Column Addition
You are at the shop with 27p in one pocket and
35p in the other — how much altogether? Once the numbers grow past a
single digit, totting up prices, scores or steps in your head gets tricky, so people invented a
neat pen-and-paper method that never lets you down.
When numbers get bigger than a single digit, adding them in your head gets
hard. There is a tidy way to do it on paper: write the numbers
stacked on top of each other, lined up by
place value — ones under
ones, tens under tens — and add one column at a time. This is
column addition, and it is just ordinary
addition done one place at a
time.
The golden rule is to line the numbers up on the right, so that the ones sit
above the ones and the tens sit above the tens. Then draw a line underneath, and you are ready
to add:
\begin{array}{r} 23 \\ {} + 45 \\ \hline {} \end{array}
The most important habit is where you start. You do not add from the
left, the way you read. You always start at the right-hand column — the ones —
and work your way leftwards into the tens. In a moment you will see why: it is so that any
spare ten you make can be handed safely to the next column.
A sum with no carrying
Start with the easy kind, where every column stays a single digit. Take
23 + 45. Add the ones first:
3 + 5 = 8 — that fits in one box, so write the
8 in the ones place. Now the tens:
2 + 4 = 6 tens, so write the 6 in the
tens place. The answer is 68:
\begin{array}{r} 23 \\ {} + 45 \\ \hline 68 \end{array}
Notice each column was added on its own and each total was a single digit, so nothing had to
move. The next sum is not so lucky.
When a column makes ten: carrying
Now try 27 + 35. Add the ones column first: the
ones are 7 and 5, and
7 + 5 = 12. But a column can only hold a single digit!
Twelve is one ten and two ones, so we write the 2
in the ones place and carry the 1 ten over to the
top of the tens column.
Carrying is exactly the
bundling
idea from place value: whenever you collect ten ones, you tie them into one
neat ten and move it to the tens column — just like trading ten single cubes
for one tall rod of ten, or ten 1p coins for a single 10p. The little
1 you write above the tens is that bundle.
Then add the tens column, remembering the carried ten:
1 + 2 + 3 = 6 tens. Write the 6,
and the total is 62:
27 + 35 = 62
See it built
Watch the sum come together one column at a time. First the numbers stack up; then the
ones are added and the ten is carried; then the tens; then the total. Step through it.
One more, with a bigger carry
Carrying always works the same, no matter how big the ones get. Take
48 + 36. The ones are 8 + 6 = 14 —
again that is too big for one box. Fourteen is one ten and four ones, so
write the 4 and carry the 1. Then the
tens: 1 + 4 + 3 = 8, giving 84:
48 + 36 = 84
You still only ever carry a single ten, even when the ones make 14, 17 or 18
— because two single digits can never add to more than 9 + 9 = 18,
which is just one ten and eight ones.
Now you try
Here is a fresh sum each time. Step through it the same way: stack by place value, add the ones
and carry the ten, then add the tens. Press Refresh for a brand-new sum to
work through.
The two mistakes that catch out almost everyone:
-
Line up by place value, not by the left edge. To add
6 + 72 the 6 is six ones, so
it goes under the 2 (the ones), never under the 7. Push every number to
the right so the ones columns agree, then add.
-
Carry the 1 — don't write the whole 13 in the ones column. If the ones
make 13, you write the 3 and carry the
1 ten. Squeezing "13" into the single ones box is the classic
slip — a box holds one digit only.
A carry is just bundling. When the ones column collects ten loose ones, you
tie them into one ten and slide it next door — exactly like swapping ten 1p coins for a
single 10p coin:
= one ten
The total amount never changes — you have simply tidied ten little things into one bigger
one. That tidy-up is the little 1 you carry to the next column.
It does not matter at all — you still line up the ones on the right. To add
5 + 43, the five cookies are all ones, so the
5 sits under the 3:
+ 43 = 48
The tens column above the 5 is just empty (worth nothing), so the
4 tens come straight down. Ones with ones, tens with tens — every
single time.
See it explained
Sal Khan adds two numbers with regrouping — exactly the carrying we just did.