Hello Maths Explorers! Let's Dive into Fractions!

Have you ever shared a pizza with friends or cut a cake for a birthday? If you have, then you've used fractions! Fractions are a super useful way to talk about parts of a whole thing. In these notes, we're going to learn about the three main types of fractions. Don't worry if this seems tricky at first – we'll use fun examples and take it one step at a time. You've got this!


What is a Fraction? (A Quick Review)

A fraction is made of two parts: a top number and a bottom number, separated by a line.

$$ \frac{\text{Numerator}}{\text{Denominator}} $$

  • The Numerator is the top number. It tells us how many parts we have.
  • The Denominator is the bottom number. It tells us how many equal parts the whole is divided into.

Example: Imagine a pizza cut into 8 equal slices. If you eat 3 slices, you have eaten $$ \frac{3}{8} $$ of the pizza! Here, 3 is the numerator and 8 is the denominator.

Key Takeaway

A fraction shows a part of a whole. The top number is what you have, and the bottom number is the total number of parts.


Meet the Three Types of Fractions

Fractions come in three main types. Let's get to know them!

1. Proper Fractions (The "Normal" Slice)

A proper fraction is a fraction where the numerator is smaller than the denominator. This means it is always less than one whole thing.

Think of it as having some slices, but not the whole pizza yet.

Examples of Proper Fractions:
  • $$ \frac{1}{4} $$ (One slice of a pizza cut into four)
  • $$ \frac{5}{8} $$ (Five slices of a pizza cut into eight)
  • $$ \frac{2}{3} $$ (Two parts of something split into three)
Quick Review Box

Proper Fraction: Top number is SMALLER than the bottom number.
Value: Always less than 1 whole.


2. Improper Fractions (The "Top-Heavy" Fraction)

An improper fraction is a fraction where the numerator is bigger than or equal to the denominator. We sometimes call these "top-heavy" because the top number is bigger! This means it is equal to 1 whole thing or more than 1 whole thing.

Think of it as having at least one whole pizza, maybe even more!

Examples of Improper Fractions:
  • $$ \frac{5}{4} $$ (You have 5 slices, but each pizza is only cut into 4. So you have one whole pizza and one extra slice!)
  • $$ \frac{8}{8} $$ (You have all 8 slices of a pizza cut into 8. That's exactly 1 whole pizza!)
  • $$ \frac{6}{3} $$ (You have 6 slices, and pizzas are cut into 3. That means you have exactly 2 whole pizzas!)
Quick Review Box

Improper Fraction: Top number is BIGGER than or EQUAL to the bottom number.
Value: 1 whole or more than 1 whole.


3. Mixed Numbers (The "Whole and a Bit")

A mixed number is a mix of a whole number and a proper fraction together. It’s another way to show a value that is more than one whole.

This is like saying "I have 1 whole pizza and 1 extra slice."

Examples of Mixed Numbers:
  • $$ 1\frac{1}{4} $$ (This means one whole and one quarter. It's the same as the improper fraction $$ \frac{5}{4} $$!)
  • $$ 3\frac{1}{2} $$ (Three whole things and a half)
  • $$ 2\frac{3}{5} $$ (Two whole things and three-fifths)

Important: A mixed number is really a whole number plus a proper fraction. So, `$$ 1\frac{1}{4} $$` is the same as `$$ 1 + \frac{1}{4} $$`.

Quick Review Box

Mixed Number: A whole number and a proper fraction side-by-side.
Value: Always more than 1 whole.


The Big Switch-Up: Converting Fractions

Improper fractions and mixed numbers are just two different ways of writing the same amount. We can switch between them! Learning how to do this is a very important maths skill.

How to Change an Improper Fraction into a Mixed Number

Let's change the improper fraction $$ \frac{7}{3} $$ into a mixed number. It's as easy as dividing!

Step 1: Divide the numerator by the denominator.

Divide 7 by 3.
$$ 7 \div 3 = 2 $$ with a remainder of 1.

Step 2: Use the results to build your mixed number.

  • The result of your division (2) becomes the big whole number.
  • The remainder (1) becomes the new numerator.
  • The denominator (3) stays the same!

So, $$ \frac{7}{3} $$ becomes $$ 2\frac{1}{3} $$!

Key Takeaway: Improper to Mixed

Divide the top by the bottom.
The answer is the Whole Number.
The remainder is the new Top Number.
The Bottom Number Stays the Same.


How to Change a Mixed Number into an Improper Fraction

Let's change the mixed number $$ 3\frac{2}{5} $$ into an improper fraction. We can use a fun trick for this called MAD!

M.A.D. stands for:
M = Multiply
A = Add
D = Denominator stays the same

Step 1 (M - Multiply): Multiply the whole number by the denominator.

The whole number is 3 and the denominator is 5.
$$ 3 \times 5 = 15 $$

Step 2 (A - Add): Add the result to the numerator.

Our result was 15, and the old numerator is 2.
$$ 15 + 2 = 17 $$
This is our new numerator!

Step 3 (D - Denominator): The denominator stays the same.

The denominator was 5, so it stays 5.

So, $$ 3\frac{2}{5} $$ becomes $$ \frac{17}{5} $$!

Key Takeaway: Mixed to Improper

Remember MAD!
1. (M)ultiply the whole number by the denominator.
2. (A)dd the result to the numerator.
3. The (D)enominator stays the same.

Did you know?

The line in a fraction, called a vinculum, is just another symbol for division! That's why we use division to turn an improper fraction into a mixed number. So $$ \frac{10}{2} $$ really means $$ 10 \div 2 $$, which equals 5!