Multiply and Divide

Introduction

Our number system is based on the number 10. This is called the base-ten place value system.

Every time we move one place to the left in a number, the value becomes ten times larger. Every time we move one place to the right, the value becomes ten times smaller.

Recognizing these patterns makes it much easier to multiply, divide, estimate, and work with large and small numbers.

The Base-Ten Number System

Our number system uses ten digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

The value of a digit depends on its position.

Each place is worth 10 times the place immediately to its right.

Place Value	Value
Thousands	1,000
Hundreds	100
Tens	10
Ones	1

Notice the pattern:

• 1 × 10 = 10
• 10 × 10 = 100
• 100 × 10 = 1,000
• 1,000 × 10 = 10,000

These are called powers of ten.

Powers of Ten

A power of ten is a number made by multiplying 10 by itself one or more times.

Power	Value
10⁰	1
10¹	10
10²	100
10³	1,000
10⁴	10,000
10⁵	100,000

Each power of ten is ten times larger than the previous one.

Multiplying by 10, 100, and 1000

When you multiply a whole number by a power of ten, every digit moves to the left.

• ×10 → move one place left.
• ×100 → move two places left.
• ×1000 → move three places left.

The number becomes larger because each digit moves into a place worth ten times as much.

Examples

Calculation	Answer
6 × 10	60
24 × 10	240
35 × 100	3,500
482 × 100	48,200
7 × 1000	7,000
125 × 1000	125,000

Notice that the digits move, not simply "zeros are added." Thinking about place value is more important than memorizing shortcuts.

Dividing by 10, 100, and 1000

When dividing by a power of ten, every digit moves to the right.

• ÷10 → move one place right.
• ÷100 → move two places right.
• ÷1000 → move three places right.

The number becomes smaller because each digit moves into a place worth one-tenth as much.

Examples

Calculation	Answer
80 ÷ 10	8
560 ÷ 10	56
4,300 ÷ 100	43
75,000 ÷ 1000	75
126,000 ÷ 1000	126

Understanding the Pattern

Look at the number 347.

Operation	Result
347	347
347 × 10	3,470
347 × 100	34,700
347 × 1000	347,000

Now divide:

Operation	Result
347	347
347 ÷ 10	34.7
347 ÷ 100	3.47
347 ÷ 1000	0.347

Each multiplication moves the digits one place to the left for every factor of 10.

Each division moves the digits one place to the right for every factor of 10.

Patterns in a Place Value Chart

Every place value is related to the next.

Place	Relationship
Ones → Tens	×10
Tens → Hundreds	×10
Hundreds → Thousands	×10
Thousands → Ten Thousands	×10

Moving the opposite direction means dividing by 10 each time.

Making Predictions Using Place Value

Understanding place value patterns helps us predict answers without lengthy calculations.

Example 1

If:

42 × 10 = 420

Then:

42 × 100 = 4,200

Example 2

If:

5,600 ÷ 100 = 56

Then:

5,600 ÷ 1000 = 5.6

Example 3

If a number is multiplied by 100 twice:

35 × 100 × 100

This is the same as:

35 × 10,000

Answer:

350,000

Recognizing these patterns makes calculations much faster.

Common Mistakes

One common mistake is thinking that multiplying by 10 simply means "adding a zero."

This works for many whole numbers, but not for all numbers.

For example:

6.5 × 10 = 65

The digits have moved one place to the left.

Similarly:

650 ÷ 10 = 65

The digits have moved one place to the right.

Thinking in terms of place value works for both whole numbers and decimals.

Real-World Applications

Understanding place value patterns is useful in many situations, including:

• Converting metric units.
• Estimating calculations.
• Working with money.
• Reading population data.
• Scientific measurements.
• Engineering calculations.
• Computer programming.
• Organizing large sets of data.

Worked Examples

Example 1

Calculate:

48 × 100

Answer:

4,800

Example 2

Calculate:

7,500 ÷ 100

Answer:

75

Example 3

Complete the pattern.

8

80

800

8,000

Answer:

Each number is multiplied by 10.

Example 4

What happens to the digits when a number is divided by 100?

Answer:

Each digit moves two places to the right, making the number one hundred times smaller.

Example 5

Predict:

250 × 1000

Answer:

250,000

Did You Know?

The metric system is based on powers of ten. This makes converting between units very simple. For example, 1 kilometre = 1,000 metres, 1 metre = 100 centimetres, and 1 kilogram = 1,000 grams. Because everything is based on powers of ten, metric conversions are much easier than converting between imperial units such as feet, inches, and miles.

Key Terms

Term	Definition
Place Value	The value of a digit determined by its position in a number.
Base-Ten System	A number system in which each place value is ten times the value of the place to its right.
Power of Ten	A number produced by multiplying 10 by itself one or more times.
Multiply	To increase a number by repeated groups or scaling.
Divide	To split a number into equal parts or reduce its size by a factor.

Key Takeaways

• Our number system is based on powers of ten.
• Every place value is ten times greater than the place to its right.
• Multiplying by 10, 100, or 1000 moves every digit to the left by one, two, or three places.
• Dividing by 10, 100, or 1000 moves every digit to the right by one, two, or three places.
• Understanding place value patterns makes calculations faster and helps predict answers.
• Place value patterns are used throughout mathematics, science, engineering, and the metric system.