each column counts. Put every quantity into the same unit before setting out the addition, then include that unit with the answer.
Money
A book costs £18.75 and a notebook costs £6.90. Their total cost is £18.75 + £6.90 = £25.65.
Measure
A 2.35 m length is joined to a 0.85 m length. Since both measurements are already in metres, the total is 2.35 m + 0.85 m = 3.20 m, or 3.2 m.
Add two chosen numbers
Align equal place values
Begin with the decimal points directly beneath one another. The empty hundredths place in 956.7 is held by a zero, so it can be read as 956.70.
Common mistakes
Wrong answers in column addition rarely come from the arithmetic. Four slips account for most of them, and each one is a failure of the arrangement rather than of the adding.
- Aligning the final digitsFor decimals of different lengths, align the decimal points. Final digits may represent different place values.
- Forgetting a regrouped amountWrite each regrouped digit above the next column and include it before adding that column.
- Writing two digits in one columnIf a column totals 13, write 3 in that column and regroup the 1 into the column to its left.
- Moving the decimal pointThe decimal point does not move during addition. Copy it straight down into the answer line.
Summary
- Align equal place values; for decimals, place each point on the same boundary.
- Begin with the furthest occupied column to the right and work left.
- Write one result digit in each column and regroup any tens into the column to the left.
- Convert quantities to the same unit before adding, and attach that unit to the total.
- Check by subtracting either addend from the total to recover the other addend.