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Class 8 Ganita Prakash · Part 2 · Chapter 3
Class 8 Ganita Prakash · Part 2 · Chapter 3
Proportional Reasoning-2
Page-wise textbook solutions (pages 55–69, Part 2) · Every step · Figures · Common mistakes · Tips
๐️ Key ideas of this chapter
- a : b and c : d are proportional when \(ad = bc\).
- Map scale (RF) 1 : 60,00,000 → 1 cm on the map = 60 km on the ground.
- Ratios can have many terms: 8 : 4 : 2 : 1. To share x in a : b : c, each part = \(x \times \dfrac{\text{term}}{a + b + c}\).
- Pie chart angle = \(\dfrac{\text{part}}{\text{total}} \times 360^\circ\).
- Direct proportion: \(\dfrac{x}{y}\) = constant. Inverse proportion: \(xy\) = constant (more workers → fewer days).
๐ Textbook Page 55
๐ Page 55Recap
Viswanath mixed 6 cups of rice with 3 cups of urad dal; Puneet mixed 4 cups with 2. Would their idlis taste the same?
1Cross-multiply 6 : 3 and 4 : 2: 6 × 2 = 12 and 3 × 4 = 12.
2Equal → proportional (both are 2 : 1).
✅ Answer: Yes, the ratios are proportional (if all other ingredients are proportional too).
๐ Textbook Page 56
๐ Page 56Think
What does RF 1 : 60,00,000 mean? Convert 60,00,000 cm to km.
1RF = map distance : real distance. 1 cm on the map = 60,00,000 cm on the ground.
260,00,000 cm ÷ 100 = 60,000 m; ÷ 1000 = 60 km.
✅ Answer: 1 cm on the map = 60 km on the ground (straight-line distance, not road distance).
๐ Page 56Map
Use the map to find the geographical distance Bengaluru–Chennai and Mangaluru–Chennai.
1Measure with a ruler on the map, then multiply each cm by 60 km.
2Real straight-line distances: Bengaluru–Chennai ≈ 290 km, Mangaluru–Chennai ≈ 590 km.
3Note: the map in the book says 'Map not to scale' — it is printed smaller, so on that page Bengaluru–Chennai measures only about 2.4 cm (≈ 145 km by the RF). On a correctly printed 1 : 60,00,000 map it would measure about 4.8 cm.
✅ Answer: About 290 km and 590 km. Different maps give about the same real distance if they are printed to their scale.
⚠️ Common mistake: Using the road distance (about 350 km) — RF gives the straight-line distance.
๐ Textbook Page 57
๐ Page 57Activity
Make a sketch of your classroom with the scale 1 : 50.
11 : 50 → 1 cm on paper = 50 cm in the room.
2Room 8 m × 6 m = 800 cm × 600 cm → 16 cm × 12 cm on paper.
3A 1.5 m desk → 3 cm.
✅ Answer: Divide every real length (in cm) by 50.
๐ Page 57Spice mix
Viswanath uses coriander : chillies : toor dal : methi = 8 : 4 : 2 : 1. Puneet has only 2 chillies. How much of each should he use?
12 chillies is half of 4 → halve every term.
28 ÷ 2 = 4, 4 ÷ 2 = 2, 2 ÷ 2 = 1, 1 ÷ 2 = 0.5
✅ Answer: 4 spoons coriander, 2 chillies, 1 spoon toor dal, ½ spoon methi: 8 : 4 : 2 : 1 :: 4 : 2 : 1 : 0.5
๐ Textbook Page 58
๐ Page 58Example 1
Red : Blue : White = 2 : 3 : 5. Yasmin has 10 L of white. How much red and blue? Total?
1White 5 parts = 10 L → 1 part = 2 L.
2Red 2 × 2 = 4 L, blue 3 × 2 = 6 L.
3Total 4 + 6 + 10 = 20 L.
✅ Answer: Red 4 L, blue 6 L; total 20 L of purple.
๐ Page 58Example 2
Concrete cement : sand : gravel = 1 : 1.5 : 3. With 3 bags of cement, how many bags of concrete?
1Multiply each term by 3: 3 : 4.5 : 9.
2Total 3 + 4.5 + 9 = 16.5
✅ Answer: 16.5 bags
๐ Textbook Page 59
๐ Page 59Example 3
110 units of concrete in the ratio 1 : 1.5 : 3. How much of each?
1One 'batch' = 1 + 1.5 + 3 = 5.5 units.
2110 ÷ 5.5 = 20 batches.
3Cement 20, sand 30, gravel 60.
✅ Answer: 20, 30 and 60 units
๐ Page 59Example 4
50 mL of purple paint in red : blue : white = 2 : 3 : 5. How much of each?
1Sum of terms = 10.
2Red = 50 × 2/10 = 10; blue = 50 × 3/10 = 15; white = 50 × 5/10 = 25
✅ Answer: Red 10 mL, blue 15 mL, white 25 mL
๐ก Tip: Part = whole × (its term ÷ sum of terms).
๐ Textbook Page 60
๐ Page 60Example 5
Construct a triangle with angles in the ratio 1 : 3 : 5.
1Sum = 9 parts = 180° → 1 part = 20°.
2Angles 20°, 60°, 100°.
3Draw AB, make 20° at A and 60° at B; the rays meet at C.
✅ Answer: 20°, 60°, 100°
๐ Page 60Figure it Out · Q1
Warm-up/cool-down : batting : bowling : fielding = 3 : 4 : 3 : 5 in a 150-minute session. Time for each?
1Sum = 15 parts → 1 part = 150 ÷ 15 = 10 min.
✅ Answer: 30 min, 40 min, 30 min, 50 min
๐ Page 60Figure it Out · Q2
Odiya : Hindi : English books = 3 : 2 : 1. There are 288 Odiya books. Hindi and English?
13 parts = 288 → 1 part = 96.
✅ Answer: Hindi 192, English 96
๐ Page 60Figure it Out · Q3
100 coins: ₹10 : ₹5 : ₹2 : ₹1 coins = 4 : 3 : 2 : 1. Total money?
1Sum = 10 parts → 1 part = 10 coins.
240 × ₹10 = 400; 30 × ₹5 = 150; 20 × ₹2 = 40; 10 × ₹1 = 10
✅ Answer: ₹600
⚠️ Common mistake: Dividing the money in 4 : 3 : 2 : 1 — the ratio is of the NUMBER of coins.
๐ Page 60Figure it Out · Q4
Construct a triangle with sides in the ratio 3 : 4 : 5. Are all such triangles congruent?
1E.g. 3 cm, 4 cm, 5 cm or 6 cm, 8 cm, 10 cm.
2All have the same shape (angles 90°, about 37°, 53°) but different sizes.
✅ Answer: No — they are similar (same shape) but not congruent unless the sizes are equal.
๐ Page 60Figure it Out · Q5
Can you construct a triangle with sides in the ratio 1 : 3 : 5?
1In any triangle, the sum of two sides must be greater than the third.
2k + 3k = 4k, which is less than 5k.
✅ Answer: No — the two shorter sides can't reach each other.
๐ Textbook Page 61
๐ Page 61Pie chart
Grades A–E: 12, 10, 8, 6, 4 students (40). Find the angles of the slices.
1Simplify 12 : 10 : 8 : 6 : 4 by HCF 2 → 6 : 5 : 4 : 3 : 2 (sum 20).
21 part = 360° ÷ 20 = 18°.
3A 108°, B 90°, C 72°, D 54°, E 36°
✅ Answer: 108°, 90°, 72°, 54°, 36°
๐ Textbook Page 62
๐ Page 62Figure it Out · Q1
360 people: summer 90, rainy 120, rest winter. Draw a pie chart.
1Winter = 360 − 210 = 150.
2Total is 360, so each person = 1°.
✅ Answer: Summer 90°, rainy 120°, winter 150°
๐ Page 62Figure it Out · Q2
Pie chart: Entertainment 50%, Sports 25%, News 15%, Information 10%.
1Angle = % × 360° ÷ 100 = % × 3.6°
250% → 180°, 25% → 90°, 15% → 54°, 10% → 36°
✅ Answer: 180°, 90°, 54°, 36°
๐ Textbook Page 63
๐ Page 63Figure it Out · Q3
Make a pie chart of your class's favourite subjects.
1Example class of 40: Language 6, Arts 4, Vocational 2, Social Science 6, PE 4, Maths 10, Science 8.
21 student = 360° ÷ 40 = 9°.
3Angles: 54°, 36°, 18°, 54°, 36°, 90°, 72° (sum 360°).
✅ Answer: Angle for a subject = (students ÷ total) × 360°. Use your own class data.
๐ Page 63Example 1
5 workers move 4500 bricks a day. How many workers for 18000 bricks a day?
1More bricks → more workers (direct).
24500 : 18000 :: 5 : x → x = 18000 × 5 ÷ 4500 = 20
✅ Answer: 20 workers
๐ Page 63Example 2
30 km/h takes 3 hours. How long at 60 km/h? Can we write 30 : 60 :: 3 : x?
1Faster → less time, so NOT direct.
2Distance = 30 × 3 = 90 km → time = 90 ÷ 60 = 1.5 h.
3Table: speed 5, 15, 30, 60 → time 18, 6, 3, 1.5. Each product is 90.
✅ Answer: 1.5 hours. No — speed and time are in inverse proportion: speed × time = 90.
๐ Textbook Page 64
๐ Page 64Check
Show x₁/x₂ = y₂/y₁ for walking (5 km/h, 18 h) and car (60 km/h, 1.5 h).
1x₁/x₂ = 5/60 = 1/12 ≈ 0.0833
2y₂/y₁ = 1.5/18 = 1/12 ≈ 0.0833
✅ Answer: Both are 1/12 — the factors are inverse.
๐ Textbook Page 65
๐ Page 65Figure it Out · Q1
Which tables show inverse proportion? (i) x 40, 80, 25, 16; y 20, 10, 32, 50 (ii) x 40, 80, 25, 16; y 20, 10, 12.5, 8 (iii) x 30, 90, 150, 10; y 15, 5, 3, 45
1Check if x × y is the same for every column.
2(i) 800, 800, 800, 800 → inverse ✔
3(ii) 800, 800, 312.5, 128 → not inverse (here y = x ÷ 2, a direct proportion)
4(iii) 450, 450, 450, 450 → inverse ✔
✅ Answer: (i) and (iii)
๐ Page 65Figure it Out · Q2
x and y are inverse: x 16, 12, __, 36; y 9, __, 48, __. Fill in.
1k = 16 × 9 = 144
2x = 12 → y = 144 ÷ 12 = 12
3y = 48 → x = 144 ÷ 48 = 3
4x = 36 → y = 144 ÷ 36 = 4
✅ Answer:
| x | 16 | 12 | 3 | 36 |
| y | 9 | 12 | 48 | 4 |
๐ Page 65Example 3
20 workers take 4 days to lay a road. How long for 10 workers?
1Inverse: 20 × 4 = 10 × y
2y = 8
✅ Answer: 8 days
๐ Textbook Page 66
๐ Page 66Example 4
2 pumps fill a tank in 18 hours. With 2 more pumps?
14 pumps: 2 × 18 = 4 × x → x = 9
✅ Answer: 9 hours
๐ Page 66Example 5
Food for 80 students lasts 15 days. 20 more join. How long now?
180 × 15 = 100 × x → x = 12
✅ Answer: 12 days
๐ Page 66Example 6
Ram cuts vegetables in 1 h, Shyam in 1.5 h. Together? Is work and time direct or inverse?
1In 1 hour: Ram does 1 job, Shyam \(\tfrac{2}{3}\) job → together \(\tfrac{5}{3}\) jobs per hour.
2Work and time (at a fixed rate) are DIRECTLY proportional: \(\tfrac{5}{3} : 1 :: 1 : x\).
3\(x = \tfrac{3}{5}\) h = 36 minutes
✅ Answer: \(\tfrac{3}{5}\) hour = 36 minutes
⚠️ Common mistake: Adding the times (2.5 h) — working together must be FASTER than either alone.
๐ Textbook Page 67
๐ Page 67Figure it Out · Q1
Which pairs are in inverse proportion? (i) taps and time to fill (ii) painters and days (iii) distance and petrol (iv) speed and time on a fixed route (v) length of cloth and price (vi) pages and reading time
1Inverse: one goes up, the other goes down by the same factor.
2(i) more taps → less time ✔ (ii) more painters → fewer days ✔ (iv) faster → less time ✔
3(iii), (v), (vi) both go up together → direct.
✅ Answer: Inverse: (i), (ii), (iv)
๐ Page 67Figure it Out · Q2
24 pencils cost ₹120. Cost of 20?
1Direct: 1 pencil = 120 ÷ 24 = ₹5
220 × 5 = 100
✅ Answer: ₹100
๐ Page 67Figure it Out · Q3
Water for 20 families lasts 6 days. 10 more families come. How long now? Assumptions?
1Inverse: 20 × 6 = 30 × x → x = 4
✅ Answer: 4 days — assuming every family uses the same amount each day and no new water is added.
๐ Page 67Figure it Out · Q4
Match each animal's sleep chart with hours from: 15, 2.5, 20, 8, 3.5, 13, 10.5, 18.
1The blue (dark) part of each ring shows the fraction of 24 hours spent sleeping.
2Smallest blue → giraffe (≈ 1/10 of the ring) → 2.5 h; elephant → 3.5 h; human (1/3) → 8 h
3Dog (a bit less than half) → 10.5 h; cat (a bit more than half) → 13 h; squirrel → 15 h
4Python (3/4) → 18 h; bat (5/6) → 20 h
✅ Answer: Giraffe 2.5, Elephant 3.5, Human 8, Dog 10.5, Cat 13, Squirrel 15, Python 18, Bat 20
๐ Textbook Page 68
๐ Page 68Figure it Out · Q5
Pie chart of transport: Walk 90°, Bus 120°, Two-wheeler 60°, Cycle 60°, Car the rest. (i) Most common? (ii) Fraction by car? (iii) 18 by car → total? Taxis? (iv) Which two are equal?
1Car = 360° − (90 + 120 + 60 + 60) = 30°.
2(i) Bus has the biggest angle.
3(ii) 30/360 = 1/12
4(iii) 1/12 of total = 18 → total 216. There is no taxi slice → 0 children use taxis.
5(iv) Cycle and two-wheeler (60° each).
✅ Answer: (i) Bus (ii) 1/12 (iii) 216 children; 0 by taxi (iv) Cycle and two-wheeler
๐ Page 68Figure it Out · Q6
3 workers paint a fence in 4 days. With 1 more worker?
13 × 4 = 4 × x → x = 3
✅ Answer: 3 days — assuming all workers work at the same speed and the work can be shared.
๐ Page 68Figure it Out · Q7
A pump fills 2 tanks in 6 hours. How long for 5 tanks?
1Direct: 1 tank = 3 h
25 × 3 = 15
✅ Answer: 15 hours
⚠️ Common mistake: Using inverse proportion — more tanks need MORE time.
๐ Page 68Figure it Out · Q8
25 rows of 12 chairs. Rearranged with 20 per row — how many rows?
1Total 25 × 12 = 300 chairs
2300 ÷ 20 = 15
✅ Answer: 15 rows
๐ Page 68Figure it Out · Q9
8 periods of 45 min. With 9 periods in the same time, how long is each?
1Total = 360 min
2360 ÷ 9 = 40
✅ Answer: 40 minutes
๐ Page 68Figure it Out · Q10
Small pump fills a tank in 3 h, large in 2 h. Together?
1Per hour: \(\tfrac{1}{3} + \tfrac{1}{2} = \tfrac{5}{6}\) tank
2Time = \(\tfrac{6}{5}\) h
✅ Answer: 1.2 hours = 1 h 12 min
๐ Page 68Figure it Out · Q11
42 machines make the toys in 63 days. How many machines for 54 days?
1Inverse: 42 × 63 = x × 54
2x = 2646 ÷ 54 = 49
✅ Answer: 49 machines
๐ Page 68Figure it Out · Q12
A car takes 2 h at 60 km/h. How long at 80 km/h?
1Distance = 120 km
2120 ÷ 80 = 1.5
✅ Answer: 1.5 hours
๐ Where do we use this?
- Maps and house plans, recipes and concrete mixes, pie charts in news and reports, planning workers, pumps and travel time.
๐ Link to higher classes
- Class 9–10: graphs of xy = k, statistics and pie charts, similar triangles, speed–time problems in science.
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