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Regrouping Place Values of Multi-Digit Numbers

This lesson explains regrouping place values of multi-digit numbers: what regrouping means, how carrying works in addition, how borrowing works in subtraction, and worked examples across tens, hundreds, and thousands.

What Does Regrouping Mean?

Regrouping is what happens when you have too many, or too few, units in one place value column to write with a single digit. Because our number system is based on groups of ten, every ten ones can be traded for one ten, every ten tens can be traded for one hundred, and every ten hundreds can be traded for one thousand. Regrouping is simply the act of making that trade so every column ends up with a single digit from 0 to 9.

Before regrouping makes sense, it helps to be comfortable with identifying the place value of a digit and reading numbers apart in a place value table, since regrouping is really just moving amounts between those columns.

Ten ones regroup into one ten because a place value column only holds digits 0 through 9.

Regrouping When Adding Multi-Digit Numbers

When you add two multi-digit numbers, you add digit by digit starting from the ones column. Whenever a column's total reaches 10 or more, you regroup: write down the ones digit of that total and carry the extra ten into the next column to the left.

Try adding \( 358 + 276 \):

Ones: \( 8 + 6 = 14 \). Write 4 in the ones place, and carry 1 ten into the tens column.
Tens: \( 5 + 7 + 1 = 13 \). Write 3 in the tens place, and carry 1 hundred into the hundreds column.
Hundreds: \( 3 + 2 + 1 = 6 \). Write 6 in the hundreds place.

The regrouped sum is \( 358 + 276 = 634 \).

Small carried digits above the tens and hundreds columns show where regrouping happened.

Regrouping When Subtracting Multi-Digit Numbers

Subtraction regroups in the opposite direction. If the top digit in a column is smaller than the bottom digit, you cannot subtract yet, so you borrow one unit from the column to its left. That borrowed unit is worth ten in the column that needed help.

Try subtracting \( 542 - 267 \):

Ones: 2 is less than 7, so borrow 1 ten from the tens column. The ones column becomes \( 12 - 7 = 5 \), and the tens digit drops from 4 to 3.
Tens: 3 is less than 6, so borrow 1 hundred from the hundreds column. The tens column becomes \( 13 - 6 = 7 \), and the hundreds digit drops from 5 to 4.
Hundreds: \( 4 - 2 = 2 \).

The regrouped difference is \( 542 - 267 = 275 \).

Regrouping Across Larger Place Values

The same trading rule keeps working as numbers grow into thousands, ten thousands, and beyond: ten hundreds regroup into one thousand, and ten thousands regroup into one ten thousand. This is the same idea used to convert between place values, since moving a group of ten from one column to the next is exactly how regrouping works. Laying a number out with base-ten blocks, as in place value models of multi-digits, can make it easier to see why the trade is always fair: the total value never changes, only how it is grouped.

Quick Tips for Regrouping

Always work column by column, starting from the ones place, whether you are adding or subtracting.
In addition, regroup only when a column total is 10 or more.
In subtraction, regroup only when the top digit in a column is smaller than the bottom digit.
A regrouped amount is always worth exactly ten of the column it moves into or out of, never any other amount.
Double-check your answer by adding the parts back together or by re-expanding the result to see it matches the original value.

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