Comparing and Ordering Numbers

A journey through the fascinating world of number relationships, where we'll discover how to compare and order integers, decimals, and fractions with confidence and precision.

Created by Om Dhani, your dedicated maths tutor | Visit odmaths.co.uk for more resources

What We'll Learn

1

Compare positive and negative integers

Understand which numbers are greater or smaller and why

2

Order decimals

Arrange decimal numbers from smallest to largest and vice versa

3

Compare and order fractions

Develop strategies for determining relative sizes of fractions

4

Use number lines

Visualize number relationships and order on a number line

5

Apply to real-world problems

Use comparison skills in practical situations

odmaths.co.uk | Om Dhani, Tutor

Why This Matters

Comparing and ordering numbers isn't just about maths—it's a skill you'll use throughout your life!

  • Understand temperature changes and weather forecasts
  • Compare prices to find the best deals
  • Track sports statistics like scores and rankings
  • Manage money in bank accounts
  • Interpret scientific measurements and data
  • Read maps with different elevations

These skills form the foundation for algebra, statistics, and advanced mathematics you'll encounter in the future.

odmaths.co.uk | Om Dhani, Tutor

Understanding Number Relationships

Greater Than (>)

When a number is larger than another number

Example: 8 > 3

Less Than (<)

When a number is smaller than another number

Example: 4 < 9

Equal To (=)

When two numbers have the same value

Example: 5 = 5

Remember: The symbol always "points" to the smaller number!

odmaths.co.uk | Om Dhani, Tutor

Positive and Negative Integers

Integers are whole numbers and their negatives. They include all positive whole numbers, zero, and negative whole numbers:

... -3, -2, -1, 0, 1, 2, 3 ...

Important Rules:

  • Numbers increase in value as you move to the right on a number line
  • Positive numbers are always greater than negative numbers
  • Zero is greater than any negative number
  • The further from zero a negative number is, the smaller its value

odmaths.co.uk | Om Dhani, Tutor

Comparing Integers: Examples

Example 1:

Compare 5 and -3

5 is positive, -3 is negative

Since any positive number is greater than any negative number:

5 > -3

Example 2:

Compare -4 and -7

Both are negative, but -4 is closer to zero

The closer a negative number is to zero, the greater it is:

-4 > -7

odmaths.co.uk | Om Dhani, Tutor

Try It Yourself: Integers

Put these integers in ascending order (smallest to largest):

-8, 4, 0, -3, 7, -5

Solution:

Ascending order: -8, -5, -3, 0, 4, 7

Remember: Move from left to right on the number line to get ascending order.

odmaths.co.uk | Om Dhani, Tutor

Understanding Decimals

Decimals are numbers that include a decimal point separating the whole number and fractional parts.

Each position in a decimal represents a specific place value:

Tens

3

Ones

4

Tenths

5

Hundredths

6

Thousandths

7

The decimal point separates the whole number part (34) from the fractional part (.567).

odmaths.co.uk | Om Dhani, Tutor

Comparing Decimals

When comparing decimals, follow these steps:

Compare the whole numbers first

If they're different, you have your answer! (7.3 > 5.9 because 7 > 5)

If whole numbers are the same, compare the tenths

3.7 > 3.5 because 7 tenths > 5 tenths

If tenths are the same, compare the hundredths

4.38 > 4.35 because 8 hundredths > 5 hundredths

Continue comparing place values until you find a difference

1.425 < 1.428 because 5 thousandths < 8 thousandths

odmaths.co.uk | Om Dhani, Tutor

Decimal Place Value Trick

To compare decimals with different numbers of decimal places, use this trick:

Add zeroes at the end to make them the same length

Example: Compare 0.8 and 0.75

  1. Rewrite 0.8 as 0.80 (adding a zero doesn't change its value)
  1. Now compare 0.80 and 0.75
  1. Since 80 hundredths > 75 hundredths, we know 0.8 > 0.75

odmaths.co.uk | Om Dhani, Tutor

Try It Yourself: Decimals

Put these decimals in descending order (largest to smallest):

3.45, 3.5, 3.45, 3.054, 3.504

Solution:

Step 1: Add zeroes to make all numbers the same length

3.450, 3.500, 3.450, 3.054, 3.504

Step 2: Compare place values until you find differences

Descending order (largest to smallest):

3.504, 3.500, 3.450, 3.450, 3.054

Note: Two numbers (3.45) are equal, so either can come first in the ordering.

odmaths.co.uk | Om Dhani, Tutor

Understanding Fractions

A fraction represents a part of a whole, written as:

  • The numerator (top number) tells us how many parts we have
  • The denominator (bottom number) tells us how many equal parts the whole is divided into

Example: In the fraction , the whole is divided into 4 equal parts, and we have 3 of those parts.

odmaths.co.uk | Om Dhani, Tutor

Comparing Fractions: Same Denominator

When fractions have the same denominator, comparing them is straightforward:

Just compare the numerators!

Example:

Compare and

Since the denominators are the same (8), compare the numerators: 3 and 5

Since 3 < 5, we know that

Quick Check:

Put these fractions in order:


Answer:

odmaths.co.uk | Om Dhani, Tutor

Comparing Fractions: Same Numerator

When fractions have the same numerator, the comparison works differently:

Compare the denominators, but in reverse!

The fraction with the smaller denominator is larger.

Example:

Compare and

The numerators are the same (2), so compare the denominators: 3 and 5

Since 3 < 5, and we're comparing denominators in reverse,

odmaths.co.uk | Om Dhani, Tutor

Comparing Fractions: Different Denominators

When fractions have different numerators AND denominators, we have three main strategies:

Find a Common Denominator

Convert both fractions so they have the same denominator, then compare numerators.

Example: vs

Convert to: vs

Therefore:

Convert to Decimals

Divide the numerator by the denominator, then compare the decimals.

Example: and

Since 0.8 > 0.714,

Cross Multiply

Multiply the numerator of each fraction by the denominator of the other.

Example: vs

3 × 3 = 9 and 4 × 2 = 8

Since 9 > 8,

odmaths.co.uk | Om Dhani, Tutor

Finding a Common Denominator

Method 1: Multiply Denominators

The simplest (but not always most efficient) way:

Example: Compare and

  1. Multiply denominators: 3 × 5 = 15
  1. Convert
  1. Convert
  1. Compare:

Method 2: Find the LCM

The Least Common Multiple of the denominators:

Example: Compare and

  1. Find LCM of 4 and 6 = 12
  1. Convert
  1. Convert
  1. Compare:

odmaths.co.uk | Om Dhani, Tutor

Cross Multiplication Method

Cross multiplication is a quick way to compare fractions without finding common denominators.

Steps:

  1. Set up the fractions: and
  1. Multiply diagonally: a × d and b × c
  1. Compare the products:
  • If a × d > b × c, then
  • If a × d < b × c, then
  • If a × d = b × c, then

Example: Compare and

Cross multiply: 4 × 9 = 36 and 7 × 5 = 35

Since 36 > 35,

odmaths.co.uk | Om Dhani, Tutor

Try It Yourself: Fractions

Put these fractions in ascending order (smallest to largest):


Solution:

Let's convert to a common denominator of 120:











Ascending order:

odmaths.co.uk | Om Dhani, Tutor

The Number Line: A Visual Tool

A number line is a powerful tool for visualizing and comparing different types of numbers.

Benefits of using a number line:

  • Shows the order and relationship between numbers clearly
  • Helps visualize distance between numbers
  • Works for all number types: integers, fractions, decimals
  • Makes it easy to see which numbers are greater or less than others
  • Helps you understand number density (how many numbers exist between others)

Numbers increase in value as you move to the right on the number line.

odmaths.co.uk | Om Dhani, Tutor

Placing Numbers on a Number Line

Example 1: Integers

Placing -3, 0, and 4 on a number line:

Example 2: Decimals

Placing 1.5, 1.75, and 2.25 on a number line:

Example 3: Fractions

Placing and on a number line:

The number line helps us see the correct ordering:

odmaths.co.uk | Om Dhani, Tutor

Mixed Numbers and Improper Fractions

Mixed Numbers

A whole number and a fraction combined:

Example: 2 = 2 +

This means 2 whole units plus of another unit.

On a number line, it's between 2 and 3, closer to 3.

Improper Fractions

Numerator is greater than or equal to denominator:

Example:

Can be converted to mixed number:

To convert: divide numerator by denominator (11 ÷ 4 = 2 remainder 3)

When comparing numbers, you can convert between these forms to make comparison easier.

odmaths.co.uk | Om Dhani, Tutor

Real-World Application: Temperature

Example: Weekly temperatures in Leeds (°C)

Monday: 3.5°C

Tuesday: -2°C

Wednesday: 0°C

Thursday: 7°C

Friday: -4.5°C

Saturday: 5.75°C

Sunday: 2.25°C

Questions:

  1. What was the coldest day?
  1. What was the warmest day?
  1. Order the days from coldest to warmest.

Answers:

  1. Friday (-4.5°C)
  1. Thursday (7°C)
  1. Friday, Tuesday, Wednesday, Sunday, Monday, Saturday, Thursday

odmaths.co.uk | Om Dhani, Tutor

Real-World Application: Money Management

Example: Comparing savings and expenses

Lucy is tracking her savings and expenses. Help her compare these amounts:

  • Shopping: £24.75
  • Savings this month: of £100
  • Bus fare: £2 per day for 12 days
  • Previous savings: £35.50
  • Money earned: £42.75

To compare, we need to convert everything to the same format:

  • Shopping: £24.75
  • Savings this month: £30.00
  • Bus fare total: £30.00
  • Previous savings: £35.50
  • Money earned: £42.75

In ascending order: £24.75 < £30.00 = £30.00 < £35.50 < £42.75

odmaths.co.uk | Om Dhani, Tutor

Common Mistakes to Avoid

Comparing Only Numerators or Only Denominators

INCORRECT: because 5 > 4

CORRECT: Must account for both parts or find common denominators

Forgetting Negative Rules

INCORRECT: -5 > -2 because 5 > 2

CORRECT: -2 > -5 because -2 is closer to zero

Misaligning Decimal Places

INCORRECT: 0.5 < 0.25 because 5 < 25

CORRECT: 0.5 = 0.50 > 0.25 (comparing by place value)

Ignoring the Number Line Direction

INCORRECT: Numbers decrease as you move right

CORRECT: Numbers increase as you move right on the number line

odmaths.co.uk | Om Dhani, Tutor

Summary: Comparing and Ordering Numbers

Integers

  • Positive > 0 > Negative
  • For negatives, those closer to zero are greater

Decimals

  • Compare place values from left to right
  • Add zeroes to match decimal places

Fractions

  • Use common denominators
  • Convert to decimals
  • Try cross multiplication

Key Skills Achieved:

You now have the tools to accurately compare and order any combination of numbers, which will help you in many areas of mathematics and everyday life. Keep practicing these skills—they form the foundation for algebra, statistics, and all advanced mathematics!

Continue your maths journey with Om Dhani at odmaths.co.uk - your trusted tutor for mathematical excellence.