⚖️ Ratio and Proportion - Formula Sheet

🎯 Basic Concepts

Ratio

a : b = a / b
Where a and b are same type quantities

Proportion

a : b = c : d  or  a/b = c/d

Continued Proportion

a : b = b : c = c : d

📊 Ratio Operations

Equivalent Ratios

a : b = (a × k) : (b × k)

Simplifying Ratios

Divide by HCF of all terms

Compound Ratio

If a : b and c : d, then compound ratio = ac : bd

Duplicate Ratio

(a : b)² = a² : b²

Sub-duplicate Ratio

√(a : b) = √a : √b

🔢 Proportion Properties

Invertendo

If a : b = c : d, then b : a = d : c

Alternendo

If a : b = c : d, then a : c = b : d

Componendo

If a : b = c : d, then (a + b) : b = (c + d) : d

Dividendo

If a : b = c : d, then (a - b) : b = (c - d) : d

⚡ Applications

Partnership

Profit sharing ratio = Investment₁ × Time₁ : Investment₂ × Time₂

Mixtures

(Cheaper : Dearer) = (Mean - Cheaper) : (Dearer - Mean)

Distribution

Share = Total × (Ratio Part) / (Sum of Ratios)

📝 Problem Types

Direct Proportion

If x increases, y increases proportionally
x₁/y₁ = x₂/y₂

Inverse Proportion

If x increases, y decreases proportionally
x₁ × y₁ = x₂ × y₂

Compound Proportion

Chain of proportional relationships

🔍 Quick Tips

1. Always simplify ratios to lowest terms
2. Convert all quantities to same units
3. Check if direct or inverse proportion
4. Use HCF to simplify complex ratios
5. Verify answers by checking consistency

Master Ratio & Proportion - Foundation for many topics! ⚖️