Ratio and Proportion - Concept of Ratio

Definition of Ratio: A ratio is a mathematical comparison of two quantities of the same kind, measured in the same units, performed by division. It tells us how many times one quantity is contained within the other. Visually, if you have a box of 3 red balls and 5 blue balls, the ratio of red to blue is $3:5$, representing the relative sizes of these two groups.

Notation and Terms: A ratio is expressed using the colon symbol ($:$). For two quantities $a$ and $b$, the ratio is written as $a:b$ (read as '$a$ is to $b$'). The first term $a$ is called the 'Antecedent' and the second term $b$ is called the 'Consequent'. This can be visualized as a fraction where the antecedent is the numerator and the consequent is the denominator.

Order of Terms: The order of terms in a ratio is extremely important. The ratio $2:3$ is not the same as $3:2$. For example, if a recipe calls for 2 cups of sugar for every 3 cups of flour, swapping them to $3:2$ would change the taste and texture of the food entirely.

Requirement of Same Units: To compare two quantities as a ratio, they must be in the same unit of measurement. If you are comparing $50$ cm to $2$ meters, you must first convert the meters to centimeters ($200$ cm) before forming the ratio $50:200$. Visually, you can imagine two line segments being measured against the same ruler to ensure a fair comparison.

Simplest Form: A ratio is said to be in its simplest or lowest form when the Highest Common Factor (HCF) of the antecedent and the consequent is $1$. For example, the ratio $10:20$ can be simplified to $1:2$ by dividing both terms by their common factor $10$. This is similar to reducing a fraction to its simplest form.

Ratios have No Units: Since a ratio is a comparison of two similar quantities (e.g., length to length, or weight to weight), the units cancel out during the division process. Therefore, a ratio is a pure number and does not have any units like kg, cm, or liters.

Equivalent Ratios: Multiplying or dividing both the antecedent and the consequent by the same non-zero number results in an equivalent ratio. For instance, $1:2$, $2:4$, and $3:6$ are all equivalent. Visually, if you look at a grid of squares, shaded $1$ out of every $2$ squares is the same proportion as shading $2$ out of every $4$ squares.

Comparison of Ratios: To compare which of two ratios is larger, convert them into fractions and make their denominators equal (like finding a common denominator). For example, to compare $2:3$ and $3:4$, compare $\frac{2}{3}$ (which is $\frac{8}{12}$) and $\frac{3}{4}$ (which is $\frac{9}{12}$).