Class 8 Ganita Prakash Part 2 Chapter 3 Proportional Reasoning-2: Complete NCERT Solutions, Examples & Figure it Out Answers Labels:

 Class 8 Ganita Prakash Part 2 Chapter 3 Proportional Reasoning-2: Complete NCERT Solutions, Examples & Figure it Out Answers

Introduction

“Proportional Reasoning-2” is Chapter 3 of Ganita Prakash Part-II (Class 8). It builds on ratios and proportions with real-life applications: maps (Representative Fraction), multi-term ratios (spice mixes, concrete), dividing wholes, pie charts, and inverse proportions (workers vs days, speed vs time). The chapter uses cross-multiplication, scaling, and visual charts to develop strong proportional thinking.

Basic Knowledge Required

• Ratio notation (a : b) and simplification
• Cross-multiplication to check proportionality
• Direct proportion (both increase/decrease by same factor)
• Area/perimeter of basic shapes (for triangle construction)
• Angle sum in triangle = 180°
• Pie chart construction (360° circle)

Key Definitions

• Representative Fraction (RF): Ratio of map distance to actual ground distance (e.g., 1 : 60,00,000 means 1 cm on map = 60 km on ground).
• Multi-term Ratio: a : b : c : d means quantities change by same factor.
• Inverse Proportion: When one quantity increases, the other decreases by the same factor; product remains constant (xy = k).

Important Formulas Used

1. Two ratios a:b and c:d proportional if $a \times d = b \times c$ or $\dfrac{a}{c} = \dfrac{b}{d}$
2. Multi-term: $\dfrac{a}{p} = \dfrac{b}{q} = \dfrac{c}{r} = \dfrac{d}{s}$
3. Dividing x in ratio p : q : r : … First part = $x \times \dfrac{p}{p+q+r+\ldots}$
4. Inverse proportion: $x_1 y_1 = x_2 y_2 = k$ or $\dfrac{x_1}{x_2} = \dfrac{y_2}{y_1}$
5. Pie chart angle = $\dfrac{\text{part}}{\text{total}} \times 360^\circ$

3.1 Proportionality – A Quick Recap

Example (Idli batter): Viswanath 6:3, Puneet 4:2 Cross-multiply: 6×2 = 12, 3×4 = 12 → proportional (same taste if other ingredients scale).

Figure it Out: None in section (recap only).

3.2 Ratios in Maps

RF 1 : 60,00,000 → 1 cm map = 60 km ground.

Geographical distance Bengaluru–Chennai: Measure map distance (say d cm) → actual = d × 60,00,000 cm = d × 60 km. (Same for Mangaluru–Chennai.)

Different maps: All give approximately same ground distance (independent of scale).

Map Making Activity: Sketch classroom at 1:50 scale (mark desks, board, etc.).

Figure it Out: Use ruler on map (answers depend on printed map scale – verify with atlas).

3.3 Ratios with More than 2 Terms

Spice mix: 8:4:2:1 Puneet (2 chillies = half) → 4:2:1:0.5 Proportional: 8:4:2:1 :: 4:2:1:0.5

Example 1 (Purple paint): White = 5 parts = 10 L → 1 part = 2 L Red = 2×2 = 4 L, Blue = 3×2 = 6 L Total = 20 L.

Example 2 (Concrete): Ratio 1:1.5:3, 3 bags cement → 3:4.5:9 Total = 16.5 bags concrete.

Figure it Out

1. Warm-up/cool-down : batting : bowling : fielding = 3:4:3:5 Total parts = 15, 150 min → each part = 10 min Warm-up = 30 min, Batting = 40 min, Bowling = 30 min, Fielding = 50 min.
2. Odiya:Hindi:English = 3:2:1, Odiya=288 Total parts=6, 1 part=96 Hindi=192, English=96.
3. ₹10:₹5:₹2:₹1 = 4:3:2:1, total coins=100 Parts=10, 1 part=10 coins ₹10 coins=40 (₹400), ₹5=30 (₹150), ₹2=20 (₹40), ₹1=10 (₹10) Total money = ₹600.
4. Sides 3:4:5 → right-angled triangle (3²+4²=5²). All such triangles similar (not congruent – different sizes).
5. Sides 1:3:5 → 1+3=4 <5 → cannot form triangle (triangle inequality fails).

3.4 Dividing a Whole in a Given Ratio

Example 3 (Concrete 110 units): Ratio 1:1.5:3 → total parts=5.5 110 ÷ 5.5 = 20 Cement=20, Sand=30, Gravel=60 units.

Example 4 (Purple 50 ml): Ratio 2:3:5, total=10 Red=50×2/10=10 ml, Blue=15 ml, White=25 ml.

Example 5 (Triangle angles 1:3:5): Total 9 parts, 180° ∠A=20°, ∠B=60°, ∠C=100°.

Figure it Out – already solved above (1–5).

3.5 A Slice of the Pie

Grade pie chart (40 students): 12:10:8:6:4 → simplify ÷2 = 6:5:4:3:2 (total 20) Each part=18° A=108°, B=90°, C=72°, D=54°, E=36°.

Figure it Out

1. Summer 90, Rainy 120, Winter 150 (total 360) Angles: Summer 90°, Rainy 120°, Winter 150°.
2. Entertainment 50%→180°, Sports 25%→90°, News 15%→54°, Info 10%→36°.
3. Collect class data → convert to angles (total 360°).

3.6 Inverse Proportions

Example 1 (Workers): 5 workers → 4500 bricks/day For 18000 bricks → 20 workers.

Example 3 (Road): 20 workers 4 days → 10 workers = 8 days.

Example 4 (Pumps): 2 pumps 18 h → 4 pumps = 9 h.

Example 5 (Food): 80 students 15 days → 100 students = 12 days.

Example 6 (Vegetables): Ram 1 h, Shyam 1.5 h → together 3/5 h.

Figure it Out

1. (i) Yes (inverse), (ii) Yes, (iii) No (direct), (iv) Yes, (v) No (direct), (vi) No (direct).
2. 24 pencils ₹120 → 20 pencils = ₹100.
3. 20 families 6 days → 30 families = 4 days (assume same consumption).
4. Human 8, Elephant 2.5, Cat 13, Dog 10.5, etc. (from charts).
5. (i) Bus, (ii) 1/6, (iii) 108 children total, 12 by taxi, (iv) Cycle & Two-wheeler (both 60°).
6. 3 workers 4 days → 4 workers = 3 days.
7. 2 tanks 6 h → 5 tanks = 15 h.
8. 25 rows ×12 = 300 chairs → 20 per row = 15 rows.
9. 8 periods 45 min = 360 min total → 9 periods = 40 min each.
10. Small 3 h, Large 2 h → together 1.2 h (6/5 h).
11. 42 machines 63 days → 54 days = 49 machines.
12. 60 km/h 2 h → 80 km/h = 1.5 h.

15 Most Asked FAQs – Proportional Reasoning-2 (Click to Expand)

1. What does RF 1:60,00,000 mean on a map?

Answer: 1 cm on map = 60 km on ground. Distance = map cm × 60 km.

2. Spice ratio 8:4:2:1, Puneet has 2 chillies. Quantities?

Answer: Half → 4:2:1:0.5 (coriander 4, chillies 2, dal 1, methi 0.5).

3. Purple paint 2:3:5, 10 L white. Red & blue?

Answer: 1 part=2 L → Red 4 L, Blue 6 L, total 20 L.

4. Concrete 1:1.5:3, 3 bags cement. Total bags?

Answer: 3:4.5:9 → total 16.5 bags.

5. 110 units concrete in 1:1.5:3. Quantities?

Answer: Parts 5.5, multiplier 20 → Cement 20, Sand 30, Gravel 60.

6. Triangle angles 1:3:5. Measures?

Answer: 20°, 60°, 100° (180° total, parts 9).

7. 360 people: Summer 90, Rainy 120, Winter 150. Pie angles?

Answer: Summer 90°, Rainy 120°, Winter 150°.

8. 20 workers 4 days road. 10 workers how many days?

Answer: Inverse → 8 days (20×4=10×8).

9. 2 pumps 18 h tank. 4 pumps time?

Answer: 9 hours (inverse, 2×18=4×9).

10. 80 students food 15 days. 100 students days?

Answer: 12 days (80×15=100×12).

11. Ram 1 h, Shyam 1.5 h vegetables. Together time?

Answer: 3/5 hour (combined rate 5/3 per hour).

12. 24 pencils ₹120. 20 pencils cost?

Answer: ₹100 (direct proportion).

13. School 8 periods 45 min. 9 periods duration?

Answer: 40 min each (total 360 min fixed).

14. Pie chart 12:10:8:6:4 students. Angles?

Answer: 108°, 90°, 72°, 54°, 36° (simplify 6:5:4:3:2 ×18°).

15. Is number of workers & days inverse proportion? Example.

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Answer: Yes. 20 workers 4 days → 10 workers 8 days (product constant 80).