Subtraction Within 1,000 with Regrouping

When subtracting three-digit numbers, you will sometimes find that a digit in the top number is smaller than the digit directly below it. When this happens, you need to regroup (also known as borrowing) from the next place value column to the left.

Let's look at how to master three-digit subtraction, including the tricky task of borrowing across zeros!

The Basic Steps of Column Subtraction

1. Line them up: Write the numbers vertically, making sure the ones, tens, and hundreds columns line up perfectly.
2. Start on the right: Always subtract the ones first, then the tens, and finally the hundreds.
3. Regroup if needed: If the top digit is smaller than the bottom digit, borrow $1$ from the column to the left.

Example 1: Standard Regrouping

Let's solve $921 - 467$.

• Step 1: Subtract the ones. We need to calculate $1 - 7$. Since $1$ is smaller than $7$, we borrow $1$ ten from the $2$ in the tens column. The $2$ becomes a $1$, and the $1$ in the ones column becomes an $11$. Now, subtract: $11 - 7 = 4$.

• Step 2: Subtract the tens. Now we have $1 - 6$ in the tens column. We can't do that, so we borrow $1$ hundred from the $9$ in the hundreds column. The $9$ becomes an $8$, and the $1$ ten becomes an $11$. Now, subtract: $11 - 6 = 5$.

• Step 3: Subtract the hundreds. Finally, subtract the hundreds: $8 - 4 = 4$.

Answer: $921 - 467 = 454$

Example 2: Borrowing Across Zeros

Sometimes, the column you need to borrow from has a zero. You have to keep moving left until you find a number to borrow from. Let's solve $800 - 356$.

• Step 1: Subtract the ones. We need to calculate $0 - 6$. We try to borrow from the tens column, but it's also a $0$! So, we move to the hundreds column.

• Step 2: Regroup across the zeros. Borrow $1$ hundred from the $8$. The $8$ becomes a $7$. The $0$ in the tens column becomes a $10$. Now, we can borrow from the tens column! Cross out the $10$ and make it a $9$. Pass that borrowed ten to the ones column, turning the $0$ ones into a $10$.

• Step 3: Subtract column by column.

  • Ones: $10 - 6 = 4$
  • Tens: $9 - 5 = 4$
  • Hundreds: $7 - 3 = 4$

Answer: $800 - 356 = 444$

Example 3: A Zero in the Middle

Let's try one more: $503 - 278$.

• Ones column: $3 - 8$. We can't do this, and the tens column has a $0$.
• Regrouping: Borrow from the $5$ in the hundreds column (it becomes a $4$). The $0$ in the tens column becomes a $10$. Now, borrow from that $10$ (it becomes a $9$) to give $10$ to the ones column. The $3$ becomes a $13$.
• Subtract:
  • Ones: $13 - 8 = 5$
  • Tens: $9 - 7 = 2$
  • Hundreds: $4 - 2 = 2$

Answer: $503 - 278 = 225$