Three-digit by three-digit subtraction with regrouping is a math skill that helps you subtract numbers from 100 to 999 when one place value is too small to subtract directly. By exchanging one hundred for ten tens, or one ten for ten ones, you keep the number’s total value the same while making the subtraction easier to solve.
People argue about this. Here's where I land on it.
Introduction to Three-Digit Subtraction
Subtraction means finding the difference between two numbers. In a three-digit subtraction problem, both numbers have a hundreds digit, a tens digit, and a ones digit. For example:
- 742 − 215
- 608 − 347
- 913 − 486
Sometimes, each digit in the top number is larger than or equal to the matching digit below it. Day to day, in that case, subtraction can be completed directly. Still, regrouping is needed when a digit on top is smaller than the digit beneath it.
Here's one way to look at it: in 724 − 358, the ones digit 4 is smaller than 8. You cannot take 8 ones away from 4 ones, so you must regroup from the tens place It's one of those things that adds up..
Regrouping is also called borrowing, although regrouping is a more accurate description. You are not removing value from the number; you are changing its form so that subtraction can be completed.
Understanding Place Value
Before learning the steps, it is important to understand the value of each digit.
In the number 642:
- 6 represents 6 hundreds, or 600.
- 4 represents 4 tens, or 40.
- 2 represents 2 ones, or 2.
The number can therefore be written as:
642 = 600 + 40 + 2
Place value makes regrouping possible. Ten ones equal one ten, and ten tens equal one hundred. What this tells us is:
- 1 hundred can be exchanged for 10 tens.
- 1 ten can be exchanged for 10 ones.
- 10 hundreds can be exchanged for 1 thousand.
When working only with three-digit numbers, the most common exchanges are from hundreds to tens and from tens to ones.
Step-by-Step Method for Regrouping
Follow these steps whenever you solve a three-digit by three-digit subtraction problem:
- Write the numbers vertically. Place the larger number on top and the smaller number underneath it.
- Align the place values. Line up the ones, tens, and hundreds columns.
- Subtract the ones column first. Regroup from the tens place if the top ones digit is too small.
- Subtract the tens column. Use the adjusted tens digit and regroup from the hundreds place if needed.
- Subtract the hundreds column.
- Check the answer. Add the difference to the number subtracted. The result should equal the original top number.
Example 1: 724 − 358
Set up the problem:
7 2 4
- 3 5 8
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Step 1: Subtract the ones
The ones column is 4 − 8. Since 4 is smaller than 8, borrow 1 ten from the tens place.
The 2 tens become 1 ten, and the 4 ones become 14 ones. Now subtract:
14 − 8 = 6
Step 2: Subtract the tens
The tens column now contains 1 ten. Because 1 is smaller than 5, borrow 1 hundred from the hundreds place Simple, but easy to overlook. Still holds up..
The 7 hundreds become 6 hundreds, and the 1 ten becomes 11 tens. Now subtract:
11 − 5 = 6
Step 3: Subtract the hundreds
The hundreds column is now:
6 − 3 = 3
The final answer is:
724 − 358 = 366
Step 4: Check the answer
Add the difference and the number subtracted:
366 + 358 = 724
The check confirms that the answer is correct.
What If There Is a Zero in the Middle?
A zero in the tens place can look confusing, but the same place-value rules still apply.
Example 2: 503 − 278
Set up the problem:
5 0 3
- 2 7 8
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Step 1: Borrow from the hundreds
The ones column is 3 − 8, so borrowing is needed. That said, the tens digit is 0 and cannot lend anything directly Took long enough..
Borrow 1 hundred from the 5 hundreds. The 5 hundreds become 4 hundreds, and the 0 tens become 10 tens.
Now one of those tens can be exchanged for 10 ones. The 10 tens become 9 tens, and the 3 ones become 13 ones.
