Multiplication and division help us work with numbers, but ratios help us understand the relationship between them. A ratio is simply a way to compare two or more quantities. Think of it as a recipe for a relationship. If you're mixing orange paint, you might use 3 parts red for every 2 parts yellow. That relationship, 3-to-2, is a ratio.

There are two main ways to think about these comparisons. Let's imagine a bowl of fruit containing 4 apples and 6 bananas. That's 10 pieces of fruit in total.

A part-to-part ratio compares one part of the group to another part. In our fruit bowl, the ratio of apples to bananas is 4 to 6. We can write this using a colon, as 4:6. For every 4 apples, there are 6 bananas.

A part-to-whole ratio compares one part of the group to the entire group. The ratio of apples to the total fruit is 4 to 10, or 4:10. This is useful for understanding what fraction of the whole is made up of one part. In this case, apples make up 4/10 of the fruit.

Notice how the order matters. The ratio of apples to bananas (4:6) is different from the ratio of bananas to apples (6:4). The quantity you mention first always comes first in the notation. This foundational idea of comparing parts was formally explored by ancient Greek mathematicians like Euclid of Alexandria in his work, Elements.

Just like fractions, ratios can often be simplified. A ratio of 4 apples to 6 bananas (4:6) describes the same fundamental relationship as 2 apples to 3 bananas (2:3). To simplify a ratio, you find the greatest common divisor (GCD) of both numbers and divide each by it. Here, the GCD of 4 and 6 is 2.

This simplified ratio gives us the core relationship. For every 2 apples, there are 3 bananas. This leads to the idea of equivalent ratios. Any ratio that can be simplified to the same base form is equivalent. So, 4:6, 6:9, and 8:12 are all equivalent to 2:3. They all represent the same proportional relationship, just at a different scale.

This ability to scale ratios up and down is incredibly powerful in the real world. Imagine you have a recipe for pancakes that serves 4 people and requires 200g of flour and 2 eggs. The ratio of flour to eggs is 200:2, which simplifies to 100:1.

What if you need to make pancakes for 10 people? You need to find an equivalent ratio. The scaling factor is the number you multiply or divide by to get from your current scale to your desired scale. To go from serving 4 people to 10, the scaling factor is $10 \div 4 = 2.5$. So you multiply each ingredient by 2.5.

• Flour: $200g \times 2.5 = 500g$
• Eggs: $2 \times 2.5 = 5$ eggs

The new ratio is 500:5, which still simplifies to 100:1. The relationship remains the same.

This same principle applies everywhere. Architectural blueprints use ratios to represent a huge building on a small piece of paper. A map uses a scale ratio (like 1:50,000) to show large distances in a manageable format. By understanding how to work with ratios, you can accurately scale any relationship up or down.

Now that we've covered the different types of ratios and how to scale them, let's review the key terms.

Ready to test your understanding?

Understanding ratios is a bridge from simple arithmetic to practical, real-world maths. It allows you to analyse relationships, maintain proportions, and scale ideas with confidence.