International Mathematics (0607) | Algebraic fractions

Mastering Algebraic Fractions (IGCSE 0607)

Welcome to the world of algebraic fractions! If the thought of fractions with 'x's and 'y's makes you nervous, don't worry. This chapter simply takes the rules you already know for normal numbers and applies them to algebra. Mastering this skill is essential for solving complex equations and handling advanced algebra later on.

Think of it this way: An algebraic fraction is just a fraction where the numerator (top) and/or the denominator (bottom) contains variables (letters).

1. Simplifying Algebraic Fractions

The golden rule for simplifying is the same as for numerical fractions: You can only cancel common factors that are multiplied together, not terms that are added or subtracted.

Step 1: Simplify by Cancelling Common Factors (Core Concept)

If the terms are simple monomials (single terms multiplied together), look for common numbers and variables in the numerator and denominator.

Example 1 (Core): Simplify \(\frac{6x^2}{3x}\)

1. Simplify the numbers: \(\frac{6}{3} = 2\).
2. Simplify the variables using index laws: \(\frac{x^2}{x} = x^{2-1} = x\).
Result: \(2x\)

Example 2 (Core): Simplify \(\frac{10ab}{15ac}\)

1. Numbers: \(\frac{10}{15} = \frac{2}{3}\).
2. Variables: Cancel the common \(a\). The \(b\) and \(c\) remain.
Result: \(\frac{2b}{3c}\)

Step 2: Simplify by Factoring (Extended Concept)

If you have expressions involving addition or subtraction (polynomials), you must factorise the numerator and the denominator first. Then, look for common bracketed factors to cancel.

Analogy: The Locked Doors
You can only take an ingredient out of a kitchen (cancel a factor) if it's sitting free on the counter. If the ingredient is locked inside a fridge or a cupboard (a factorised bracket), you have to take the whole fridge or cupboard! You cannot cancel part of a bracket.

Procedure for Factoring and Simplifying:

1. Factorise Fully: Factor the numerator and the denominator completely.
2. Identify Common Factors: Look for identical bracketed expressions.
3. Cancel: Cancel the common factors (brackets).

Example 3 (Extended): Simplify \(\frac{x^2 - 4}{x^2 + x - 6}\)

1. Factorise Numerator (Difference of Squares):
\(x^2 - 4 = (x - 2)(x + 2)\)
2. Factorise Denominator (Quadratic):
\(x^2 + x - 6 = (x + 3)(x - 2)\)
3. Rewrite the fraction: \(\frac{(x - 2)(x + 2)}{(x + 3)(x - 2)}\)
4. Cancel: The common factor \((x - 2)\) cancels out.
Result: \(\frac{x + 2}{x + 3}\)

Key Takeaway for Simplifying: Always factorise first! If you see a plus or minus sign, put on your "factoring hat" before attempting to cancel anything.

2. Multiplication and Division of Algebraic Fractions

These operations are generally easier than addition or subtraction because you don't need a common denominator.

2.1 Multiplication

The rule is simple: Multiply the numerators together, and multiply the denominators together.

Step-by-Step Multiplication:

1. (Optional but Recommended) Factorise: Factorise all four parts (two numerators, two denominators).
2. Cancel Diagonally: Cancel any common factors between any numerator and any denominator.
3. Multiply: Multiply the remaining numerators and remaining denominators.

Example 4 (Extended): Work out \(\frac{x}{3} \times \frac{x - 5}{2}\)

Since nothing can be factored or cancelled here:

Multiply tops: \(x(x - 5)\)
Multiply bottoms: \(3 \times 2 = 6\)
Result: \(\frac{x^2 - 5x}{6}\) (or \(\frac{x(x - 5)}{6}\))

Example 5 (Extended): Work out \(\frac{2x}{x+1} \times \frac{x^2+x}{4}\)

1. Factorise the top right: \(x^2+x = x(x+1)\).
Rewrite: \(\frac{2x}{x+1} \times \frac{x(x+1)}{4}\)
2. Cancel: The factor \((x+1)\) cancels. The 2 and 4 simplify to \(\frac{1}{2}\).
Rewrite: \(\frac{x}{1} \times \frac{x}{2}\)
3. Multiply: \(x \times x = x^2\). \(1 \times 2 = 2\).
Result: \(\frac{x^2}{2}\)

2.2 Division

The rule for dividing fractions is: Keep, Change, Flip (KCF).

• Keep the first fraction.
• Change the division sign to multiplication.
• Flip (take the reciprocal of) the second fraction.

Once you've done KCF, treat it exactly like a multiplication problem (Steps 1, 2, 3 above).

Example 6 (Extended): Work out \(\frac{3a}{4} \div \frac{9a}{10}\)

1. KCF: \(\frac{3a}{4} \times \frac{10}{9a}\)
2. Cancel:
\(\rightarrow\) The \(3\) in the top cancels with the \(9\) on the bottom (leaving \(3\)).
\(\rightarrow\) The \(4\) on the bottom and \(10\) on the top simplify by dividing by 2 (leaving \(2\) and \(5\)).
\(\rightarrow\) The \(a\) cancels.
3. Multiply the remainders: \(\frac{1}{2} \times \frac{5}{3}\)
Result: \(\frac{5}{6}\)

Key Takeaway for Mult/Div: Division becomes multiplication (KCF). Always try to simplify factors diagonally before doing the final multiplication to keep your numbers small!

3. Addition and Subtraction of Algebraic Fractions

This is often the trickiest operation, as you must have a Common Denominator.

Imagine trying to add apples and oranges—you can't! You need to convert them into a common item, like 'pieces of fruit'. In algebra, you need to convert fractions to have the same denominator before combining them.

Step-by-Step Addition/Subtraction:

1. Find the LCM: Determine the Lowest Common Multiple (LCM) of the denominators. This will be the new common denominator.
2. Adjust Numerators: Multiply the numerator of each fraction by the factor needed to turn its denominator into the LCM.
3. Combine: Write the entire expression over the single common denominator.
4. Simplify: Expand and collect like terms in the numerator. (If possible, try to factorise the final numerator to see if it simplifies with the denominator, though often it won't).

Finding the LCM (The Key Skill)

The LCM must contain all unique factors from every denominator.

Example 7 (Extended - Simple LCM): Work out \(\frac{x}{3} + \frac{x - 4}{2}\)

1. LCM: The LCM of 3 and 2 is 6.
2. Adjust:
\(\rightarrow\) Multiply first fraction by \(\frac{2}{2}\): \(\frac{x \times 2}{3 \times 2} = \frac{2x}{6}\)
\(\rightarrow\) Multiply second fraction by \(\frac{3}{3}\): \(\frac{(x - 4) \times 3}{2 \times 3} = \frac{3(x - 4)}{6}\)
3. Combine: \(\frac{2x + 3(x - 4)}{6}\)
4. Simplify Numerator: \(2x + 3x - 12 = 5x - 12\)
Result: \(\frac{5x - 12}{6}\)

Example 8 (Extended - Algebraic LCM): Work out \(\frac{1}{x - 2} + \frac{x + 1}{x - 3}\)

1. LCM: Since the denominators share no common factors, the LCM is \((x - 2)(x - 3)\).
2. Adjust:
\(\rightarrow\) Multiply first fraction by \(\frac{x - 3}{x - 3}\): \(\frac{1(x - 3)}{(x - 2)(x - 3)}\)
\(\rightarrow\) Multiply second fraction by \(\frac{x - 2}{x - 2}\): \(\frac{(x + 1)(x - 2)}{(x - 3)(x - 2)}\)
3. Combine (Carefully!):
\(\frac{(x - 3) + (x + 1)(x - 2)}{(x - 2)(x - 3)}\)
4. Simplify Numerator: Expand the brackets in the numerator.
\((x - 3) + (x^2 - 2x + x - 2)\)
\((x - 3) + (x^2 - x - 2)\)
Combine like terms: \(x^2 + (x - x) + (-3 - 2) = x^2 - 5\)
Result: \(\frac{x^2 - 5}{(x - 2)(x - 3)}\)

Example 9 (Extended - Subtraction): Work out \(\frac{5}{x + 1} - \frac{2}{x + 4}\)

1. LCM: \((x + 1)(x + 4)\)
2. Adjust:
\(\rightarrow\) First fraction: \(\frac{5(x + 4)}{(x + 1)(x + 4)}\)
\(\rightarrow\) Second fraction: \(\frac{2(x + 1)}{(x + 1)(x + 4)}\)
3. Combine: \(\frac{5(x + 4) - 2(x + 1)}{(x + 1)(x + 4)}\)
4. Simplify Numerator: (Remember to distribute the \(-2\))
\((5x + 20) - (2x + 2)\)
\(5x + 20 - 2x - 2\)
\(3x + 18\)
Result: \(\frac{3x + 18}{(x + 1)(x + 4)}\)

Quick Review Summary: The Four Operations