How to subtract multi-digit numbers with regrouping

Where it goes wrong

Try \(64 - 29\). In the ones column you need \(4 - 9\), which doesn't work with whole numbers, so a lot of people quietly flip it to \(9 - 4 = 5\) and write \(45\). The real answer is \(35\).

Regrouping, which older textbooks and most parents call borrowing, is how you handle a column where the top digit is smaller than the bottom one. It's usually taught as a set of moves to memorize. This page shows the reason behind the moves, because once you see it, even the tricky cases, like regrouping across zeros, stop being tricky.

Why regrouping works

The number \(64\) is \(6\) tens and \(4\) ones. A ten is worth \(10\) ones, so you can "break open" one ten and write the same number as \(5\) tens and \(14\) ones:

Nothing about the value changed. Now every column can be subtracted on its own:

That's all regrouping is: trading one of a bigger unit for ten of the next smaller unit. Nothing is really borrowed, and nothing has to be paid back.

The standard algorithm, step by step

A habit that saves a lot of mistakes: when you regroup, cross out the old digit and write the new one small above it. When you finish, you can see every trade you made.

Regrouping across zeros

The case that trips up almost everyone is when the column you need to regroup from is a zero. It has nothing to give, so you keep moving left until you reach a digit that isn't zero.

For round numbers like \(1{,}000\), \(5{,}000\) or \(600\) there's a shortcut that avoids regrouping completely. Subtract from one less, then add \(1\) back. Subtracting from \(999\) or \(4{,}999\) never needs regrouping, because every digit is a \(9\).

Check your answer two ways

Estimate first. Before you start, round: \(8{,}142 - 3{,}576\) is about \(8{,}100 - 3{,}600 = 4{,}500\). The exact answer, \(4{,}566\), is close. If you had gotten \(5{,}566\) or \(466\), the estimate would have warned you right away.

Decimals and money

Same algorithm, one extra rule: line up the decimal points, and fill in zeros so both numbers have the same number of decimal places. A whole number like \(20\) has an invisible decimal point at the end: \(20 = 20.00\).

Mental math strategies

For smaller numbers you often don't need columns at all.

• Count up, the way a cashier makes change. For \(64 - 29\): from \(29\) to \(30\) is \(1\), from \(30\) to \(64\) is \(34\). Total: \(35\).
• Round and adjust. \(29\) is almost \(30\): \(64 - 30 = 34\), but you subtracted \(1\) too many, so add it back: \(35\).
• Same difference. Add the same amount to both numbers: \(64 - 29 = 65 - 30 = 35\). The gap between two numbers doesn't change if you shift both.

Common mistakes

Frequently asked questions

Is borrowing the same as regrouping?

Yes. "Regrouping" is the term most US schools use today, because it describes what happens (one ten is regrouped as ten ones) and because nothing is ever paid back. "Borrowing" is the older name for the same thing. In addition, the reverse move is called carrying.

What if the bottom number is bigger than the top number?

Then the answer is negative. Subtract the other way around and put a minus sign in front: \(29 - 64 = -(64 - 29) = -35\).

Why do we start with the ones place?

Because regrouping moves value from a column to the one on its right. If you started on the left, you might discover later that a column you already finished needed to give something away, and you'd have to redo it.

Practice problems