Multiplication of Whole Numbers

What You’ll Learn

In this lesson you’ll start with what multiplication really means using small numbers, then build up to multi-digit multiplication with the standard column method.

The Basics

Multiplication is a shortcut for repeated addition. Instead of adding the same number over and over, you multiply.

If you buy 4 packs of gum with 3 sticks each, you could add $3 + 3 + 3 + 3 = 12$. Or just say $4 \times 3 = 12$. Same answer, less work.

Here are a few single-digit examples:

• $2 \times 5 = 10$ - two groups of five
• $3 \times 4 = 12$ - three groups of four
• $6 \times 7 = 42$
• $8 \times 9 = 72$

Some handy patterns to notice:

• Anything times 1 stays the same: $7 \times 1 = 7$
• Anything times 0 is 0: $5 \times 0 = 0$
• Order doesn’t matter: $3 \times 4 = 4 \times 3 = 12$ (this is called the commutative property)
• Multiplying by 10 just adds a zero: $6 \times 10 = 60$

If you don’t have every multiplication fact memorized, that’s fine. You can always build up: $7 \times 8$ is the same as $7 \times 4 + 7 \times 4 = 28 + 28 = 56$. With practice, the common ones stick.

Here’s the full multiplication table from 1 to 12. The green diagonal shows the perfect squares (a number times itself). Use this as a reference until the facts become second nature.

×	1	2	3	4	5	6	7	8	9	10	11	12
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144

Multi-Digit Multiplication

Once single-digit multiplication feels natural, the same idea scales up. You just multiply one digit at a time and add the results. This is called long multiplication (or the standard algorithm), and it’s the classic pencil-and-paper method.

How It Works

Write the numbers vertically, lined up by place value. Then:

1. Multiply the top number by the ones digit of the bottom number.
2. Write that result (the first partial product).
3. Drop down a row, place a zero in the ones column (because you’re now multiplying by tens).
4. Multiply the top number by the tens digit.
5. Add the partial products together.

If any single multiplication gives 10 or more, write the ones digit and carry the tens digit to the next column, just like addition.

Step-by-Step: $756 \times 32$

$$\begin{array}{rrrrr} & & \scriptsize{1} & \scriptsize{1} & \\ & & 7 & 5 & 6 \\ \times & & & 3 & 2 \\ \hline & 1 & 5 & 1 & 2 \\ 2 & 2 & 6 & 8 & 0 \\ \hline 2 & 4 & 1 & 9 & 2 \end{array}$$

Here’s what happened in each row:

Row 1 - Multiply 756 by the ones digit (2):

• $2 \times 6 = 12$ - write 2, carry 1
• $2 \times 5 = 10$, plus the carry = 11 - write 1, carry 1
• $2 \times 7 = 14$, plus the carry = 15 - write 15

First partial product: $1{,}512$

Row 2 - Multiply 756 by the tens digit (3), shifted one place left:

• Place a 0 in the ones column (because 3 is in the tens place)
• $3 \times 6 = 18$ - write 8, carry 1
• $3 \times 5 = 15$, plus the carry = 16 - write 6, carry 1
• $3 \times 7 = 21$, plus the carry = 22 - write 22

Second partial product: $22{,}680$

Add the rows: $1{,}512 + 22{,}680 = 24{,}192$

Estimate to check: $750 \times 30 = 22{,}500$. In the right ballpark.

Why This Works

Long multiplication uses the distributive property under the hood:

$$\begin{aligned} 756 \times 32 &= 756 \times 2 + 756 \times 30 \\ &= 1{,}512 + 22{,}680 \\ &= 24{,}192 \end{aligned}$$

Each partial product handles one digit of the bottom number, and the shifting (adding zeros) accounts for place value. The method works for any size numbers. Three digits times three digits just means three partial products instead of two.

A Smaller Example: $23 \times 14$

$$\begin{array}{rrrr} & & \scriptsize{1} & \\ & & 2 & 3 \\ \times & & 1 & 4 \\ \hline & & 9 & 2 \\ & 2 & 3 & 0 \\ \hline & 3 & 2 & 2 \end{array}$$

Here’s what happened in each row:

Row 1 - Multiply 23 by the ones digit (4):

• $4 \times 3 = 12$ - write 2, carry 1
• $4 \times 2 = 8$, plus the carry = 9 - write 9

First partial product: $92$

Row 2 - Multiply 23 by the tens digit (1), shifted one place left:

• Place a 0 in the ones column (because 1 is in the tens place)
• $1 \times 3 = 3$ - write 3
• $1 \times 2 = 2$ - write 2

Second partial product: $230$

Add the rows: $92 + 230 = 322$

Estimate to check: $20 \times 15 = 300$. In the right ballpark.

Worked Examples

Multiply $456 \times 32$

$$\begin{array}{rrrrr} & & \scriptsize{1} & \scriptsize{1} & \\ & & 4 & 5 & 6 \\ \times & & & 3 & 2 \\ \hline & & 9 & 1 & 2 \\ 1 & 3 & 6 & 8 & 0 \\ \hline 1 & 4 & 5 & 9 & 2 \end{array}$$

Estimate to check: $450 \times 30 = 13{,}500$. Close to 14,592, so our answer is reasonable.

Here’s what happened in each row:

Row 1 - Multiply 456 by the ones digit (2):

• $2 \times 6 = 12$ - write 2, carry 1
• $2 \times 5 = 10$, plus the carry = 11 - write 1, carry 1
• $2 \times 4 = 8$, plus the carry = 9 - write 9

First partial product: $912$

Row 2 - Multiply 456 by the tens digit (3), shifted one place left:

• Place a 0 in the ones column (because 3 is in the tens place)
• $3 \times 6 = 18$ - write 8, carry 1
• $3 \times 5 = 15$, plus the carry = 16 - write 6, carry 1
• $3 \times 4 = 12$, plus the carry = 13 - write 13

Second partial product: $13{,}680$

Add the rows: $912 + 13{,}680 = 14{,}592$

Real-World Applications

Multiplication calculates totals:

• 12 items at 7 dollars each = 84 dollars
• Area of a room: $12 \times 15 = 180$ square feet for flooring
• Weekly pay: 40 hours $\times$ 18 dollars/hour = 720 dollars

It scales recipes, budgets, or work estimates quickly.

You’ve Got This

Multiplication is just adding groups. If you get stuck on a big problem, break it into tens and ones. Practice with small multipliers first, line up your columns neatly, and double-check by estimating. You’ll handle everyday calculations with ease.

Quiz

What is 34 �- 6?

• A.194
• B.204
• C.214
• D.180

Multiply 125 �- 40.

• A.5,200
• B.500
• C.5,000
• D.4,000

A store sells 28 packs of batteries at 15 dollars each. Total cost?

• A.430 dollars
• B.400 dollars
• C.380 dollars
• D.420 dollars

What is 213 �- 23?

• A.4,989
• B.5,000
• C.4,899
• D.4,799

What is 50 �- 60?

• A.3,000
• B.300
• C.3,600
• D.110