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
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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.
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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!
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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
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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

Think of temperature: -4°C is warmer (greater) than -7°C!
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Try It Yourself: Integers
Put these integers in ascending order (smallest to largest):
-8, 4, 0, -3, 7, -5

Take a moment to work this out before looking at the solution!
Solution:
Ascending order: -8, -5, -3, 0, 4, 7
Remember: Move from left to right on the number line to get ascending order.
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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).
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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
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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

Adding zeroes at the end of a decimal doesn't change its value. 0.5 = 0.50 = 0.500
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Try It Yourself: Decimals
Put these decimals in descending order (largest to smallest):
3.45, 3.5, 3.45, 3.054, 3.504

Remember to align the decimal points and add zeroes as needed!
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.
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Understanding Fractions
A fraction represents a part of a whole, written as:
\frac{\text{numerator}}{\text{denominator}}
  • 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 \frac{3}{4}, the whole is divided into 4 equal parts, and we have 3 of those parts.
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Comparing Fractions: Same Denominator
When fractions have the same denominator, comparing them is straightforward:
Just compare the numerators!
Example:
Compare \frac{3}{8} and \frac{5}{8}
Since the denominators are the same (8), compare the numerators: 3 and 5
Since 3 < 5, we know that \frac{3}{8} < \frac{5}{8}
Quick Check:
Put these fractions in order:
\frac{7}{10}, \frac{3}{10}, \frac{9}{10}
Answer: \frac{3}{10} < \frac{7}{10} < \frac{9}{10}
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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 \frac{2}{3} and \frac{2}{5}
The numerators are the same (2), so compare the denominators: 3 and 5
Since 3 < 5, and we're comparing denominators in reverse, \frac{2}{3} > \frac{2}{5}

Think about it: With \frac{2}{3}, you get 2 out of 3 equal pieces (larger pieces). With \frac{2}{5}, you get 2 out of 5 equal pieces (smaller pieces).
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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: \frac{2}{5} vs \frac{3}{8}
Convert to: \frac{16}{40} vs \frac{15}{40}
Therefore: \frac{2}{5} > \frac{3}{8}
Convert to Decimals
Divide the numerator by the denominator, then compare the decimals.
Example: \frac{4}{5} = 0.8 and \frac{5}{7} ≈ 0.714
Since 0.8 > 0.714, \frac{4}{5} > \frac{5}{7}
Cross Multiply
Multiply the numerator of each fraction by the denominator of the other.
Example: \frac{3}{4} vs \frac{2}{3}
3 × 3 = 9 and 4 × 2 = 8
Since 9 > 8, \frac{3}{4} > \frac{2}{3}
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Finding a Common Denominator
Method 1: Multiply Denominators
The simplest (but not always most efficient) way:
Example: Compare \frac{2}{3} and \frac{3}{5}
  1. Multiply denominators: 3 × 5 = 15
  1. Convert \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}
  1. Convert \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}
  1. Compare: \frac{10}{15} > \frac{9}{15}
Method 2: Find the LCM
The Least Common Multiple of the denominators:
Example: Compare \frac{3}{4} and \frac{5}{6}
  1. Find LCM of 4 and 6 = 12
  1. Convert \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
  1. Convert \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}
  1. Compare: \frac{9}{12} < \frac{10}{12}
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Cross Multiplication Method
Cross multiplication is a quick way to compare fractions without finding common denominators.
Steps:
  1. Set up the fractions: \frac{a}{b} and \frac{c}{d}
  1. Multiply diagonally: a × d and b × c
  1. Compare the products:
  • If a × d > b × c, then \frac{a}{b} > \frac{c}{d}
  • If a × d < b × c, then \frac{a}{b} < \frac{c}{d}
  • If a × d = b × c, then \frac{a}{b} = \frac{c}{d}
Example: Compare \frac{4}{7} and \frac{5}{9}
Cross multiply: 4 × 9 = 36 and 7 × 5 = 35
Since 36 > 35, \frac{4}{7} > \frac{5}{9}
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Try It Yourself: Fractions
Put these fractions in ascending order (smallest to largest):
\frac{3}{8}, \frac{1}{2}, \frac{2}{3}, \frac{5}{12}, \frac{7}{10}

Choose a method that works best for you: common denominator, conversion to decimals, or cross multiplication.
Solution:
Let's convert to a common denominator of 120:
\frac{3}{8}
\frac{45}{120}
\frac{1}{2}
\frac{60}{120}
\frac{5}{12}
\frac{50}{120}
\frac{7}{10}
\frac{84}{120}
\frac{2}{3}
\frac{80}{120}
Ascending order: \frac{3}{8}, \frac{5}{12}, \frac{1}{2}, \frac{2}{3}, \frac{7}{10}
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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.
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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 \frac{1}{4}, \frac{1}{2}, and \frac{3}{4} on a number line:
The number line helps us see the correct ordering: \frac{1}{4} < \frac{1}{2} < \frac{3}{4}
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Mixed Numbers and Improper Fractions
Mixed Numbers
A whole number and a fraction combined:
Example: 2\frac{3}{4} = 2 + \frac{3}{4}
This means 2 whole units plus \frac{3}{4} 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: \frac{11}{4}
Can be converted to mixed number: \frac{11}{4} = 2\frac{3}{4}
To convert: divide numerator by denominator (11 ÷ 4 = 2 remainder 3)
When comparing numbers, you can convert between these forms to make comparison easier.
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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
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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: \frac{3}{10} of £100
  • Bus fare: £2\frac{1}{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
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Common Mistakes to Avoid
Comparing Only Numerators or Only Denominators
INCORRECT: \frac{5}{8} > \frac{4}{5} 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
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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.