To simplify ratios quickly, students can use a few helpful techniques. These methods make understanding and working with ratios easier in Year 8 Mathematics.

1. Finding the Greatest Common Factor (GCF)
The first method is to find the GCF of the numbers in the ratio. The GCF is the biggest number that can evenly divide both numbers.

For example, in the ratio 12:16, the GCF is 4. To simplify, divide both numbers by the GCF:

$$\frac{12 \div 4}{16 \div 4} = \frac{3}{4}$$

2. Using Prime Factorization
Another way is to break down each number into its prime factors. Then, you can cancel out any common factors.

For example, with the ratio 18:24:

• The prime factorization shows: 18 = 2 × 3² and 24 = 2³ × 3.
• Cancelling the common factors gives:

$$\frac{2 × 3²}{2³ × 3} = \frac{3}{4}$$

3. Dividing by Common Divisors
Students can also simplify ratios by finding common divisors. For instance, in the ratio 30:45, both numbers can be divided by 15:

$$\frac{30 \div 15}{45 \div 15} = \frac{2}{3}$$

4. Using a Number Line or Bar Model
Visual tools can make ratios easier to understand. By putting ratios on a number line or drawing bar models, students can see the relationships better. This way of learning helps make the idea of ratios clearer and easier to simplify.

5. Cross-Multiplying (For Comparison)
While this doesn’t directly simplify a ratio, cross-multiplying helps to compare them quickly. For example, to compare the ratios 2:3 and 4:5, you multiply:

2 × 5 and 3 × 4.
Since 10 < 12, this shows that 2:3 is smaller than 4:5.

By using these techniques, students can get better at simplifying ratios. Learning these methods helps them feel more confident and accurate when working with ratio problems. This skill is very important as they continue to learn math.