Class 8 Maths Chapter 7 Proportional Reasoning-1 Visual Infographic | NCF 2023 Ganita Prakash
Chapter 7: Proportional Reasoning - 1
Understanding Multiplicative Scaling, Ratios, Trairasika (Rule of Three), Unequal Sharing & Unit Conversions
๐ผ️ 1. Multiplicative Scaling
Similarity in ChangeTwo rectangular images look similar only when width and height change by the same multiplicative factor, not by the same additive difference.
• Image A (60 × 40) to C (30 × 20): Both scaled by × 1/2 → Similar!
• Image A (60 × 40) to B (40 × 20): Both −20 mm → Distorted!
Proportionality is fundamentally multiplicative.
⚖️ 2. Ratios & Simplest Form
HCF ReductionA ratio a : b indicates that for every a units of the first quantity, there are b units of the second.
- Simplest Form: Divide all terms by their HCF (e.g., 60 : 40 ÷ 20 = 3 : 2).
- Proportion Symbol (::): When two ratios have identical simplest forms, they are in proportion: a : b :: c : d.
- Adding Terms Traps: Adding the same constant to both terms destroys the ratio!
๐ 3. Trairฤลika (Rule of Three)
Aryabhata's AlgorithmAryabhata (499 CE) formulated the classical method for solving 4 proportional quantities when 3 are known:
Cross Multiplication Law:
If a : b :: c : d, then a × d = b × c → d = (b × c) / a.
๐ฐ 4. Sharing in a Given Ratio
Part-Whole SplittingTo divide a total quantity x into a ratio of m : n:
First Part = m × [x / (m + n)]
Second Part = n × [x / (m + n)]
Example: Divide ₹4,500 in ratio 2 : 3.
Parts = 2 + 3 = 5.
Share 1 = 2/5 × 4500 = ₹1,800; Share 2 = 3/5 × 4500 = ₹2,700.
⚡ 5. When Rule of Three Fails
Inverse ProportionalityNot all real-life relations are directly proportional! Check if quantities grow together or move inversely:
Speed & Travel Time:
If speed increases from 50 km/h to 75 km/h, time decreases! Direct Trairasika (50 : 2 :: 75 : x) fails.
Work & Time: Oxen vs Tractor ploughing time.
Always verify whether multiplying one factor increases or decreases the other.
๐ 6. Essential Unit Conversions
Standard MultipliersRatios require identical measurement units before comparison:
- Length: 1 m = 3.281 ft.
- Area: 1 acre = 43,560 sq. ft; 1 ha = 10,000 sq. m = 2.471 acres.
- Volume: 1 L = 1,000 mL = 1,000 cc.
- Mass: 1 tonne = 1,000 kg.
- Temperature: °F = (9/5 × °C) + 32.
๐ Master Proportional Problem Solving Matrix
Step-by-step applications of Trairasika, unit matching, and ratio modeling:
| Problem Scenario | Proportion Setup | Unit Adjustment / Calculation | Final Solution |
|---|---|---|---|
| Car Speed & Distance | 150 min : 90 km :: 240 min : x km | Convert 4 hours → 240 min. x = (240 × 90) / 150 | 144 km |
| Tea Cost Comparison | Himachal: 200 g : ₹200 Meghalaya: 1000 g : ₹800 |
Convert 1 kg to 1000 g. Himachal 1 kg = ₹1,000 | Himachal is costlier (₹1000 vs ₹800/kg) |
| Lilavati Saffron Problem | (3/7) niska : 2.5 pala :: 9 niska : x pala | x = (2.5 × 9) / (3/7) = 22.5 × 7 / 3 | 52.5 palas |
| Gold Mass from Volume | Water 2 : Gold 37 :: 1 kg : x kg | 1 L water = 1 kg. x = (37 × 1) / 2 | 18.5 kg |
| Sand-Cement Mixture | Initial (40 kg, 3 : 1) → 30 kg Sand, 10 kg Cement. New 5 : 2 :: 30 : x |
New Cement = (30 × 2) / 5 = 12 kg. Added = 12 − 10 |
Add 2 kg Cement |
| Cupro-Nickel Coin | Ratio Cu:Ni = 3 : 1, Mass = 7.74 g. Cu = 5.805 g, Ni = 1.935 g |
Cu cost = ₹5.26, Ni cost = ₹2.59. Total cost = 5.26 + 2.59 |
₹7.85 Metal Cost |
☕ 7. Real-Life Qualitative Reasoning
The Filter Coffee Concentration Model:
Concentration depends strictly on the ratio of decoction to milk:
- Regular Coffee: 15 mL : 35 mL = 3 : 7 (ratio value ≈ 0.43).
- Stronger Coffee: 20 mL : 30 mL = 2 : 3 (ratio value ≈ 0.67 > 0.43).
- Lighter Coffee: 10 mL : 40 mL = 1 : 4 (ratio value = 0.25 < 0.43).
Population Crowding: Compare density (People per sq. km) rather than raw totals: Mumbai (36,364/sq. km) is denser than Delhi (20,216/sq. km)!
๐งฉ 8. Binairo (Takuzu) Logic Rules
Binairo Puzzle: A pure balance grid played with horizontal (−) and vertical (|) markers:
- Equal Balance: Every row and column must contain an equal number of horizontal and vertical bars.
- No Triples: No more than two identical symbols can be placed adjacent to each other.
- Unique Layout: Every row is completely unique; every column is completely unique.
⚠️ Common Pitfalls & Conceptual Traps
Mismatched Unit Trap
Never setup proportions across mismatched units (e.g., mixing minutes with hours, or grams with kilograms). Always convert to a common base unit first!
Additive Age Scaling Trap
Ages do not scale proportionally over time! If a mother is 10 times older than her child today (30 : 3), 9 years later their ages become 39 : 12 = 13 : 4, not 10 : 1!
Packaging Economy Illusion
Larger supermarket packages are not always strictly proportional in price to sachets due to packaging, overheads, and bulk discount pricing structures.
