Introduction to Algebraic Fractions

Does the thought of fractions make you nervous? Does algebra make you want to close your textbook? Don’t worry! Algebraic fractions are simply fractions that have letters (variables) in them. The best part is that they follow the exact same rules as the normal fractions you have been using for years. If you can simplify \(\frac{4}{8}\) or add \(\frac{1}{2} + \frac{1}{3}\), you already have the skills to master this chapter!

In this guide, we will learn how to simplify these expressions and how to add, subtract, multiply, and divide them. For those taking the Extended papers, we will also look at how to solve equations involving these fractions.

1. Simplifying Algebraic Fractions

To simplify an algebraic fraction, we want to make it as "small" as possible by cancelling out factors that appear on both the top (numerator) and the bottom (denominator).

The Golden Rule: Factorise First!

You can only cancel out factors (things being multiplied). You cannot cancel out terms (things being added or subtracted).

Example (The wrong way): In the expression \(\frac{x + 5}{5}\), you cannot just cross out the \(5\)s. This is a common mistake! Because the \(5\) on top is being added, it is not a factor.

Example (The right way): Simplify \(\frac{6x^2}{3x}\).
1. Look at the numbers: \(6 \div 3 = 2\).
2. Look at the letters: \(x^2 \div x = x\).
3. The answer is \(2x\).

Extended Content: Simplifying Quadratic Fractions

If you see a quadratic expression (like \(x^2 + 5x + 6\)), you must factorise it into two brackets before you try to simplify.

Step-by-Step Example: Simplify \(\frac{x^2 - 9}{x^2 + 5x + 6}\)
1. Factorise the top: This is a "difference of two squares": \((x - 3)(x + 3)\).
2. Factorise the bottom: Find numbers that multiply to \(6\) and add to \(5\): \((x + 2)(x + 3)\).
3. Write it out: \(\frac{(x - 3)(x + 3)}{(x + 2)(x + 3)}\)
4. Cancel: Both top and bottom have \((x + 3)\). Cross them out!
5. Final Answer: \(\frac{x - 3}{x + 2}\)

Key Takeaway: Always look for common factors or quadratic patterns to factorise before you start cancelling.

2. Multiplication and Division

Multiplying and dividing algebraic fractions is often easier than adding them because you don't need a common denominator!

Multiplication

Just like normal fractions, multiply the tops together and the bottoms together.
\(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

Example: \(\frac{3x}{4} \times \frac{2}{x^2} = \frac{6x}{4x^2}\).
Now simplify: \(\frac{6}{4}\) becomes \(\frac{3}{2}\), and \(x\) over \(x^2\) leaves an \(x\) on the bottom. Answer: \(\frac{3}{2x}\).

Division

Use the "Keep-Change-Flip" rule:
1. Keep the first fraction.
2. Change the \(\div\) sign to a \(\times\) sign.
3. Flip the second fraction upside down (this is called the reciprocal).

Example: \(\frac{x}{5} \div \frac{x^2}{10}\)
1. Keep: \(\frac{x}{5}\)
2. Change: \(\times\)
3. Flip: \(\frac{10}{x^2}\)
4. Multiply: \(\frac{10x}{5x^2}\)
5. Simplify: \(\frac{2}{x}\)

3. Addition and Subtraction

This is where many students get stuck, but if you remember the "Common Denominator" rule, you’ll be fine! To add or subtract, the bottom numbers must be the same.

How to find a Common Denominator

The easiest way is to multiply the two denominators together.

Step-by-Step Example: Simplify \(\frac{3}{x} + \frac{2}{y}\)
1. The common denominator is \(x \times y = xy\).
2. Adjust the first fraction: Multiply top and bottom by \(y\): \(\frac{3y}{xy}\).
3. Adjust the second fraction: Multiply top and bottom by \(x\): \(\frac{2x}{xy}\).
4. Add the tops: \(\frac{3y + 2x}{xy}\).

Handling Linear Denominators (Extended)

If the bottoms are expressions like \((x+1)\) and \((x-2)\), the common denominator is \((x+1)(x-2)\).

Example: \(\frac{5}{x+1} - \frac{3}{x-2}\)
Common denominator: \((x+1)(x-2)\)
New expression: \(\frac{5(x-2) - 3(x+1)}{(x+1)(x-2)}\)
Expand the top: \(\frac{5x - 10 - 3x - 3}{(x+1)(x-2)}\)
Simplify: \(\frac{2x - 13}{(x+1)(x-2)}\)

Quick Tip: Be very careful with the minus sign in the middle! It applies to the entire numerator of the second fraction.

4. Solving Fractional Equations (Extended Only)

When you have an equation with algebraic fractions, your goal is to "get rid" of the fractions as quickly as possible. We do this by multiplying every term by the common denominator.

Method: Cross-Multiplication

If you have one fraction equal to another fraction, like \(\frac{A}{B} = \frac{C}{D}\), you can cross-multiply: \(A \times D = B \times C\).

Example: Solve \(\frac{2}{x+3} = \frac{1}{x-1}\)
1. Cross-multiply: \(2(x - 1) = 1(x + 3)\)
2. Expand: \(2x - 2 = x + 3\)
3. Solve for \(x\): \(x = 5\)

Method: Multiplying through

If the equation is more complex, like \(\frac{1}{x} + \frac{1}{2} = \frac{3}{x}\):
1. Find the common denominator of all terms (\(2x\)).
2. Multiply every single term by \(2x\):
\(2x(\frac{1}{x}) + 2x(\frac{1}{2}) = 2x(\frac{3}{x})\)
3. The denominators cancel out: \(2 + x = 6\)
4. Solve: \(x = 4\)

Common Mistakes to Avoid

  • Illegal Cancelling: Never cancel terms that are separated by \(+\) or \(-\) signs. Factorise first!
  • Sign Errors: When subtracting fractions, remember to distribute the negative sign to everything in the second numerator.
  • Forgetting the Bottom: When adding/subtracting, don't just add the denominators. They stay the same once you have found the common one!
  • Calculator Use: Remember, Paper 1 and Paper 2 are non-calculator. You must be comfortable doing these manipulations by hand.

Did you know? Algebraic fractions are used in physics to calculate the total resistance in parallel circuits and in medicine to determine how quickly a drug leaves the bloodstream!

Quick Review Key Takeaways:

1. Simplify: Factorise everything then cancel common brackets.
2. Multiply/Divide: Tops with tops, bottoms with bottoms. Flip the second for division.
3. Add/Subtract: Find a common denominator by multiplying the bottoms.
4. Equations: Multiply by the denominator to "clear" the fractions.