Algebraic fractions (Higher Tier)

Understanding Algebraic Fractions

Algebraic fractions are simply fractions wherein the numerator, the denominator or both are algebraic expressions.
The principles for handling algebraic fractions are largely the same as those for handling numerical fractions.
Rules for addition, subtraction, multiplication, and division all apply to algebraic fractions.
Fractions should always be simplified wherever possible. This generally involves factoring the numerator and denominator and cancelling common factors.

Addition and Subtraction of Algebraic Fractions

To add or subtract algebraic fractions, ensure they have a common denominator. This may involve multiplying one or both fractions by a suitable expression.
Once a common denominator is established, simply add or subtract the numerators.
Example: To add (x / 3) + (2x / 6), change the fractions to a common denominator, i.e., (2x / 6) + (2x / 6). The answer is 4x / 6, which can be simplified to 2x / 3.

Multiplication and Division of Algebraic Fractions

To multiply two algebraic fractions, multiply the two numerators together for the new numerator and multiply the two denominators together for the new denominator. Simplify if possible.
Division of algebraic fractions involves flipping the second fraction and then multiplying.
Example: To divide (2x / 3) / (4/ x), the operation becomes a multiplication: (2x / 3) * (x / 4). The answer is 2x^2 / 12, which can be simplified to x^2 / 6.

Solving Equations with Algebraic Fractions

To solve equations involving algebraic fractions, clear fractions by multiplying through by the denominator.
Use the cross-multiplication method for solving equations.
Start by simplifying the equation and then cross-multiply. Solve for the variable.

Application of Algebraic Fractions

Algebraic fractions come up in many contexts, from simple mathematical modelling to complex problem solving.
Mastering their manipulation can simplify computations in these contexts and is a key algebraic skill in higher mathematics.