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Mathematics: Class 8 (Part 2) — Chapter 3: Proportional Reasoning-2

3.1 Proportionality—A Quick Recap

Core Concepts
  • Proportional Relationship: When two or more related quantities change by the same factor.
  • Cross-Multiplication Method: Two ratios a : b and c : d are proportional if:
    a × d = b × c or a/c = b/d
3.2 Ratios in Maps

Representative Fraction (RF)
  • Representative Fraction (RF): An expression showing the ratio between a distance on the map and the corresponding actual geographical distance on the ground.
  • Example: An RF of 1 : 60,00,000 means 1 cm on the map represents 60,00,000 cm (or 60 km) of actual geographical distance.
3.3 Ratios with More than 2 Terms

Multi-term Ratios
  • Ratios can have multiple terms (e.g., a : b : c : d) representing the relative proportions of multiple components.
  • Proportionality: Two multi-term ratios are proportional if:
    a : b : c : d :: p : q : r : s which implies a/p = b/q = c/r = d/s
3.4 Dividing a Whole in a Given Ratio

Division Formula
  • To divide a total quantity x in the ratio a : b : c : ..., the individual parts are calculated as:
    • Part 1 = x × [a / (a + b + c + ...)]
    • Part 2 = x × [b / (a + b + c + ...)]
    • Part 3 = x × [c / (a + b + c + ...)]
3.5 A Slice of the Pie (Pie Charts)

Constructing Pie Charts
  • Pie Chart: A circular statistical graphic divided into slices to illustrate numerical proportion.
  • Angle Calculation: The angle for each category is proportional to its value relative to the total sum, mapped against the total circle angle of 360°.
    Sector Angle = (Category Value / Total Value) × 360°
3.6 Inverse Proportions

Direct vs. Inverse Proportions
  • Direct Proportion: Two quantities change by the same factor in the same direction (as one increases, the other increases).
  • Inverse Proportion: Two quantities change by the same factor in opposite directions (as one increases, the other decreases by the same factor).
    Mathematically: x × y = k (where k is a constant).
  • Proportionality Relation: If x and y are inversely proportional, then:
    x₁ × y₁ = x₂ × y₂ or x₁/x₂ = y₂/y₁
Chapter Summary