Regrouping in Addition and Subtraction

The idea in one sentence

Ten of anything in one column becomes one of the next column over — that trade is the only thing regrouping does.

The trade

Write \(47 + 38\) in columns and start at the ones. Seven ones plus eight ones is fifteen ones. But a column only holds a single digit, and fifteen needs two.

So you trade. Ten of those fifteen ones bundle into one ten and move next door; five ones stay behind. The little 1 written above the tens column is not a mysterious mark — it is one bundle of ten, relocated to the column where tens are counted.

Four tens, plus three tens, plus the traded one, is eight tens. Five ones stay. The answer is 85, and every step was a trade rather than a rule.

Borrowing is the same trade run backwards

Now try \(52 - 27\). In the ones column you need to take 7 from 2, and there isn’t enough. So you go next door and break one ten back into ten ones.

The 5 tens become 4 tens, and the 2 ones become 12 ones. Now \(12 - 7 = 5\), and \(4 - 2 = 2\). The answer is 25.

Notice that the number never changed. 52 was five tens and two ones; after the trade it is four tens and twelve ones. Both descriptions are 52. Regrouping rearranges a number without resizing it, which is exactly why it’s allowed.

Why the columns must line up

Every digit has a place, and the place decides what the digit is worth. The 4 in 47 is not four — it is forty. If you slide the numbers out of alignment and add a tens digit to a ones digit, you are adding forty to eight and calling it twelve.

This is why adding 47 and 8 trips people up more than adding 47 and 38. There is no digit under the 4, and the instinct is to slide the 8 leftwards to fill the gap. The 8 is ones. It belongs under the 7.

Common mix-ups

• Forgetting to add the carried ten. It’s written small and above the line, which makes it easy to skip. Add it first, before the two digits beneath it.
• Borrowing without reducing the neighbour. If the ones column gained ten, the tens column must lose one. Both halves of the trade or neither.
• Subtracting the small digit from the big one regardless of which is on top. \(52 - 27\) is not “2 from 7 is 5” in the ones column. The top number is the one you’re taking from, and if it’s too small, you trade.
• Borrowing across a zero and stopping there. To take from a 0 in the tens, first trade a hundred into that zero, making it 10, and then trade from it.

Check yourself

Work out \(63 + 29\) and \(40 - 16\). For each one, say where a trade happened.

\(63 + 29 = 92\): nine ones plus three is twelve, so ten ones traded up and the tens became \(6 + 2 + 1 = 9\). \(40 - 16 = 24\): there were no ones to take from, so one ten traded down into ten ones, leaving three tens and ten ones.