Simplifying Algebraic Fractions - GCSE Maths - Steps & Examples - Free Printable

Let's solve each expression step by step. We'll simplify the algebraic fractions using rules of multiplication and division of fractions.

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Left Column: Multiplication Problems

1. $\frac{3}{5} \times \frac{x}{y}$

Multiply numerators and denominators:
$$
= \frac{3x}{5y}
$$

2. $\frac{x}{y} \times \frac{3}{5}$

Same as above (commutative property):
$$
= \frac{3x}{5y}
$$

3. $\frac{x}{y} \times \frac{3}{6}$

Simplify $\frac{3}{6} = \frac{1}{2}$:
$$
= \frac{x}{y} \times \frac{1}{2} = \frac{x}{2y}
$$

4. $\frac{x}{y} \times \frac{3}{6x}$

Simplify $\frac{3}{6x} = \frac{1}{2x}$:
$$
= \frac{x}{y} \times \frac{1}{2x} = \frac{1}{2y} \quad \text{(the } x \text{ cancels)}
$$

5. $\frac{x}{xy} \times \frac{3}{6}$

Simplify $\frac{x}{xy} = \frac{1}{y}$, and $\frac{3}{6} = \frac{1}{2}$:
$$
= \frac{1}{y} \times \frac{1}{2} = \frac{1}{2y}
$$

6. $\frac{7x}{xy} \times \frac{3}{6}$

Simplify $\frac{7x}{xy} = \frac{7}{y}$, and $\frac{3}{6} = \frac{1}{2}$:
$$
= \frac{7}{y} \times \frac{1}{2} = \frac{7}{2y}
$$

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Right Column: Division and Multiplication Problems

1. $\frac{7x}{xy} \times \frac{3}{6y}$

Simplify $\frac{7x}{xy} = \frac{7}{y}$, and $\frac{3}{6y} = \frac{1}{2y}$:
$$
= \frac{7}{y} \times \frac{1}{2y} = \frac{7}{2y^2}
$$

2. $\frac{7x}{xy} \div \frac{3}{6y}$

First, simplify both fractions:
- $\frac{7x}{xy} = \frac{7}{y}$
- $\frac{3}{6y} = \frac{1}{2y}$

Now divide:
$$
\frac{7}{y} \div \frac{1}{2y} = \frac{7}{y} \times \frac{2y}{1} = 7 \times 2 = 14
$$

(The $y$ cancels out.)

3. $\frac{7x}{xy} \div \frac{6y}{3}$

Simplify:
- $\frac{7x}{xy} = \frac{7}{y}$
- $\frac{6y}{3} = 2y$

So:
$$
\frac{7}{y} \div 2y = \frac{7}{y} \times \frac{1}{2y} = \frac{7}{2y^2}
$$

4. $\frac{7x}{xy} \times \frac{6y}{3}$

Simplify:
- $\frac{7x}{xy} = \frac{7}{y}$
- $\frac{6y}{3} = 2y$

So:
$$
\frac{7}{y} \times 2y = 7 \times 2 = 14
$$

(The $y$ cancels.)

5. $\frac{7x}{xy} \times \frac{6y}{3} \times \frac{4y}{14}$

Step-by-step simplification:

- $\frac{7x}{xy} = \frac{7}{y}$
- $\frac{6y}{3} = 2y$
- $\frac{4y}{14} = \frac{2y}{7}$

Now multiply:
$$
\frac{7}{y} \times 2y \times \frac{2y}{7}
$$

Let's compute step by step:

- $\frac{7}{y} \times 2y = 14$
- $14 \times \frac{2y}{7} = 2 \times 2y = 4y$

So result is:
$$
4y
$$

6. $\frac{7x}{xy} \div \frac{6y}{3} \times \frac{4y}{14}$

Again:
- $\frac{7x}{xy} = \frac{7}{y}$
- $\frac{6y}{3} = 2y$
- $\frac{4y}{14} = \frac{2y}{7}$

So:
$$
\frac{7}{y} \div 2y \times \frac{2y}{7}
$$

First do the division:
$$
\frac{7}{y} \div 2y = \frac{7}{y} \times \frac{1}{2y} = \frac{7}{2y^2}
$$

Then multiply by $\frac{2y}{7}$:
$$
\frac{7}{2y^2} \times \frac{2y}{7} = \frac{1}{y} \quad \text{(7 and 2 cancel)}
$$

Final answer: $\frac{1}{y}$

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✔ Final Answers (Simplified):

#### Left Column:
1. $\frac{3x}{5y}$
2. $\frac{3x}{5y}$
3. $\frac{x}{2y}$
4. $\frac{1}{2y}$
5. $\frac{1}{2y}$
6. $\frac{7}{2y}$

#### Right Column:
1. $\frac{7}{2y^2}$
2. $14$
3. $\frac{7}{2y^2}$
4. $14$
5. $4y$
6. $\frac{1}{y}$

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Let me know if you'd like these arranged in a table or need further explanation!