A ratio compares quantities of the same kind and shows how much there is of one quantity compared to another. Ratios can be written in the form a : b and can be simplified, scaled, and used to calculate unknown amounts.

To simplify a ratio, divide all parts by their highest common factor (HCF).

$$
12 : 18 = 2 : 3
$$

This keeps the relationship the same while using smaller numbers.

To use a ratio to divide a quantity, first add the parts of the ratio to find the total number of parts. Then divide the total quantity by this number to find the value of one part.

For example, divide \(40\) in the ratio \(3 : 5\):

$$
3 + 5 = 8
$$

$$
40 \div 8 = 5
$$

Each part is worth \(5\), so the final amounts are:

$$
3 \times 5 = 15, \quad 5 \times 5 = 25
$$

Ratios can also be used to scale quantities. If a ratio is increased or decreased by the same factor, the relationship stays the same.

$$
2 : 7 \rightarrow 4 : 14
$$

Both parts have been multiplied by \(2\).

In real-life contexts, ratios often appear in problems involving recipes, maps, mixtures, and sharing.

For example, if paint is mixed in the ratio \(1 : 4\) (red : white) and there are \(10\) litres of white paint, the scale factor is:

$$
10 \div 4 = 2.5
$$

So the amount of red paint needed is:

$$
1 \times 2.5 = 2.5
$$

Always make sure the ratio matches the quantities given. Mixing up the order of the ratio terms is a common source of errors, even when the calculations are correct.