Multiplication With Regrouping: Methods and Examples

What is multiplication with regrouping?

Multiplication with regrouping is an algorithm used when the product of digits in any place-value column is $10$ or greater. Because the base-ten number system only allows a single digit from $0$ to $9$ in each column, the extra value cannot stay in that position.

Instead, the extra value is carried over to the next larger place-value column on the left. The remaining digit is written below the line in the current column. This renaming process continues until all digits in the multiplicand have been multiplied.

Why regrouping works

Regrouping works because our number system is based on groups of ten. Ten ones are equivalent to one ten, and ten tens are equivalent to one hundred.

When a multiplication step produces a result of $10$ or more, we separate the value by its place value. For example, if multiplying the ones column yields $24$, that represents $2$ tens and $4$ ones. The $4$ remains in the ones column, while the $2$ tens are carried over to join the other tens in the next column.

An alternative way to visualize this process without carrying digits is the area model for multiplication.

Set up by place value

To multiply accurately, set up the calculation in vertical columns based on place value.

Write the larger number, called the multiplicand, on the top row. Write the smaller number, called the multiplier, on the bottom row. Align the numbers so that their ones digits are in the same vertical column on the far right. The tens and hundreds digits will form their own columns to the left.

Draw a horizontal line below the multiplier to separate the problem from the partial products and the final answer.

Step-by-step method

Before applying this method, it is helpful to master multiplying by 10, 100 and 1,000, as it reinforces how digits shift across place-value columns.

To complete a multiplication problem with regrouping:

1. Multiply the bottom digit by the ones digit of the top number.
2. If the product is $9$ or less, write it below the line in the ones column.
3. If the product is $10$ or more, write the ones digit below the line and write the tens digit above the next column to the left.
4. Multiply the bottom digit by the next digit to the left in the top number.
5. Add the carried digit to this new product.
6. Write the final sum below the line, carrying again if the sum is $10$ or more.

Multiply a multi-digit number by one digit

When multiplying a larger number by a single digit, apply the regrouping steps from right to left.

Multiply the multiplier by the ones digit first, carrying any extra value to the tens column. Then, multiply the multiplier by the tens digit and add the carried value. If the new sum requires regrouping, carry its extra value to the hundreds column. Repeat this cycle until you have multiplied the bottom digit by every digit in the top number.

Once you are comfortable multiplying by a single digit, you can apply the exact same regrouping rules to long multiplication with larger multi-digit numbers.

Worked examples

The following vertical calculations show how carried values are recorded during multiplication.

Example 1: Multiplying a two-digit number

Question: What is $48 \times 3$?

Method:

1. Setup the numbers vertically.
2. Multiply $8$ ones by $3$: $8 \times 3 = 24$. Write $4$ in the ones column and carry the $2$ tens.
3. Multiply $4$ tens by $3$: $4 \times 3 = 12$.
4. Add the carried $2$: $12 + 2 = 14$. Write $14$ below the line.

Answer: $144$.

Check: You can estimate using compatible numbers: $50 \times 3 = 150$. The product $144$ is close to the estimate.

Example 2: Multiplying a three-digit number

Question: What is $156 \times 5$?

Method:

1. Multiply $6$ ones by $5$: $6 \times 5 = 30$. Write $0$ and carry $3$.
2. Multiply $5$ tens by $5$: $5 \times 5 = 25$.
3. Add the carried $3$: $25 + 3 = 28$. Write $8$ and carry $2$.
4. Multiply $1$ hundred by $5$: $1 \times 5 = 5$.
5. Add the carried $2$: $5 + 2 = 7$. Write $7$.

Answer: $780$.

Check: $150 \times 5 = 750$. The answer $780$ is reasonable.

Example 3: Carrying to a new place value

Question: What is $89 \times 7$?

Method:

1. Multiply $9 \times 7 = 63$. Write $3$ and carry $6$.
2. Multiply $8 \times 7 = 56$.
3. Add the carried $6$: $56 + 6 = 62$.
4. Since there are no more digits to multiply, write the complete $62$ below the line, placing the $6$ in the hundreds column.

Answer: $623$.

Check: Using compatible numbers: $90 \times 7 = 630$. The product $623$ is slightly less, which is exactly correct for $89 \times 7$.

Common mistakes

Many mistakes in regrouping occur because the steps are applied in the wrong order.

• Adding before multiplying: A frequent error is adding the carried digit to the top number before performing the multiplication. Always multiply the bottom and top digits first, then add the carried digit to that product.
• Forgetting to add the carried digit: You must remember to add the extra amount you placed above the column. Writing the carried digit clearly and crossing it out after using it can help prevent this mistake.
• Misplacing the carried digit: Always carry the tens digit to the next column on the left, not the ones digit. If the product is $24$, carry the $2$, not the $4$.
  

Always multiply first, then add the carried digit.

Frequently asked questions

How do I regroup when multiplying two-digit numbers by two-digit numbers?

The regrouping process is exactly the same. You multiply the ones digit of the bottom number across the top number, regrouping as needed. Then, you place a zero as a placeholder in the ones column and multiply the tens digit of the bottom number across the top number, regrouping again as needed. Finally, you add the two partial products together.

How does regrouping work with decimal numbers?

When multiplying with decimals, you apply the same regrouping steps as whole numbers. Ignore the decimal point while multiplying. After you find the final product, count the total number of decimal places in both original numbers, and place the decimal point in the answer so it has the same number of decimal places.

How can I be sure my regrouped answer is correct?

You can verify your final product by checking multiplication and division relationships. Divide your final answer by the multiplier; if the result matches the multiplicand, your calculation is correct.