Maths Study Notes: Fractions (Numbers and the Number System)

Welcome to the World of Fractions!

Hello future Mathematicians! Welcome to one of the most fundamental chapters in the GCSE curriculum: Fractions. Don't worry if fractions sometimes feel like a puzzle; we are going to break down every concept step-by-step.
Fractions are just a way of representing a part of a whole. Understanding them is crucial, not just for passing exams, but for everyday life—think cooking, sharing food, or calculating discounts!

1. The Anatomy of a Fraction

A fraction has two main parts separated by a line:

Key Components

\( \frac{\text{Numerator}}{\text{Denominator}} \)

  • The Numerator (The number on top): This tells you how many parts you have.
  • The Denominator (The number on the bottom): This tells you how many equal parts the whole is divided into.

Analogy: If you cut a pizza into 8 slices (Denominator = 8) and you eat 3 of them (Numerator = 3), you ate \( \frac{3}{8} \) of the pizza.

Quick Takeaway: The denominator is the total number of pieces. If the denominators are different, the pieces are different sizes!

2. Types of Fractions

There are three main types of fractions you must be able to recognise and convert between.

Type A: Proper Fractions

In a Proper Fraction, the Numerator is smaller than the Denominator.
They are always less than 1.
Example: \( \frac{2}{5} \), \( \frac{1}{10} \)

Type B: Improper Fractions

An Improper Fraction (sometimes called a top-heavy fraction) has a Numerator that is equal to or larger than the Denominator.
They are always equal to or greater than 1.
Example: \( \frac{7}{4} \), \( \frac{10}{3} \)

Type C: Mixed Numbers

A Mixed Number is a combination of a whole number and a proper fraction.
Example: \( 1 \frac{3}{4} \), \( 5 \frac{1}{2} \)

Converting Between Types

It is essential to be able to switch between Improper Fractions and Mixed Numbers, especially before adding or subtracting.

1. Mixed Number to Improper Fraction (Making it Top-Heavy)

Step 1: Multiply the Whole Number by the Denominator.
Step 2: Add the Numerator to your answer from Step 1.
Step 3: Put this new number over the original Denominator.

Example: Convert \( 3 \frac{1}{5} \)
1. \( 3 \times 5 = 15 \)
2. \( 15 + 1 = 16 \)
3. Result: \( \frac{16}{5} \)

2. Improper Fraction to Mixed Number

Step 1: Divide the Numerator by the Denominator.
Step 2: The whole number answer is the new Whole Number part of the mixed fraction.
Step 3: The Remainder is the new Numerator. Keep the original Denominator.

Example: Convert \( \frac{7}{3} \)
1. \( 7 \div 3 = 2 \) remainder \( 1 \).
2. Whole number is 2.
3. Remainder (1) is the new numerator. Denominator stays 3.
Result: \( 2 \frac{1}{3} \)

Quick Review: Improper fractions (\( \frac{7}{4} \)) and Mixed numbers (\( 1 \frac{3}{4} \)) represent the same value.

3. Equivalent Fractions and Simplifying

Equivalent Fractions

Equivalent Fractions look different but represent the exact same amount.
Example: \( \frac{1}{2} \) is the same as \( \frac{2}{4} \) and \( \frac{5}{10} \).

Rule: To find an equivalent fraction, you must multiply or divide both the Numerator and the Denominator by the same number.

Example: To change \( \frac{2}{3} \) into twelfths, we need the denominator to be 12. Since \( 3 \times 4 = 12 \), we must also multiply the numerator by 4: \( \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \).

Simplifying Fractions (Reducing to Lowest Terms)

Simplifying means finding the equivalent fraction that has the smallest possible numerator and denominator. This is usually required for final answers in exams.

Step 1: Find the Highest Common Factor (HCF) of the Numerator and the Denominator.
Step 2: Divide both the Numerator and the Denominator by the HCF.

Example: Simplify \( \frac{18}{24} \)
1. Factors of 18: 1, 2, 3, 6, 9, 18.
2. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
3. The HCF is 6.
4. \( \frac{18 \div 6}{24 \div 6} = \frac{3}{4} \).

Struggling Tip: If you can't find the HCF right away, just keep dividing by small common factors (like 2, 3, or 5) until you can't divide anymore!

4. Adding and Subtracting Fractions

This is often where students run into trouble! Remember this golden rule: You can only add or subtract fractions if they have the same denominator (the same size pieces).

Fractions with the Same Denominator

Simply add or subtract the numerators. Keep the denominator the same.
Example: \( \frac{4}{9} + \frac{2}{9} = \frac{4 + 2}{9} = \frac{6}{9} \)
(Don't forget to simplify! \( \frac{6 \div 3}{9 \div 3} = \frac{2}{3} \))

Fractions with Different Denominators

Step 1: Find the Lowest Common Multiple (LCM) of the denominators. This will be your new Common Denominator.
Step 2: Convert both fractions into equivalent fractions using the new common denominator.
Step 3: Add or subtract the new numerators.
Step 4: Simplify the final answer if necessary.

Example: Calculate \( \frac{1}{4} + \frac{2}{3} \)
1. LCM of 4 and 3 is 12. (New denominator).
2. Convert \( \frac{1}{4} \): \( \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \)
3. Convert \( \frac{2}{3} \): \( \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \)
4. Add: \( \frac{3}{12} + \frac{8}{12} = \frac{11}{12} \)

Handling Mixed Numbers in Addition/Subtraction

Crucial Strategy: Always convert all Mixed Numbers into Improper Fractions before attempting to add or subtract. This avoids common errors.
Example: \( 1 \frac{1}{2} - \frac{3}{4} \)
Convert \( 1 \frac{1}{2} \) to \( \frac{3}{2} \).
Find LCM of 2 and 4 (which is 4).
\( \frac{3 \times 2}{2 \times 2} - \frac{3}{4} = \frac{6}{4} - \frac{3}{4} = \frac{3}{4} \)

Common Mistake to Avoid: DO NOT add the denominators! \( \frac{1}{2} + \frac{1}{4} \text{ is NOT } \frac{2}{6} \).

5. Multiplying Fractions

Good news! Multiplication is the easiest operation because you do not need a common denominator.

The Multiplication Rule

Step 1: Convert any Mixed Numbers to Improper Fractions.
Step 2: Multiply the Numerators together.
Step 3: Multiply the Denominators together.
Step 4: Simplify the result.

\( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \)

Example: Calculate \( \frac{2}{5} \times \frac{3}{4} \)
Numerator: \( 2 \times 3 = 6 \)
Denominator: \( 5 \times 4 = 20 \)
Result: \( \frac{6}{20} \). Simplify by dividing by 2: \( \frac{3}{10} \).

Cross-Cancelling (A Time Saver!)

Before multiplying, you can simplify diagonally (cross-cancel). If a numerator and a denominator share a common factor, divide them both by that factor first.

Using the example above: \( \frac{2}{5} \times \frac{3}{4} \)
The 2 (top-left) and the 4 (bottom-right) are both divisible by 2.
\( \frac{1}{5} \times \frac{3}{2} = \frac{1 \times 3}{5 \times 2} = \frac{3}{10} \)
This gives the simplified answer immediately!

6. Dividing Fractions

Division looks tricky, but we use a brilliant trick: we turn it into multiplication!

The Division Rule: Keep, Change, Flip (KCF)

Step 1: Keep the first fraction as it is.
Step 2: Change the division sign (\( \div \)) to a multiplication sign (\( \times \)).
Step 3: Flip the second fraction (find its Reciprocal—swap the numerator and denominator).
Step 4: Multiply the resulting fractions as normal.

Example: Calculate \( \frac{3}{5} \div \frac{2}{3} \)
1. Keep \( \frac{3}{5} \)
2. Change \( \div \) to \( \times \)
3. Flip \( \frac{2}{3} \) to \( \frac{3}{2} \)
4. Multiply: \( \frac{3}{5} \times \frac{3}{2} = \frac{9}{10} \)

Did you know? When dividing by a fraction, you are essentially finding out how many times that smaller part fits into the first part.

7. Fractions of Amounts

Finding a fraction of a whole number (or an amount) is a very common application of multiplication.

Method: Divide by the Denominator, Multiply by the Numerator

To find \( \frac{a}{b} \) of a number (N):

Step 1: Divide the whole amount (N) by the Denominator (b). This finds the value of one part.
Step 2: Multiply that result by the Numerator (a). This gives you the value of the required number of parts.

Example: Find \( \frac{3}{4} \) of $36.
1. Divide by the denominator: \( \$36 \div 4 = \$9 \) (This is \( \frac{1}{4} \))
2. Multiply by the numerator: \( \$9 \times 3 = \$27 \)
Answer: \( \frac{3}{4} \) of $36 is $27.

Alternatively, you can treat the whole number as a fraction over 1 and multiply: \( \frac{3}{4} \times \frac{36}{1} = \frac{108}{4} = 27 \)

Final Review & Key Reminders

Summary Box for Operations

OperationRuleConversion First?
Addition/SubtractionFind Common Denominator (LCM). Add/Subtract Numerators.YES (Convert Mixed to Improper)
MultiplicationMultiply Numerators. Multiply Denominators. (Use Cross-Cancelling!)YES (Convert Mixed to Improper)
DivisionKCF (Keep, Change, Flip). Then multiply.YES (Convert Mixed to Improper)

Crucial Exam Reminder: Unless otherwise specified, always present your final answer in its simplest form. If the question started with mixed numbers, sometimes they prefer the answer as a mixed number too, but a simplified improper fraction is usually acceptable unless specified.

You've mastered the fundamentals of fractions! Keep practicing the steps—they are always the same!