Ratio and Proportion

Overview & Fundamental Concept

Pointwise AISSEE Class 6 Mathematics guide for Ratio and Proportion (Topic 5). Learn ratio definitions, unit standardization, dividing quantities, product of extremes equals product of means, and mean proportional calculations.

• Definition of Ratio: A ratio is a mathematical comparison of the sizes of two quantities of the same kind and expressed in the same unit by the process of division.
• Notation & Terminology: The ratio of quantity $a$ to quantity $b$ is written as $a : b$ (read as '$a$ is to $b$') or as the fraction $\frac{a}{b}$. The first term $a$ is called the Antecedent, and the second term $b$ is called the Consequent ($b \ne 0$).
• Ratio has NO Units: Because a ratio is a comparison of two quantities in identical units, the units cancel out completely. A ratio is a pure abstract number.
• Order Matters: The ratio $a : b$ is NOT the same as $b : a$ (unless $a = b$). For instance, if boys to girls is $3 : 2$, girls to boys is $2 : 3$.
• Unit Harmonization Requirement: Two quantities can only be compared when converted into identical units. For example, comparing $40\text{ cm}$ to $2\text{ m}$: first convert $2\text{ m} = 200\text{ cm}$, giving ratio $40 : 200 = 1 : 5$ (never $40 : 2$).

• Part-to-Whole Ratio Logic: If an amount $Q$ is to be shared between two persons in the ratio $a : b$:
• • Sum of Ratio Terms: Total equal parts $= a + b$.
• • First Person's Share: $\frac{a}{a + b} \times Q$.
• • Second Person's Share: $\frac{b}{a + b} \times Q$.
• Three-Part Extended Ratios: When dividing among three entities in ratio $a : b : c$, total parts $= a + b + c$. Each share is $\frac{\text{term}}{\text{sum of terms}} \times Q$.

• Definition of Proportion: An equality of two ratios is called a proportion. If two ratios $a : b$ and $c : d$ are equal, the four numbers $a, b, c, d$ are in proportion, written as: $$a : b :: c : d \quad \text{or} \quad \frac{a}{b} = \frac{c}{d}$$
• Extremes and Means: In the proportion $a : b :: c : d$, the first and fourth terms ($a$ and $d$) are called Extremes (outer terms). The second and third terms ($b$ and $c$) are called Means (middle terms).
• The Fundamental Law of Proportion: $$\text{Product of Extremes} = \text{Product of Means} \iff a \times d = b \times c$$
• Finding Missing Term: If any one of the four terms is unknown, use the cross-multiplication identity: $x = \frac{b \times c}{d}$ or $x = \frac{a \times d}{b}$.
• Continued Proportion & Mean Proportional: Three numbers $a, b, c$ are said to be in continued proportion if: $$\frac{a}{b} = \frac{b}{c} \iff b^2 = a \times c \implies b = \sqrt{a \times c}$$ Here, $b$ is called the Mean Proportional between $a$ and $c$, and $c$ is called the Third Proportional.