Ch 07 - Proportional Reasoning-18th Ganita Prakash

• Definition: A ratio $a: b$ mathematically represents that for every $a$ units of the first quantity, there are strictly $b$ units of the second quantity.
• Terminology: The individual numbers $a$ and $b$ inside the expression are formally designated as the terms of the ratio. (Specifically, $a$ is the antecedent and $b$ is the consequent).

Boundary Conditions / Domain:

• $a, b \in \mathbb{R}$.
• Strict Constraint: The consequent $b \neq 0$, because a ratio $a:b$ is algebraically equivalent to the fraction $\frac{a}{b}$, and division by zero is undefined.

• The Concept: For an image, object, or quantity to remain proportionally similar when scaled, its dimensions must change by the exact same multiplicative factor.
• The Trap: Adding or subtracting the same constant value to both width and height (e.g., $W - 20$, $H - 20$) fundamentally distorts the shape because the relative mathematical ratio is destroyed.

Boundary Conditions / Domain:

• The multiplicative scale factor $k$ must be strictly $k > 0$ to preserve positive physical dimensions.

Component Anatomy:

• $a, b$: The given integer terms of the original ratio.
• $\text{HCF}(a,b)$: The Highest Common Factor (Greatest Common Divisor) of the two terms.

Boundary Conditions / Domain:

• $a, b \in \mathbb{Z}^+$ (Standard ratios involve positive integers).
• Strict Constraint: $b \neq 0$.