Ch 03 - Proportional Reasoning-28th Ganita Prakash Part 2

• A proportional relationship exists when two or more related quantities change strictly by the same factor.
• If the quantities are represented by a ratio $a:b$, any valid proportional state must be a scaled multiple $ka:kb$.
• Example: Mixing 2 cups of rice to 1 cup of dal ($2:1$). If you use 6 cups of rice (multiplied by 3), you must use 3 cups of dal to maintain proportionality.

Boundary Conditions / Domain:

• The scaling factor $k$ strictly cannot equal $0$ ($k \neq 0$).
• For physical quantities (like mass, distance, items), the values must strictly be positive ($a > 0, b > 0$).

• Direct Proportion: Two quantities move strictly in the same direction by the same factor.
  • Test: If $x$ doubles, $y$ doubles.
  • Mathematical Signature: Their Quotient is constant ($x / y = k$).
• Inverse Proportion: Two quantities move strictly in opposite directions by the inverse factor.
  • Test: If $x$ doubles, $y$ halves.
  • Mathematical Signature: Their Product is constant ($x \times y = k$).

Boundary Conditions / Domain:

• In both relationships, $x \neq 0$ and $y \neq 0$ to avoid undefined operations or collapsing the constant.

Component Anatomy:

• $a, b$: The terms of the first ratio ($a:b$).
• $c, d$: The terms of the second ratio ($c:d$).

Boundary Conditions / Domain:

• $b \neq 0$ and $d \neq 0$ (Denominators strictly cannot be zero).
• Cross-multiplication is strictly a test for Direct Proportion equivalence.