GCSE Maths

Numbers
32 Topics | 1 Quiz
Integers
Prime Numbers and Composite Numbers
Positive and Negative Numbers
Order of Operations
Square Numbers and Cube Numbers
The Laws of Indices
Square Roots and Cube Roots
Triangular Numbers
The Fibonacci Sequence
Factors
Multiples
The Highest Common Factor
Fractions
Adding and Subtracting Fractions
Multiplication of Fractions
Division of Fractions
Fractional Indices
Decimal Places
Fractions and Decimals
Expressing Fractions as Decimal Numbers
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Algebra
29 Topics | 1 Quiz
Algebraic Notation
Algebraic Vocabulary
Collecting Like Terms
Substituting Values into Formulae
Rearranging Equations
Algebraic Fractions
Expanding Brackets and Simplifying
Expanding Double Brackets
Factorisation
Factorising Quadratic Expressions
Factorising Quadratics with a Coefficient of x Squared Greater than 1
The Quadratic Formula
The Difference of Two Squares
Subject of Formula
Solving Linear Equations
Quadratic Equations
Completing the Square
Sequences and Nth Term
Arithmetic and Geometric Sequences
Quadratic Sequences
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Ratio, Proportion and Rates of Change
7 Topics
Ratios
Scaling Ratios
Direct Proportion
Inverse Proportion
Reverse Percentages
Units and Conversions
Scale Factors
Geometry and Measures
15 Topics
2D Shapes
Types of angles
Angle Properties
Angle in Parallel Lines
Triangles
Angles in Triangles
Polygons
Lines of Symmetry
Bearings
Loci and constructions
Constructing Triangles
Transformations
Enlargement
Plans and elevations
Areas and Volumes in Map Scale Problems
Probability
6 Topics
Introduction to probability
Finding the sum of probabilities
Conditional probability
Set Notation
Venn diagrams
The Complement of a Set
Statistics
2 Topics
Population and samples
Representing data
Vectors
Mensuration and Calculation
7 Topics
Area
Volume
Congruence and similarity
Pythagoras’ theorem
Sine and cosine rule
Area of a triangle
Standard units of measure
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Algebraic Fractions

GCSE Maths Algebra Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. Like regular fractions, algebraic fractions consist of a numerator (the top part) and a denominator (the bottom part), separated by a horizontal line or a division symbol.

They are used to represent ratios, proportions, or divisions involving variables, constants, and coefficients. For example, \frac{(2x + 3)}{(x - 1)} is an algebraic fraction.

Simplifying Algebraic Fractions

To simplify an algebraic fraction, factor the numerator and denominator and cancel out any common factors. This process reduces the fraction to its simplest form.

For example, consider the algebraic fraction \frac{(6x^2 + 3x)}{(3x)}:

Step 1: Factor the numerator and denominator:

\frac{(3x(2x + 1))}{(3x)}

Step 2: Cancel out the common factor 3x:

(2x + 1)

Operations with Algebraic Fractions

Addition and subtraction

To add or subtract algebraic fractions, find a common denominator, then add or subtract the numerators.

For example, to add \frac{(2x + 3)}{(x - 1)} and \frac{(x + 2)}{(x + 1)}:

  • Step 1: Find a common denominator, which is (x - 1)(x + 1)
  • Step 2: Add the numerators: (2x + 3)(x + 1) + \frac{(x + 2)}{(x - 1)}
  • Step 3: Simplify the result: \frac{(2x^2 + 5x + 3)}{(x^2 - 1)}

Multiplication

To multiply algebraic fractions, multiply the numerators together and the denominators together. Then simplify the result, if possible.

For example, to multiply \frac{(2x + 3)}{(x - 1)} by \frac{(x + 1)}{(3x)}:

Step 1: Multiply the numerators:

(2x + 3)(x + 1)

Step 2: Multiply the denominators:

4(x - 1)(3x)

Step 3: Simplify the result:

\frac{(2x^2 + 5x + 3)}{(3x^2 - 3x)}

Division

To divide algebraic fractions, invert the divisor (flip the numerator and denominator), then multiply the fractions and simplify the result.

For example, to divide \frac{(2x + 3)}{(x - 1)} by \frac{(x + 2)}{(x + 1)}:

Step 1: Invert the divisor:

\frac{(x + 1)}{(x + 2)}

Step 2: Multiply the fractions:

\frac{((2x + 3)}{(x - 1))} \times \frac{(x + 1)}{(x + 2)}

Step 3: Simplify the result:

\frac{(2x^2 + x + 3)}{(x^2 + x - 2)}

Solving Equations with Algebraic Fractions

When solving equations with algebraic fractions, clear fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. Then, simplify and solve the resulting equation.

For example, solve the equation \frac{(2x)}{(x - 1)} = \frac{(x + 3)}{(x + 1)}:

Step 1: Multiply both sides by the LCM, (x - 1)(x + 1):

(2x)(x + 1) = (x + 3)(x - 1)

Step 2: Simplify and solve:

2x^2 + 2x = x^2 - x - 3

x^2 + 3x + 3 = 0

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