Comparing Quantities - Equivalent Ratios

A ratio is a comparison of two quantities of the same kind by division, expressed as $a:b$ or $\frac{a}{b}$, where $a$ is the antecedent and $b$ is the consequent. Visually, if a basket has 3 apples and 5 oranges, the ratio $3:5$ represents the relative size of the apple group compared to the orange group.

Equivalent ratios are ratios that represent the same relationship or value. You can find them by multiplying or dividing both the numerator and denominator of a ratio by the same non-zero number. Visually, if you have a pattern of 1 blue tile for every 2 red tiles ($1:2$), doubling the pattern to 2 blue and 4 red tiles ($2:4$) keeps the visual proportion identical.

To check if two ratios are equivalent, convert them to their simplest form by dividing both terms by their Highest Common Factor (HCF). If the simplified fractions are the same, the ratios are equivalent. For example, $15:20$ simplifies to $3:4$ because $15 \div 5 = 3$ and $20 \div 5 = 4$.

Ratios can be compared by converting them into fractions with a common denominator. For instance, to compare $2:3$ and $3:4$, represent them as $\frac{8}{12}$ and $\frac{9}{12}$. This allows you to see on a number line that $3:4$ is larger than $2:3$.

The Cross-Multiplication Method is a quick way to verify equivalence. For two ratios $\frac{a}{b}$ and $\frac{c}{d}$, they are equivalent if $a \times d = b \times c$. Imagine an 'X' shape connecting the top of one fraction to the bottom of the other; the products must be equal.

When two ratios are equivalent, they are said to be in proportion. This is denoted as $a:b :: c:d$. In a proportional relationship, the product of the means ($b$ and $c$) equals the product of the extremes ($a$ and $d$).

Equivalent ratios are used to scale quantities up or down while maintaining the same balance. This is common in map scales where $1 \text{ cm}$ on paper represents a much larger distance like $100 \text{ km}$ in reality, maintaining a constant ratio of $1:10,000,000$.