Multiplying whole numbers

Multiplication is a way of adding a number to itself multiple times. It's a faster way to do repeated addition. When we multiply two numbers, we're finding the total amount when we have a certain number of equal groups. Multiplying whole numbers starts with a solid knowledge of the multiplication table. The more you know your multiplication table, the easier it will be for you to perform multiplication of any size!

Here's why multiplication tables are so important:

• They are the building blocks for all multiplication
• They allow for quick mental calculations
• They help you spot errors in your work
• They make more complex multiplication much easier

Key Terms in Multiplication

• Factors: The numbers being multiplied together
• Multiplicand: The number being multiplied (the number in each group)
• Multiplier: The number of groups
• Product: The result of the multiplication

For example, in the multiplication 5 × 3 = 15:

• 5 and 3 are the factors
• 5 is the multiplicand (we have 5 in each group)
• 3 is the multiplier (we have 3 groups)
• 15 is the product

Single Digit Multiplication Examples

The easiest multiplication we can perform is with single digits because all we need is a good remembrance of the multiplication table. This we can see in the examples below.

4

× 3

12

6

× 5

30

7

× 4

28

9

× 6

54

An Example of Multi-digit Multiplication Without Regrouping

Example: 231 × 2

2	3	1
×		2
4	6	2

Steps:

1. 2 × 1 = 2
2. 2 × 3 = 6
3. 2 × 2 = 4

Multiplying a three-digit number by a one-digit may be a little bit more fun when there is regrouping. The following is a multiplication of three-digit by a one digit-number (247 × 3) with regrouping.

Please study this example carefully since other examples will build on this one!

Understanding Regrouping in Multiplication

Before we set up our multiplication vertically, let's understand what happens when we need to regroup (carry) numbers. We'll break down the process using 247 × 3 as an example.

Breaking Down 247 × 3

Step 1: Break into place values

247 = 200 + 40 + 7

We'll multiply each part by 3

↓

Step 2: Multiply the ones (7 × 3)

7 × 3 = 21

This is where our first regrouping happens!

• Write down 1 in the ones place
• Carry the 2 to the tens place

↓

Step 3: Multiply the tens (40 × 3) and add carried value

40 × 3 = 120

120 + 20 (carried) = 140

• Write down 4 in the tens place
• Carry the 1 to the hundreds place

↓

Step 4: Multiply the hundreds (200 × 3) and add carried value

200 × 3 = 600

600 + 100 (carried) = 700

Write down 7 in the hundreds place

↓

Final Result: 741

Now we can set this up vertically to make it easier to solve!

You can set it up vertically and get the same answer faster with less writing

Example: 247 × 3

1	2
2	4	7
×		3
7	4	1

Steps:

1. 3 × 7 = 21; Write 1, carry 2
2. 3 × 4 = 12, plus carried 2 = 14; Write 4, carry 1
3. 3 × 2 = 6, plus carried 1 = 7

Multiplying a three-digit number by a two-digit number with carry

Example: 456 × 27

First Partial Product (× 7)

3	3	4
	4	5	6
			7
3	1	9	2

Steps:

1. 7 × 6 = 42: Write 2, carry 4
2. 7 × 5 = 35, plus carried 4 = 39: Write 9, carry 3
3. 7 × 4 = 28, plus carried 3 = 31: Write 1, carry 3
4. Write the carried 3

Second Partial Product (× 20)

	1	1
	4	5	6
		2
0	9	1	2	0

Steps:

1. 2 × 6 = 12: Write 2, carry 1
2. 2 × 5 = 10, plus carried 1 = 11: Write 1, carry 1
3. 2 × 4 = 8, plus carried 1 = 9: Write 9
4. Add final 0 because we're actually multiplying by 2. In other words since we are multiplying by 20, the digit in the ones place does not exist. But we can put a 0 as a placeholder

Final Addition

1		1
	3	1	9	2
0	9	1	2	0
1	2	3	1	2

Steps:

1. 2 + 0 = 2
2. 9 + 2 = 11: Write 1, carry 1
3. 1 + 1 + 1 = 3
4. 3 + 9 = 12: Write 2, carry 1
5. 0 + 1 = 1

Key Points:

• Each partial product is calculated separately with its own carry numbers
• The 2 is placed under the 5 because we're multiplying by 20 (2 tens)
• Red numbers show carries within multiplication steps
• Green numbers show carries during final addition
• Final answer: 456 × 27 = 12,312