Multiplying One-Digit Numbers by Multiples of 10 - Third Grade Mathematics

Once you know your basic one-digit multiplication facts, you hold the key to multiplying larger numbers by multiples of 10 (such as 20, 30, 40, 50, 60, 70, 80, and 90). In this chapter, you will discover the underlying place value structure of multiplying by tens and master the fast mental math technique that allows you to multiply numbers like 4 x 60 in seconds.

What Is a Multiple of 10?

A multiple of 10 is any number created by multiplying a whole number by 10. Multiples of 10 always end with at least one zero in the ones place: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120...

You can think of every multiple of 10 as a collection of ten-rods:
- 20 = 2 tens

- 30 = 3 tens

- 50 = 5 tens

- 80 = 8 tens


The Place Value Logic Behind Multiplying by Multiples of 10

Consider the problem: 3 x 40.

Instead of writing 40 as a plain number, rephrase it using place value units: 3 x (4 tens)

Multiply the single-digit numbers first: 3 x 4 = 12

Now attach the unit name: 12 tens!

What is the value of 12 tens? 12 tens = 120!

Equation:       3 x 40 = 120
Thinking:       3 x 4 tens = 12 tens = 120

Visualizing with Base-Ten Blocks:
- Group 1: [||||||||||] [||||||||||] [||||||||||] [||||||||||] (4 tens)

- Group 2: [||||||||||] [||||||||||] [||||||||||] [||||||||||] (4 tens)

- Group 3: [||||||||||] [||||||||||] [||||||||||] [||||||||||] (4 tens)
Total rods: 12 rods of 10 = 120 units!

The Fast Mental Math Rule

To multiply a one-digit number by a multiple of 10:
1. Identify the basic fact: multiply the non-zero digits together.

2. Count the zero in the multiple of 10.

3. Attach that zero to the end of the basic fact product.


Example 1: 5 x 60
- Basic fact: 5 x 6 = 30
- Attach the zero: 300!
- Answer: 5 x 60 = 300.
(Notice that 30 already had a zero from 5 x 6, so attaching the tens zero makes 300!)

Example 2: 7 x 40
- Basic fact: 7 x 4 = 28
- Attach the zero: 280
- Answer: 7 x 40 = 280.

Example 3: 9 x 80
- Basic fact: 9 x 8 = 72
- Attach the zero: 720
- Answer: 9 x 80 = 720.

Using the Associative Property to Explain Why It Works

You can prove this rule mathematically using the Associative Property of Multiplication:

Problem: 4 x 50
Step 1: Write 50 as (5 x 10)
        4 x (5 x 10)
Step 2: Regroup using the Associative Property:
        (4 x 5) x 10
Step 3: Multiply inside parentheses:
        20 x 10
Step 4: Multiply by 10:
        200!

This mathematical proof shows that our zero shortcut is not magic—it is pure mathematical logic.

Chapter Practice Exercises

  1. Find the product using the basic fact and place value: a. 3 x 20 b. 6 x 30 c. 4 x 50 d. 8 x 20 e. 7 x 50
  2. Solve: a. 9 x 30 b. 5 x 80 c. 6 x 70 d. 8 x 40 e. 7 x 90
  3. Complete the number sentence: a. 4 x ___ = 240 b. ___ x 50 = 350 c. 8 x ___ = 640
  4. A movie theater has 6 rows of seats with 30 seats in each row. How many seats are in the theater?
  5. A sports shop orders 7 boxes of tennis balls. Each box contains 40 tennis balls. How many tennis balls did the shop order in all?

Solutions and Step-by-Step Answers

  1. Products: a. 3 x 20: 3 x 2 tens = 6 tens = 60. b. 6 x 30: 6 x 3 tens = 18 tens = 180. c. 4 x 50: 4 x 5 tens = 20 tens = 200. d. 8 x 20: 8 x 2 tens = 16 tens = 160. e. 7 x 50: 7 x 5 tens = 35 tens = 350.
  2. Mental math: a. 9 x 30: 9 x 3 = 27 -> 270. b. 5 x 80: 5 x 8 = 40 -> 400. c. 6 x 70: 6 x 7 = 42 -> 420. d. 8 x 40: 8 x 4 = 32 -> 320. e. 7 x 90: 7 x 9 = 63 -> 630.
  3. Missing factors: a. 4 x 60 = 240 (since 24 / 4 = 6). b. 7 x 50 = 350 (since 35 / 5 = 7). c. 8 x 80 = 640 (since 64 / 8 = 8).
  4. 6 rows of 30 seats: 6 x 30 = 180 seats.
  5. 7 boxes of 40 balls: 7 x 40 = 280 tennis balls.