Class 8 Maths Chapter 3 Proportional Reasoning-2 Visual Infographic | NCF 2023 Ganita Prakash
Chapter 3: Proportional Reasoning - 2
Multi-Term Ratios, Representative Fractions (RF), Pie Chart Central Angles, and Inverse Variations
🗺️ 1. Representative Fraction (RF)
Map Scale RatioA Representative Fraction (RF) represents the exact ratio between a distance measured on a map and actual geographical ground distance:
RF = Map Distance / Ground Distance
• Scale 1 : 60,00,000 means 1 cm on map = 60,00,000 cm on ground.
• Conversion: 60,00,000 cm = 60,000 m = 60 km.
🧪 2. Multi-Term Ratios
Compound ScalingRatios easily extend beyond 2 terms (e.g., a : b : c : d). When scaled, every single term must change by the exact same multiplier k:
Spice Mix Example: Coriander : Chilli : Dal : Methi = 8 : 4 : 2 : 1.
If scaled by 1/2: 4 : 2 : 1 : 0.5.
🧱 3. Concrete & Paint Division
Part-Whole AllocationTo divide a total quantity x in the ratio a : b : c : ...
Term Value = x × [ Term / (Sum of all terms) ]
Concrete (1 : 1.5 : 3 for 110 units):
Sum = 1 + 1.5 + 3 = 5.5 units.
• Cement = (1/5.5) × 110 = 20 units
• Sand = (1.5/5.5) × 110 = 30 units
• Gravel = (3/5.5) × 110 = 60 units
🥧 4. Pie Chart Central Angles
360° Circular ProportionsEach sector angle in a pie chart is directly proportional to its category's share of the whole (360°):
Bus 120° (1/3), Walk 90° (1/4), Cycle 60°, Car 30°
⚖️ 5. Direct vs Inverse Proportions
Mathematical Comparison| Property | Direct Variation | Inverse Variation |
|---|---|---|
| Behavior | Both increase/decrease together | One increases → other decreases |
| Constant | Quotient: x / y = k | Product: x × y = k |
| Ratio Law | x1 / x2 = y1 / y2 | x1 / x2 = y2 / y1 |
⏱️ 6. Combined Work & Pipe Rates
Reciprocal AdditionWhen multiple agents work together, add their rates per unit time (reciprocals):
Two Pumps Example: Small (3 hrs) & Large (2 hrs):
Combined Time = (3 × 2) / (3 + 2) = 6/5 = 1.2 hours (1 hr 12 min)!
🔍 Master Proportional Reasoning Applications Matrix
Standard real-world problem archetypes across multi-term allocation, geometry, and inverse relationships:
| Problem Scenario | Proportion Model | Working & Step-by-Step Logic | Final Solution |
|---|---|---|---|
| Triangle Angles (1 : 3 : 5) | Direct Angle Split of 180° | Sum of terms = 1 + 3 + 5 = 9 parts Angle 1 = (1/9)×180°, Angle 2 = (3/9)×180°, Angle 3 = (5/9)×180° |
20°, 60°, 100° |
| Library Books (Odiya : Hindi : Eng = 3 : 2 : 1) | Multi-Term Direct Scaling | 3 parts = 288 books → 1 part = 288 / 3 = 96 books Hindi = 2 × 96, English = 1 × 96 |
Hindi: 192, English: 96 |
| Coin Ratio Value (₹10:₹5:₹2:₹1 = 4:3:2:1) | Total 100 coins divided | Parts = 4+3+2+1 = 10 → 40(₹10), 30(₹5), 20(₹2), 10(₹1) Value = 400 + 150 + 40 + 10 |
Total = ₹600 |
| Hostel Food Provisions | Inverse Variation: Persons × Days = k | Initial: 80 students × 15 days = 1200 student-days New: 80 + 20 = 100 students → Days = 1200 / 100 |
12 Days |
| Speed & Travel Time | Inverse Variation: Speed × Time = k | Distance = 60 km/h × 2 hrs = 120 km New Speed = 80 km/h → Time = 120 / 80 = 1.5 hrs |
1 hr 30 min |
| Factory Machine Scheduling | Inverse Variation: Machines × Days = k | 42 machines × 63 days = 2646 machine-days Days = 54 → Machines = 2646 / 54 |
49 Machines |
📐 7. Triangle Side Ratios & Inequality
Can any given 3-term ratio form the sidelengths of a triangle?
The sum of any two sides must be strictly greater than the third side (a + b > c).
- Ratio 3 : 4 : 5: 3 + 4 = 7 > 5 → Valid Triangle (Right-angled).
- Ratio 1 : 3 : 5: 1 + 3 = 4 < 5 → IMPOSSIBLE! The shorter segments cannot meet.
Triangles with identical side ratios are similar, but only congruent if absolute lengths match.
🐾 8. Sector Proportions: Animal Sleep
Reading 24-hour circular dial proportions into daily sleep hours:
- Cat (Half of dial shaded → 1/2 of 24 hrs): 12 to 14 hrs.
- Python / Snake (Three-quarters shaded → 3/4 of 24 hrs): 18 hrs.
- Bat (Five-sixths shaded → 5/6 of 24 hrs): 20 hrs.
- Cow / Giraffe (Small slice → 1/12 of 24 hrs): 2 to 3.5 hrs.
⚠️ Common Pitfalls & Conceptual Traps
Direct Rule on Inverse Tasks
Applying direct cross-multiplication (a : b :: c : d) to workers or speed problems gives absurd results! If 3 workers take 4 days, 4 workers take (3 × 4)/4 = 3 days, not more days!
Map RF Unit Consistency
RF ratios are pure unitless proportions (1 : 60,00,000 means 1 unit to 60,00,000 identical units). Always convert ground measurements to cm before ratio setup!
Combined Time Addition
Two workers working together NEVER take the sum of their individual times (1 hr + 1.5 hr ≠ 2.5 hrs). Add their work rates: 1 + 2/3 = 5/3 rate → Total Time = 3/5 hr (36 mins).
