Algebraic Fractions Revision Notes for AQA GCSE Maths

Algebraic fractions

Working with algebraic fractions might seem daunting at first, but the good news is that all the rules you've learned for numerical fractions apply here too. The main difference is that these fractions contain variables (letters) alongside numbers, making them more versatile for solving mathematical problems.

Simplifying algebraic fractions

The process of simplifying algebraic fractions involves reducing them to their simplest form by cancelling common factors from both the numerator and denominator. This technique helps make expressions easier to work with and understand.

When you simplify, you need to look for factors that appear in both the top and bottom of the fraction. You can cancel these common factors just like you would with numerical fractions. The key is to treat each variable individually and cancel as much as possible.

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Sometimes, you might need to factorise expressions before you can spot the common factors. This is particularly important when working with quadratic expressions or more complex algebraic terms. Methods like DOTS (Difference of Two Squares) can be very helpful for factorising expressions before simplification.

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Worked Example: Simplifying with Variables

When simplifying expressions with variables like $\frac{x^2y^3}{xy^2}$ , you would cancel the common $x$ and $y^2$ factors, leaving you with $xy$ .

Step 1: Identify common factors in numerator and denominator

Numerator: $x^2y^3$
Denominator: $xy^2$

Step 2: Cancel the common factors

Cancel one $x$ from $x^2$ and $x$ : leaves $x$ in numerator
Cancel $y^2$ from $y^3$ and $y^2$ : leaves $y$ in numerator

Step 3: Write the simplified result $\frac{x^2y^3}{xy^2} = xy$

When dealing with more complex expressions involving brackets, factorising first often reveals hidden common factors that can be cancelled.

Multiplying and dividing algebraic fractions

Multiplying algebraic fractions follows the same straightforward rule as numerical fractions. You simply multiply the numerators together and multiply the denominators together, then simplify the result if possible.

The process becomes systematic when you:

Multiply all terms in the numerators to create the new numerator
Multiply all terms in the denominators to create the new denominator
Look for opportunities to simplify the resulting fraction

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Division requires an additional step but follows a familiar pattern. To divide by a fraction, you turn the second fraction upside down (find its reciprocal) and then multiply. This "flip and multiply" method transforms division into multiplication, making the calculation more manageable.

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Worked Example: Multiplying Algebraic Fractions

To multiply $\frac{2x}{3y} \times \frac{4z}{5x}$ :

Step 1: Multiply numerators and denominators $\frac{2x \times 4z}{3y \times 5x} = \frac{8xz}{15xy}$

Step 2: Simplify by cancelling common factors Cancel the common $x$ : $\frac{8z}{15y}$

Remember that division by a fraction is the same as multiplication by its reciprocal, so once you've flipped the second fraction, you proceed exactly as you would for multiplication.

Adding and subtracting algebraic fractions

Adding and subtracting algebraic fractions requires more careful consideration than multiplication or division. The fundamental rule is that you can only add or subtract fractions when they share the same denominator.

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The Three-Step Process for Adding/Subtracting:

Find the common denominator - Identify an expression that both denominators can be converted to
Adjust each fraction - Multiply both numerator and denominator by the required factor
Combine the numerators - Add or subtract the numerators while keeping the denominator unchanged

Finding the common denominator is your first priority. You need to identify an expression that both denominators can be converted to. This often involves finding the lowest common multiple of the denominators, which might be their product if they share no common factors.

Adjusting each fraction comes next. You multiply both the numerator and denominator of each fraction by whatever factor is needed to achieve the common denominator. This process doesn't change the value of the fractions - it simply expresses them in equivalent forms that can be combined.

Combining the numerators is the final step. Once both fractions have identical denominators, you can add or subtract the numerators while keeping the denominator unchanged. This gives you a single fraction that may need further simplification.

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Worked Example: Adding Algebraic Fractions

To add $\frac{3}{x+3} + \frac{2}{x-2}$ :

Step 1: Find the common denominator Common denominator = $(x+3)(x-2)$

Step 2: Adjust each fraction $\frac{3}{x+3} = \frac{3(x-2)}{(x+3)(x-2)} = \frac{3x-6}{(x+3)(x-2)}$

$\frac{2}{x-2} = \frac{2(x+3)}{(x-2)(x+3)} = \frac{2x+6}{(x+3)(x-2)}$

Step 3: Add the numerators $\frac{3x-6}{(x+3)(x-2)} + \frac{2x+6}{(x+3)(x-2)} = \frac{3x-6+2x+6}{(x+3)(x-2)} = \frac{5x}{(x+3)(x-2)}$

When working with denominators like $(x + 3)$ and $(x - 2)$ , the common denominator will typically be their product: $(x + 3)(x - 2)$ . You then need to multiply the first fraction by the factor that's missing from its denominator, and do the same for the second fraction.

Important reminders about common mistakes

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Common Mistake: Incorrect Cancellation

A frequent error involves attempting to cancel terms incorrectly. Remember that you can only cancel factors (terms that are multiplied together), not terms that are added or subtracted. For instance, you cannot cancel $x$ from both the numerator and denominator in an expression like $\frac{x + 1}{x}$ , as the $x$ in the numerator is being added to 1, not multiplied.

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Work with Complete Fractions

Always work with complete fractions rather than trying to manipulate individual parts. When you multiply or divide, ensure you're applying the operation to entire numerators and denominators, not just selected terms within them.

Summary

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Key Points to Remember:

Algebraic fractions follow the same fundamental rules as numerical fractions - the presence of variables doesn't change the basic principles
Simplification involves cancelling common factors from numerator and denominator, often requiring factorisation first
Multiplication means multiplying numerators together and denominators together, then simplifying if possible
Division requires flipping the second fraction before multiplying
Addition and subtraction need a common denominator before you can combine the numerators
Only cancel factors (multiplied terms), never terms that are added or subtracted

Algebraic Fractions (AQA GCSE Maths): Revision Notes