Lesson Objectives
  • Demonstrate an understanding of place value
  • Demonstrate an understanding of single-digit multiplication
  • Learn how to set up multi-digit multiplication problems in a vertical format
  • Learn how to perform multi-digit multiplication with regrouping (carrying)

How to Multiply Multi-Digit Whole Numbers with Regrouping (Carrying)


At this point, you should be fairly comfortable with the single-digit multiplication facts covered in our properties of multiplication lesson. In that lesson, we saw a basic times table for the numbers 1 - 9. We also learned several properties of multiplication such as: the commutative property of multiplication, the associative property of multiplication, the identity property of 1, the multiplication property of 0, and the distributive property of multiplication. Once we have mastered our single-digit multiplication facts, it’s time to move on to multiplying multi-digit whole numbers.
When we multiply multi-digit whole numbers together, we generally use a process known as vertical multiplication. This process will allow us to break our multi-digit multiplication problem down into a series of single-digit multiplication problems. In order to completely understand the process, it is imperative to have a good understanding of place value.

Vertical Multiplication

  • Set up the vertical multiplication by stacking the factors vertically and lining up the digits by place value. Although multiplication is commutative (order is not important) we want to place the number with more digits on top. If the two numbers have the same number of digits, either number can be on top.
  • Draw a multiplication symbol "x" to the left of the bottom number and a horizontal line underneath the bottom number.
  • We start our multiplication with the ones' place digit (rightmost) in the bottom column. We will multiply this digit by each digit in the top number working right to left. After each multiplication, we write the individual answers below the horizontal line working right to left. The placement here is very important to maintain the proper place value.
    • If the result of a particular multiplication is larger than 9, we will use regrouping (carrying). When this occurs, write the right digit of the number down into the answer. We then carry the left digit above the next column to the left. This digit will be added to the result of the next multiplication.
  • We continue our multiplication by shifting to the next digit left in the bottom number. We will repeat the process of multiplying this digit by each digit in the top number and regrouping (carrying) when needed. The most important thing here is to start the placement of the answers from this multiplication on a new row and one place to the left. This is done to ensure proper place value in our answer.
  • Once we have completed the process and multiplied each number in the bottom row by each number in the top row, we are ready to add. We now set up a vertical addition and find the sum of the amounts in the answer section.
Let's take a look at a few examples:
Example 1: Multiply 113 x 12
  • Set up the vertical multiplication by stacking the factors vertically and lining up the digits by place value. Although multiplication is commutative (order is not important) we want to place the number with more digits on top. If the two numbers have the same number of digits, either number can be on top.
  • Draw a multiplication symbol "x" to the left of the bottom number and a horizontal line underneath the bottom number.
  • Multiplying 113 x 12, Step 1 and 2
  • We start our multiplication with the ones' place digit (rightmost) in the bottom column. We will multiply this digit by each digit in the top number working right to left. After each multiplication, we write the individual answers below the horizontal line working right to left. The placement here is very important to maintain the proper place value.
    • If the result of a particular multiplication is larger than 9, we will use regrouping (carrying). When this occurs, write the right digit of the number down into the answer. We then carry the left digit above the next column to the left. This digit will be added to the result of the next multiplication.
    Multiplying 113 x 12, Step 3
    • Multiply starting with the rightmost digit of the bottom number 2. This digit will multiply each digit of the top number.
    • 2 x 3 = 6
    • 2 x 1 = 2
    • 2 x 1 = 2
    • After each multiplication, the result is written directly below the horizontal line working right to left.
  • We continue our multiplication by shifting to the next digit left in the bottom number. We will repeat the process of multiplying this digit by each digit in the top number and regrouping (carrying) when needed. The most important thing here is to start the placement of the answers from this multiplication on a new row and one place to the left. This is done to ensure proper place value in our answer. Multiplying 113 x 12, Step 4
    • Move one digit left in the bottom number (1). This digit will multiply each digit of the top number.
    • We start a new row to write our answers. We start the answers one place left (tens' place) and work right.
    • 1 x 3 = 3
    • 1 x 1 = 1
    • 1 x 1 = 1
  • Once we have completed the process and multiplied each number in the bottom row by each number in the top row, we are ready to add. We now set up a vertical addition and find the sum of the amounts in the answer section. Multiplying 113 x 12, Step 5
    • Find the sum of 226 and 1130.
    • 6 + 0 = 6 (or you can just bring the 6 down)
    • 2 + 3 = 5
    • 2 + 1 = 3
    • 1 + 0 = 1 (or you can just bring the 1 down)
    • 226 + 1130 = 1356
113 x 12 = 1356
Let's try an example with carrying involved.
Example 2: Multiply 305 x 49
  • Set up the vertical multiplication by stacking the factors vertically and lining up the digits by place value. Although multiplication is commutative (order is not important) we want to place the number with more digits on top. If the two numbers have the same number of digits, either number can be on top.
  • Draw a multiplication symbol "x" to the left of the bottom number and a horizontal line underneath the bottom number.
  • Multiplying 305 x 49, Step 1 and 2
  • We start our multiplication with the ones' place digit (rightmost) in the bottom column. We will multiply this digit by each digit in the top number working right to left. After each multiplication, we write the individual answers below the horizontal line working right to left. The placement here is very important to maintain the proper place value.
    • If the result of a particular multiplication is larger than 9, we will use regrouping (carrying). When this occurs, write the right digit of the number down into the answer. We then carry the left digit above the next column to the left. This digit will be added to the result of the next multiplication.
    Multiplying 305 x 49, Step 3
    • Multiply starting with the rightmost digit of the bottom number 9. This digit will multiply each digit of the top number.
    • 9 x 5 = 45
    • Since 45 is a two digit number, place the 5 directly below and carry the 4 into the next column left
    • 9 x 0 = 0
    • We then add the 4 which was carried over. This results in 0 + 4 which is 4. This will be written directly below
    • 9 x 3 = 27
    • Since there are no more columns we can simply write the 27 and move on to the next step
  • We continue our multiplication by shifting to the next digit left in the bottom number. We will repeat the process of multiplying this digit by each digit in the top number and regrouping (carrying) when needed. The most important thing here is to start the placement of the answers from this multiplication on a new row and one place to the left. This is done to ensure proper place value in our answer. Multiplying 305 x 49, 4
    • Move one digit left in the bottom number (4). This digit will multiply each digit of the top number.
    • We start a new row to write our answers. We start the answers one place left (tens' place) and work right.
    • 4 x 5 = 20
    • Since 20 is a two digit number, place the 0 below and carry the 2 into the next column left
    • 4 x 0 = 0
    • We then add the 2 which was carried over. This results in 0 + 2 which is 2. This will be written below
    • 4 x 3 = 12
    • Since there are no more columns we can simply write the 12 and move on to the next step
  • Once we have completed the process and multiplied each number in the bottom row by each number in the top row, we are ready to add. We now set up a vertical addition and find the sum of the amounts in the answer section. Multiplying 305 x 49, Step 5
    • Find the sum of 2745 and 12,200.
    • 5 + 0 = 5 (or you can just bring the 5 down)
    • 4 + 0 = 4
    • 7 + 2 = 9
    • 2 + 2 = 4
    • 1 + 0 = 1 (or you can just bring the 1 down)
    • 2745 + 12,200 = 14,945
305 x 49 = 14,945