Lesson · M.1.1
Addition and subtraction with regrouping
How to add and subtract multi-digit whole numbers with regrouping, the skill under nearly every dosage and intake and output problem on the HESI A2 math subtest.
By the end of this lesson you can
- You can add multi-digit whole numbers and carry a digit into the next place value without losing track of it.
- You can subtract multi-digit whole numbers and borrow from a higher place value when a column will not subtract cleanly.
- You can check a regrouped answer quickly enough to catch a slip before it costs you a question.
Regrouping is what you do when a column of digits adds to ten or more, or when a column does not have enough to subtract from. You carry a ten into the next column on addition, and you borrow a ten from the next column on subtraction. Almost every whole number problem on the HESI A2 math subtest that looks harder than it should turns out to be a regrouping step done in a hurry.
Place value, quickly
Every digit in a number sits in a column, and each column is worth ten times the column to its right. In 4,096 the 4 is in the thousands column, the 0 is in the hundreds column, the 9 is in the tens column, and the 6 is in the ones column. Regrouping only works because ten of anything in one column equals one of the next column up: ten ones make a ten, ten tens make a hundred, and so on.
Addition with carrying
Add the columns from right to left. When a column's total reaches ten or more, write down only the ones digit of that total and carry the tens digit into the next column, adding it in with that column's digits.
You can carry more than one digit in a long addition, and you can carry into a column that already has a carried digit waiting in it. Add every digit sitting in a column, including anything carried in, before you decide whether that column itself needs to carry.
Carrying, in one line. Ten in a column becomes a one moved into the column immediately to the left.
Subtraction with borrowing
Subtract from right to left as well. When the top digit in a column is smaller than the digit you need to take away, borrow a ten from the column to its left: that column's digit drops by one, and the column you are working in gains ten.
A borrow can chain across several columns, exactly the way it did above when the tens column had nothing to spare and had to reach into the hundreds column first. Work one column at a time and you will never lose the thread.
Regrouping is not a shortcut and not a trick. It is what makes place value consistent when a column runs short or runs over.
A fast check
Once you have an answer, run a rough estimate against it. Round each starting number to its leading digit, do the simple version, and compare.
| Original problem | Rounded estimate | Exact answer | Estimate confirms it? |
|---|---|---|---|
| 168 + 275 | 200 + 300 = 500 | 443 | yes, close to 500 |
| 512 - 238 | 500 - 200 = 300 | 274 | yes, close to 300 |
If your exact answer is far off from the rough estimate, you have likely mis-carried or mis-borrowed a column, and it is worth reworking the problem rather than moving on.
Practise this until the carrying and borrowing steps are automatic, then move on to multiplication, where the same place value habits carry over directly.
- Line up digits by place value first. A misaligned column is the most common source of a wrong answer, not a wrong method.
- Carrying moves a ten from one column into the next column to the left.
- Borrowing moves a ten from a higher column into the column that cannot subtract.
- Work right to left, one column at a time, and write the carried or borrowed digit down rather than holding it in your head.
- A wrong total from a supply count or a fluid tally is a real error on the floor, so this is worth doing slowly until it is automatic.
Check yourself
0 / 51What does carrying do when you are adding multi-digit whole numbers?
2What happens during borrowing when you are subtracting multi-digit whole numbers?
3Add 246 and 187 using carrying. What is the total?
4Subtract 156 from 423 using borrowing. What is the result?
5A learner adds 356 and 178. They write the ones column as 6 plus 8 equals 14, so they write 4 and carry 1, then the tens column as 5 plus 7 equals 12, so they write 2 and carry 1, then the hundreds column as 1 plus 3 plus 1 equals 5, for a final answer of 524. Where did the error happen?