Mastering Fraction Operations for GCSE Mathematics

Understanding Fractions

A fraction represents a part of a whole. It consists of a numerator (top number) and a denominator (bottom number). For example, in the fraction 3⁄5, 3 is the numerator and 5 is the denominator.

Types of Fractions

• Proper fractions have a numerator smaller than the denominator, e.g. 2⁄5
• Improper fractions have a numerator greater than or equal to the denominator, e.g. 7⁄3
• Mixed numbers consist of a whole number part and a fractional part, e.g. 21⁄4

Converting Fractions

It's important to be able to convert between improper fractions and mixed numbers:

Example: Converting 11⁄4 to a mixed number

• Divide the numerator (11) by the denominator (4): 11 ÷ 4 = 2 remainder 3
• The whole number part is 2, and the fractional part is 3⁄4
• Therefore, 11⁄4 = 23⁄4

Fraction of an Amount

Finding a fraction of an amount involves multiplying the amount by the fraction:

Example: Find 3⁄5 of 45

• 3⁄5 × 45 = (3 ÷ 5) × 45 = 0.6 × 45 = 27
• Therefore, 3⁄5 of 45 is 27.

Fraction Operations

Addition and Subtraction

To add or subtract fractions, the denominators must be the same. If not, find the least common multiple (LCM) of the denominators and convert fractions to equivalent fractions with the LCM denominator.

Example: 1⁄3 + 1⁄6

• LCM of 3 and 6 is 6
• Convert 1⁄3 to 2⁄6
• So 1⁄3 + 1⁄6 = 2⁄6 + 1⁄6 = 3⁄6 = 1⁄2

Multiplication

To multiply fractions, multiply the numerators and multiply the denominators:

Example: 2⁄3 × 5⁄6

• 2⁄3 × 5⁄6 = (2 × 5)⁄(3 × 6)
• = 10⁄18 = 5⁄9

Division

To divide by a fraction, multiply by its reciprocal (flip the second fraction):

Example: 3⁄4 ÷ 1⁄6

• Reciprocal of 1⁄6 is 6⁄1
• 3⁄4 ÷ 1⁄6 = 3⁄4 × 6⁄1 = 18⁄4 = 9⁄2 = 41⁄2

Key Points

• Convert fractions to a common denominator before adding/subtracting
• Multiply straight across for multiplication of fractions
• To divide by a fraction, multiply by its reciprocal
• Simplify fractions where possible