Two ratios are considered equivalent if their cross-products are equal.

To determine whether two ratios are equivalent, you can utilize the cross-multiplication method. For instance, let’s examine the ratios $\frac{a}{b}$ and $\frac{c}{d}$. To check for equivalence, multiply the numerator of the first ratio by the denominator of the second ratio, and the denominator of the first ratio by the numerator of the second ratio. If the resulting products are equal, then the ratios are equivalent. In mathematical terms, this can be expressed as:

Let’s illustrate this with an example. Consider the ratios $\frac{2}{3}$ and $\frac{4}{6}$. To determine if they are equivalent, we perform the cross-multiplication: calculate $2 \times 6$ and $3 \times 4$. Both calculations yield a product of $12$, demonstrating that the ratios $\frac{2}{3}$ and $\frac{4}{6}$ are indeed equivalent.

Another method for verifying the equivalence of two ratios is by simplifying them. If both ratios reduce to the same simplest form, they are equivalent. For example, the ratio $\frac{4}{6}$ simplifies to $\frac{2}{3}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $2$. Since $\frac{2}{3}$ is already in its simplest form, and both ratios simplify to the same fraction, we conclude that they are equivalent.

Grasping these methods will empower you to confidently determine whether two ratios are equivalent, which is an essential skill in GCSE Maths.