What are algebraic fractions and how do they differ from regular fractions?

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions, such as variables and constants. Unlike regular fractions, which only involve numbers, algebraic fractions require understanding of algebraic operations to simplify, add, subtract, multiply, or divide them. This concept is crucial in the IB MYP Mathematics curriculum as it builds foundational skills in algebra.

How do you simplify algebraic fractions?

To simplify algebraic fractions, you need to factor both the numerator and the denominator and then cancel out any common factors. For example, in the fraction (x^2 - 4)/(x^2 - 2x), you would factor the numerator as (x + 2)(x - 2) and the denominator as x(x - 2), then cancel the common factor (x - 2), resulting in (x + 2)/x. Simplifying algebraic fractions is a key skill in the IB MYP Mathematics program.

How can you add or subtract algebraic fractions?

To add or subtract algebraic fractions, you must first find a common denominator, similar to adding or subtracting numerical fractions. Once you have a common denominator, adjust the numerators accordingly, perform the addition or subtraction, and simplify the result if possible. For example, to add (1/x) + (2/x^2), you would find a common denominator of x^2, rewrite the fractions as (x/x^2) + (2/x^2), and then add them to get (x + 2)/x^2. Mastery of this process is essential in the IB MYP Mathematics curriculum.

What is the process for multiplying and dividing algebraic fractions?

Multiplying algebraic fractions involves multiplying the numerators together and the denominators together, then simplifying the resulting fraction. For division, you multiply by the reciprocal of the divisor. For example, to multiply (x/3) * (4/x^2), you multiply the numerators (x * 4) and the denominators (3 * x^2) to get 4x/3x^2, which simplifies to 4/3x. Understanding these operations is crucial for success in the IB MYP Mathematics curriculum.

What are common mistakes to avoid when working with algebraic fractions?

Common mistakes include not factoring completely, failing to find a common denominator when adding or subtracting, and incorrectly canceling terms that are not common factors. It's important to remember that only factors, not terms, can be canceled. For example, in the fraction (x^2 + x)/(x + 1), you cannot cancel the x terms directly; instead, factor the numerator to x(x + 1) and then cancel the (x + 1) factor. Avoiding these mistakes is important for mastering algebraic fractions in the IB MYP Mathematics curriculum.

Why is "Cancelling a TERM rather than a FACTOR. E.g. $\dfrac{x + 3}{x + 5} = \dfrac{3}{5}$." a common mistake in Algebraic Fractions?

Why it happens: Treating +x on top and bottom as cancellable. How to avoid it: Only FACTORS (parts of a product) can cancel. In x + 3, the x is being ADDED, not multiplied. No cancellation.

Why is "Forgetting to distribute the minus when subtracting fractions." a common mistake in Algebraic Fractions?

Why it happens: Combining numerators carelessly. How to avoid it: a/c - b/c = a - b/c — and b might be a whole expression. Use brackets: a - (b)/c.

Why is "Expanding before cancelling." a common mistake in Algebraic Fractions?

Why it happens: Default arithmetic instinct. How to avoid it: FACTORISE FIRST. The whole point is to spot common factors that cancel. Expanding obscures them.

AlgebraIB MYP Maths Extended Level

Algebraic Fractions

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Algebraic Fractions — IB MYP Mathematics (Extended): the four operations on fractions with variables

Fractions with algebra in them behave just like numerical fractions — but you have to factorise first to spot common factors. This MYP Mathematics Extended note covers simplification, addition, subtraction, multiplication and division of algebraic fractions.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Simplify algebraic fractions.
MYP Mathematics A — Add, subtract, multiply, divide algebraic fractions.
MYP Mathematics C — Show factorisation as a step in the working.

Simplifying algebraic fractions

Factorise top and bottom; cancel common FACTORS.

An algebraic fraction has algebra somewhere — top, bottom or both.

To simplify:

FACTORISE the numerator.
FACTORISE the denominator.
CANCEL any factor common to both.

Worked example. Simplify $\dfrac{6x^2 y}{9xy^2}$ .

$\dfrac{6x^2 y}{9xy^2} = \dfrac{\cancel{3} \cdot 2 \cdot x \cdot \cancel{x} \cdot \cancel{y}}{\cancel{3} \cdot 3 \cdot \cancel{x} \cdot \cancel{y} \cdot y} = \dfrac{2x}{3y}$ .

Worked example. Simplify $\dfrac{x^2 - 9}{x^2 + 4x + 3}$ .

Factorise top: $x^2 - 9 = (x-3)(x+3)$ (difference of squares).
Factorise bottom: $x^2 + 4x + 3 = (x+1)(x+3)$ .
Cancel common $(x+3)$ : $\dfrac{(x-3)(x+3)}{(x+1)(x+3)} = \dfrac{x-3}{x+1}$ .

Crucial rule: you can only cancel FACTORS, not TERMS. $\dfrac{x + 3}{x + 7}$ does NOT simplify — $x$ is a TERM, not a factor.

Factorise top AND bottom first.
Cancel only common FACTORS.
Common TERMS (in sums) cannot be cancelled.

Adding and subtracting

Common denominator, then combine numerators.

Just like numerical fractions — find a common denominator first.

Worked example. $\dfrac{2}{x} + \dfrac{3}{x+1}$ .

Common denominator: $x(x+1)$ .
Rewrite: $\dfrac{2(x+1)}{x(x+1)} + \dfrac{3x}{x(x+1)}$ .
Combine: $\dfrac{2(x+1) + 3x}{x(x+1)} = \dfrac{2x + 2 + 3x}{x(x+1)} = \dfrac{5x + 2}{x(x+1)}$ .

Worked example. $\dfrac{3}{x-2} - \dfrac{1}{x+2}$ .

Common denominator: $(x-2)(x+2)$ .
$\dfrac{3(x+2)}{(x-2)(x+2)} - \dfrac{1(x-2)}{(x-2)(x+2)}$ .
$= \dfrac{3(x+2) - (x-2)}{(x-2)(x+2)} = \dfrac{3x + 6 - x + 2}{x^2 - 4} = \dfrac{2x + 8}{x^2 - 4}$ .
Optionally factorise: $\dfrac{2(x+4)}{(x-2)(x+2)}$ .

Be careful with minus signs. When subtracting, distribute the minus across ALL terms in the numerator.

Common denominator = product (or LCM) of denominators.
Multiply each fraction so it has the common denominator.
Combine numerators carefully — watch the minus signs.

Multiplying and dividing

Factor first, then cancel across.

Multiplication: $\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}$ . Factorise and cancel BEFORE multiplying out.

Worked example. $\dfrac{x+2}{x-3} \times \dfrac{(x-3)(x+1)}{x+2}$ .

Cancel $(x+2)$ in top with $(x+2)$ in bottom of second.
Cancel $(x-3)$ in bottom with $(x-3)$ in top of second.
Left with $\dfrac{x+1}{1} = x+1$ .

Division: $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{ad}{bc}$ . 'Flip the second, multiply.'

Worked example. $\dfrac{x^2 - 4}{x^2 + 5x} \div \dfrac{x - 2}{x + 5}$ .

Flip the second: $\dfrac{x^2 - 4}{x^2 + 5x} \times \dfrac{x + 5}{x - 2}$ .
Factorise everything: $\dfrac{(x-2)(x+2)}{x(x+5)} \times \dfrac{x+5}{x-2}$ .
Cancel $(x-2)$ and $(x+5)$ .
Left with $\dfrac{x+2}{x}$ .

A key Extended-level skill: factorising BEFORE multiplying makes cancellations obvious. If you expand first you'll get a horrendous mess and lose marks.

Multiply: factorise everything, cancel across, then combine.
Divide: flip the second fraction, then multiply.
Cancel FACTORS, not terms.

How it’s examined

Criterion A: combine/simplify algebraic fractions. Criterion C: clear factorisation step.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

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Step-by-step worked examples — Algebraic Fractions

Step-by-step solutions to past-paper-style questions on algebraic fractions, written exactly the way a tutor would explain them at the board.

1Simplify by factoring
Building confidence• simplify
Question
Simplify $\dfrac{x^2 - 16}{x^2 + 6x + 8}$ .
Step-by-step solution
1. Step 1
  Top: $x^2 - 16 = (x-4)(x+4)$ (difference of squares).
2. Step 2
  Bottom: $x^2 + 6x + 8 = (x+2)(x+4)$ (trinomial).
3. Step 3
  Cancel common factor $(x+4)$ : $\dfrac{(x-4)(x+4)}{(x+2)(x+4)} = \dfrac{x-4}{x+2}$ .
Answer
$\dfrac{x-4}{x+2}$
2Add two algebraic fractions
Building confidence• addition
Question
Compute $\dfrac{1}{x} + \dfrac{2}{x+1}$ .
Step-by-step solution
1. Step 1
  Common denominator: $x(x+1)$ .
2. Step 2
  Convert: $\dfrac{1(x+1)}{x(x+1)} + \dfrac{2x}{x(x+1)}$ .
3. Step 3
  Combine: $\dfrac{(x+1) + 2x}{x(x+1)} = \dfrac{3x + 1}{x(x+1)}$ .
Answer
$\dfrac{3x + 1}{x(x+1)}$
3Multiply algebraic fractions
Building confidence• multiplication
Question
Simplify $\dfrac{x+3}{x-2} \times \dfrac{x^2 - 4}{x+3}$ .
Step-by-step solution
1. Step 1
  Factor: $x^2 - 4 = (x-2)(x+2)$ .
2. Step 2
  Rewrite: $\dfrac{x+3}{x-2} \times \dfrac{(x-2)(x+2)}{x+3}$ .
3. Step 3
  Cancel $(x+3)$ and $(x-2)$ . Result: $x + 2$ .
Answer
$x + 2$
4Divide algebraic fractions
Stretch• division
Question
Simplify $\dfrac{x^2 + 5x + 6}{x^2 - 4} \div \dfrac{x + 3}{x - 2}$ .
Step-by-step solution
1. Step 1
  Flip the second: $\dfrac{x^2 + 5x + 6}{x^2 - 4} \times \dfrac{x - 2}{x + 3}$ .
2. Step 2
  Factor: $x^2 + 5x + 6 = (x+2)(x+3)$ and $x^2 - 4 = (x-2)(x+2)$ .
3. Step 3
  Rewrite: $\dfrac{(x+2)(x+3)}{(x-2)(x+2)} \times \dfrac{x-2}{x+3}$ .
4. Step 4
  Cancel $(x+2)$ , $(x+3)$ , $(x-2)$ . Result: $1$ .
Answer
$1$

Key Definitions and Keywords — Algebraic Fractions

Definitions to memorise and the exact keywords mark schemes credit for algebraic fractions answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Algebraic fraction
Examiner keyword
A fraction whose numerator and/or denominator is a polynomial.
Common denominator
A shared denominator (usually the product or LCM of the originals) that allows two fractions to be added or subtracted.

Common Mistakes and Misconceptions — Algebraic Fractions

The traps other students keep falling into on algebraic fractions questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Cancelling a TERM rather than a FACTOR. E.g. $\dfrac{x + 3}{x + 5} = \dfrac{3}{5}$ .
Why it happens
Treating $+x$ on top and bottom as cancellable.
How to avoid it
Only FACTORS (parts of a product) can cancel. In $x + 3$ , the $x$ is being ADDED, not multiplied. No cancellation.
✕Forgetting to distribute the minus when subtracting fractions.
Why it happens
Combining numerators carelessly.
How to avoid it
$\dfrac{a}{c} - \dfrac{b}{c} = \dfrac{a - b}{c}$ — and $b$ might be a whole expression. Use brackets: $\dfrac{a - (b)}{c}$ .
✕Expanding before cancelling.
Why it happens
Default arithmetic instinct.
How to avoid it
FACTORISE FIRST. The whole point is to spot common factors that cancel. Expanding obscures them.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Algebraic Fractions — frequently asked questions

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Algebraic Fractions

Short Study Notes in the form of Slides

Short Study Notes — Algebraic Fractions

Detailed Notes

What you’ll learn

How it’s examined

1Simplify by factoring

2Add two algebraic fractions

3Multiply algebraic fractions

4Divide algebraic fractions

Algebraic fraction

Common denominator

✕Cancelling a TERM rather than a FACTOR. E.g. x+3x+5=35\dfrac{x + 3}{x + 5} = \dfrac{3}{5}x+5x+3​=53​.

✕Forgetting to distribute the minus when subtracting fractions.

✕Expanding before cancelling.

Practice questions

Past paper style quiz

Algebraic Fractions — frequently asked questions

✕Cancelling a TERM rather than a FACTOR. E.g. $\dfrac{x + 3}{x + 5} = \dfrac{3}{5}$ .