class8-Chapter 7 part 1 Proportional Reasoning-1 -solutions

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Class 8 Ganita Prakash · Chapter 7
Proportional Reasoning-1

Page-wise textbook solutions (pages 159–178) · Every step · Figures · Common mistakes · Tips

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🗝️ Key ideas of this chapter

• Ratio a : b — for every a units of the first quantity there are b units of the second. Order matters.
• Simplest form: divide both terms by their HCF. 60 : 40 = 3 : 2.
• Proportional (a : b :: c : d) — both terms change by the same multiplying factor; check with \(ad = bc\).
• Rule of Three (Trairāśika): \(d = \dfrac{b \times c}{a}\) — but only when both quantities increase together.
• Sharing x in m : n → \(m \times \dfrac{x}{m+n}\) and \(n \times \dfrac{x}{m+n}\).
• Always use the same units before comparing ratios.

📖 Textbook Page 159

📖 Page 159Think

Which images look similar and which ones look different? (Images A–E of the tiger.)

1Look at the shape, not the size. A, C and D look like the same picture made bigger or smaller.

2B looks stretched (long and thin). E looks squashed (fat).

✅ Answer: A, C and D look similar. B and E look different (distorted).

📖 Textbook Page 160

📖 Page 160Think

Why do B and E look different from A, C and D? Use the table of widths and heights.

1Table: A 60 × 40, B 40 × 20, C 30 × 20, D 90 × 60, E 60 × 60 (in mm).

2A → C: width 60 → 30 (÷ 2), height 40 → 20 (÷ 2). Same factor → similar.

3A → B: width 60 → 40 (× ⅔), height 40 → 20 (× ½). Different factors → not similar.

4Subtracting the same number (20 mm from both) does NOT keep the shape. Multiplying both by the same number does.

Gold = similar (width : height = 3 : 2). B is 2 : 1 and E is 1 : 1, so they look stretched or squashed.

✅ Answer: Images look similar when width and height change by the same multiplying factor. B and E did not.

⚠️ Common mistake: Thinking 'both became 20 mm smaller' means similar. Similarity needs the same MULTIPLYING factor, not the same difference.

📖 Page 160Check

By what factors did the width and height of image D change compared to image A? Are the factors the same?

1Width: 60 → 90, factor = 90 ÷ 60 = \(\tfrac{3}{2}\)

2Height: 40 → 60, factor = 60 ÷ 40 = \(\tfrac{3}{2}\)

✅ Answer: Both factors are \(\tfrac{3}{2}\) (1.5) — the same. So D is similar to A.

📖 Textbook Page 161

📖 Page 161Think

By what factor should we multiply the ratio 60 : 40 (image A) to get 90 : 60 (image D)?

1\(60 \times \tfrac{3}{2} = 90\) and \(40 \times \tfrac{3}{2} = 60\)

✅ Answer: By \(\tfrac{3}{2}\).

📖 Page 161Simplest form

Write the ratios of images A, B, C, D, E in simplest form.

1Divide both terms by their HCF.

2A 60 : 40 → HCF 20 → 3 : 2

3C 30 : 20 → HCF 10 → 3 : 2

4D 90 : 60 → HCF 30 → 3 : 2

5B 40 : 20 → HCF 20 → 2 : 1

6E 60 : 60 → HCF 60 → 1 : 1

✅ Answer: A, C, D = 3 : 2 (proportional); B = 2 : 1; E = 1 : 1. So 60 : 40 :: 30 : 20 :: 90 : 60.

💡 Tip: Ratio a : b means 'for every a units of the first there are b units of the second'. The order matters: 3 : 2 is not 2 : 3.

📖 Textbook Page 162

📖 Page 162Example 1

Are the ratios 3 : 4 and 72 : 96 proportional? (What is the HCF of 72 and 96?)

13 : 4 is already in simplest form.

2HCF of 72 and 96: 72 = 2³ × 3², 96 = 2⁵ × 3 → HCF = 2³ × 3 = 24.

372 ÷ 24 = 3, 96 ÷ 24 = 4 → 3 : 4

✅ Answer: Yes, 3 : 4 :: 72 : 96. (HCF = 24)

📖 Page 162Example 2

Kesang used 10 spoons of sugar for 6 glasses of lemonade. To make 18 glasses with the same sweetness, how many spoons are needed? How do we find the factor of change?

1Write the proportion 6 : 10 :: 18 : ?

2Factor of change = 18 ÷ 6 = 3.

3Multiply the second term by the same factor: 10 × 3 = 30.

	Before	After (× 3)
Glasses	6	18
Spoons of sugar	10	30

✅ Answer: 30 spoons of sugar (6 : 10 :: 18 : 30).

⚠️ Common mistake: Adding 12 to both terms (6 + 12 = 18, 10 + 12 = 22) — that changes the sweetness. Multiply, don't add.

📖 Textbook Page 163

📖 Page 163Example 3

Nitin built a 60 ft wall with 3 bags of cement; Hari built a 40 ft wall with 2 bags. Is Nitin right to worry that Hari's wall is weaker?

1Compare length : bags of cement.

2Nitin 60 : 3 = 20 : 1 (÷ 3)

3Hari 40 : 2 = 20 : 1 (÷ 2)

4Both are 20 ft per bag → proportional.

✅ Answer: No. Both walls used cement in the same ratio 20 : 1, so they are equally strong.

📖 Page 163Example 4

A school has a teacher : student ratio of 5 : 170. Count in your school and compare.

15 : 170 → divide by 5 → 1 : 34 (one teacher for every 34 students).

2Example: a school with 24 teachers and 720 students → 24 : 720 = 1 : 30.

31 : 30 is not equal to 1 : 34, so they are not proportional. (Use your own school's numbers.)

✅ Answer: Simplify your school's ratio and compare it with 1 : 34. They are proportional only if the simplest forms match.

📖 Page 163Example 5

Measure the width and height of your blackboard. Draw a rectangle in your notebook with the same ratio. Do your classmates' rectangles look the same?

1Example: blackboard 240 cm × 120 cm → 240 : 120 = 2 : 1.

2Notebook: 8 cm × 4 cm, or 6 cm × 3 cm (still 2 : 1).

3All rectangles with ratio 2 : 1 look the same shape, though sizes differ.

✅ Answer: Different sizes, same shape — because the ratios are proportional.

📖 Textbook Page 164

📖 Page 164Example 6

When Neelima was 3, her mother was 10 times her age. What is the ratio of their ages? What will it be when Neelima is 12? Does it stay the same?

1Mother's age = 10 × 3 = 30 → 3 : 30 = 1 : 10.

29 years later: Neelima 12, mother 39 → 12 : 39.

3HCF 3 → 4 : 13.

✅ Answer: 1 : 10 then 4 : 13. The ratio changes, because adding the same number of years does not keep a ratio.

💡 Tip: Adding (or subtracting) the same number to both terms usually changes the ratio.

📖 Page 164Example 7

Fill in the missing numbers so that each ratio is proportional to 14 : 21: ___ : 42, 6 : ___, 2 : ___.

1___ : 42 → 42 = 21 × 2, so first term = 14 × 2 = 28.

26 : ___ → factor = 6 ÷ 14 = \(\tfrac{3}{7}\) (a fraction!), second term = \(21 \times \tfrac{3}{7} = 9\).

32 : ___ → 14 ÷ 7 = 2, so 21 ÷ 7 = 3.

The factor need not be a whole number: 14 × 3/7 = 6, so 21 × 3/7 = 9.

✅ Answer: 28 : 42, 6 : 9, 2 : 3

📖 Page 164Filter coffee

Regular coffee is 15 mL decoction : 35 mL milk. Strong is 20 : 30, light is 10 : 40. Why is one stronger and one lighter?

1Regular 15 : 35 = 3 : 7 → decoction is 15 out of 50 = 30%.

2Strong 20 : 30 = 2 : 3 → 20 out of 50 = 40% decoction (more coffee).

3Light 10 : 40 = 1 : 4 → 10 out of 50 = 20% decoction (more milk).

✅ Answer: More decoction per mL of milk → stronger; less → lighter.

📖 Textbook Page 165

📖 Page 165Table

Mark each coffee as Regular (3 : 7), Strong or Light: 300 : 600, 150 : 500, 200 : 400, 24 : 56, 100 : 300.

1Compare each with 3 : 7 by finding decoction as a fraction of milk: regular = 3/7 ≈ 0.43.

2300 : 600 = 1 : 2 = 0.5 → more than 0.43 → Strong

3150 : 500 = 3 : 10 = 0.3 → Light

4200 : 400 = 1 : 2 = 0.5 → Strong

524 : 56 = 3 : 7 → Regular

6100 : 300 = 1 : 3 ≈ 0.33 → Light

✅ Answer:

Decoction (mL)	Milk (mL)	Type
300	600	Strong
150	500	Light
200	400	Strong
24	56	Regular
100	300	Light

📖 Page 165Figure it Out · Q1

Circle the true proportions: (i) 4 : 7 :: 12 : 21 (ii) 8 : 3 :: 24 : 6 (iii) 7 : 12 :: 12 : 7 (iv) 21 : 6 :: 35 : 10 (v) 12 : 18 :: 28 : 12 (vi) 24 : 8 :: 9 : 3

1Use cross multiplication: a : b :: c : d is true only if a × d = b × c.

2(i) 4 × 21 = 84, 7 × 12 = 84 ✔

3(ii) 8 × 6 = 48, 3 × 24 = 72 ✘

4(iii) 7 × 7 = 49, 12 × 12 = 144 ✘

5(iv) 21 × 10 = 210, 6 × 35 = 210 ✔

6(v) 12 × 12 = 144, 18 × 28 = 504 ✘

7(vi) 24 × 3 = 72, 8 × 9 = 72 ✔

✅ Answer: True: (i), (iv) and (vi).

💡 Tip: Simplest forms also work: (iv) both are 7 : 2; (vi) both are 3 : 1.

📖 Page 165Figure it Out · Q2

Give 3 ratios that are proportional to 4 : 9.

1Multiply both terms by the same number: × 2, × 3, × 4.

✅ Answer: 8 : 18, 12 : 27, 16 : 36 (many more answers possible, e.g. 40 : 90).

📖 Page 165Figure it Out · Q3

Fill in the missing numbers proportional to 18 : 24: 3 : __, 12 : __, 20 : __, 27 : __.

118 : 24 = 3 : 4 in simplest form (HCF 6). So the second term is always \(\tfrac{4}{3}\) of the first.

23 → 4

312 → 16

420 → \(20 \times \tfrac{4}{3} = \tfrac{80}{3} = 26\tfrac{2}{3}\)

527 → 36

✅ Answer: 3 : 4, 12 : 16, 20 : \(26\tfrac{2}{3}\), 27 : 36

⚠️ Common mistake: Expecting every answer to be a whole number — 20 : 80/3 is correct.

📖 Textbook Page 166

📖 Page 166Figure it Out · Q4

Which rectangles (A, B, C, D, E) are similar? Measure and compare their ratios.

1Measure the short side and long side of each rectangle with a scale (turn the book for the slanted ones).

2Approximate ratios (short : long): A ≈ 1 : 3, B ≈ 2 : 3, C ≈ 2 : 5, D ≈ 1 : 3, E ≈ 2 : 5.

3Same ratio → similar.

Measured: A ≈ 1 : 3, D ≈ 1 : 3 (gold, similar); C ≈ 2 : 5, E ≈ 2 : 5 (green, similar); B ≈ 3 : 2 (different).

✅ Answer: A and D are similar (about 1 : 3). C and E are similar (about 2 : 5). B is not similar to any other.

💡 Tip: A rectangle that is turned (rotated) is still the same shape. Always compare short side : long side.

📖 Page 166Figure it Out · Q5

Draw a smaller and a bigger rectangle with the same width : height ratio as the given rectangle. Are your classmates' rectangles the same?

1Measure the given rectangle, e.g. 6 cm × 4 cm → 3 : 2.

2Smaller: divide both by 2 → 3 cm × 2 cm. Bigger: multiply both by 1.5 → 9 cm × 6 cm.

3Classmates may choose other factors (e.g. 4.5 cm × 3 cm, 12 cm × 8 cm).

All three have width : height = 3 : 2, so they look the same shape. (Example sizes — measure your own book's rectangle.)

✅ Answer: Sizes may differ, but they are NOT wrong as long as width : height stays 3 : 2 — they will all look the same shape.

📖 Page 166Figure it Out · Q6

What is the ratio of grey bricks to coloured bricks in walls (a) and (b)? Give the simplest form.

1Find ONE repeating unit of the pattern and count only inside it.

2(a) Unit is 5 bricks wide, 3 rows: red bricks 3 + 2 + 1 = 6, grey 15 − 6 = 9 → 9 : 6 = 3 : 2

3(b) Unit is 4 bricks wide, 7 rows: orange 1 + 2 + 2 + 2 + 2 + 2 + 1 = 12, grey 28 − 12 = 16 → 16 : 12 = 4 : 3

The first repeating unit of each wall has a dark outline. Count inside ONE unit.

✅ Answer: (a) 3 : 2 (b) 4 : 3

⚠️ Common mistake: Counting the whole picture — the edges are cut off. Count one complete repeating unit.

📖 Textbook Page 167

📖 Page 167Figure it Out · Q7

Measure a friend's head, torso, arms and legs. Write head : torso, torso : arms, torso : legs. Draw a figure with the same ratios. Does it look more realistic?

1Example measurements: head 22 cm, torso 55 cm, arm 66 cm, leg 88 cm.

2head : torso = 22 : 55 = 2 : 5

3torso : arms = 55 : 66 = 5 : 6

4torso : legs = 55 : 88 = 5 : 8

5Drawing: if torso = 5 cm, head = 2 cm, arms = 6 cm, legs = 8 cm.

✅ Answer: Answers depend on your measurements. Yes — drawings with proportional body parts look much more realistic.

📖 Page 167Example 8

120 students need 15 kg of rice. Only 80 came. How much rice should be cooked? What is the factor of change?

1120 : 15 :: 80 : ?

2Factor = \(\tfrac{80}{120} = \tfrac{2}{3}\)

3\(15 \times \tfrac{2}{3} = 10\)

✅ Answer: 10 kg of rice (factor ⅔).

📖 Textbook Page 168

📖 Page 168Cross multiplication

Show that if a : b :: c : d then ad = bc. How did Āryabhaṭa state this (Rule of Three)?

1Same factor f: c = fa and d = fb, so \(f = \tfrac{c}{a} = \tfrac{d}{b}\).

2Multiply both sides by ab: bc = ad.

3Āryabhaṭa: pramāṇa (a) : phala (b) :: ichchhā (c) : ichchhāphala (d).

4'Multiply the phala by the ichchhā and divide by the pramāṇa': \(d = \tfrac{b \times c}{a}\).

✅ Answer: ad = bc, so the unknown fourth term is \(d = \tfrac{bc}{a}\).

💡 Tip: Product of the outer terms = product of the inner terms.

📖 Textbook Page 169

📖 Page 169Example 9

A car travels 90 km in 150 minutes. At the same speed, how far in 4 hours? Is 150 : 90 :: 4 : ? the right way?

1No — the units differ (minutes and hours). 4 hours = 240 minutes.

2150 : 90 :: 240 : x

3150x = 240 × 90 → \(x = \tfrac{21600}{150} = 144\)

✅ Answer: 144 km

⚠️ Common mistake: Mixing minutes and hours in the same proportion. Convert to the same unit first.

📖 Textbook Page 170

📖 Page 170Example 10

Himachal: 200 g of tea for ₹200. Meghalaya: 1 kg for ₹800. Are weight : price proportional? Which tea is more expensive?

1Same units: Himachal 200 : 200 = 1 : 1; Meghalaya 1000 : 800 = 5 : 4 → not proportional.

2Price of 1 kg in Himachal: 200 g is ⅕ kg, so \(\tfrac{1}{5}x = 200 \Rightarrow x = 1000\).

3Himachal ₹1000 per kg; Meghalaya ₹800 per kg.

✅ Answer: Not proportional. Himachal tea is more expensive (₹1000 vs ₹800 per kg).

📖 Page 170Activity 1

Find the ingredients of your favourite dish for your family (say 4 people). Find the quantities for 15 guests.

1Example (4 people): 2 cups rice, 1 cup dal, 4 cups water.

2Factor = 15 ÷ 4 = 3.75.

3Rice 2 × 3.75 = 7.5 cups, dal 3.75 cups, water 15 cups.

✅ Answer: Multiply every ingredient by (guests ÷ family size).

📖 Page 170Figure it Out · Q1

The Earth travels about 940 million km around the Sun in a year. How far in a week?

1940 million = 94,00,00,000 km. One year ≈ 52 weeks.

2940000000 : 52 :: x : 1 → \(x = \tfrac{940000000}{52} \approx 18076923\)

3(Using 365 days: \(940000000 \times \tfrac{7}{365} \approx 18027397\) — also about 18 million.)

✅ Answer: About 1.8 crore km (≈ 18 million km) in a week.

📖 Textbook Page 171

📖 Page 171Figure it Out · Q2

A mason needs about 1450 bricks for a 10 ft wall. How many bricks for the whole house (outer walls + inner wall)?

1Big rectangle: (9 + 15) ft by 12 ft → perimeter 2(24 + 12) = 72 ft.

2Inner wall between rooms: 12 ft.

3Small room below: 6 + 9 + 9 = 24 ft (its top side is already part of the big rectangle).

4Total wall = 72 + 12 + 24 = 108 ft.

510 : 1450 :: 108 : x → \(x = \tfrac{1450 \times 108}{10} = 15660\)

Outer walls 72 + 24 = 96 ft (navy) and inner wall 12 ft (red) → 108 ft of wall.

✅ Answer: About 15,660 bricks.

⚠️ Common mistake: Counting the wall shared by the small room and the big room twice.

📖 Page 171Think

Puneeth's father takes 2 hours at 50 km/h. How long at 75 km/h? Can we write 50 : 2 :: 75 : __?

1Distance = 50 × 2 = 100 km.

2At 75 km/h: time = 100 ÷ 75 = \(1\tfrac{1}{3}\) h = 1 h 20 min.

3Faster speed → LESS time. 50 : 2 :: 75 : 3 would give MORE time, which is wrong.

✅ Answer: 1 h 20 min. It cannot be solved by the Rule of Three (speed and time move in opposite directions).

⚠️ Common mistake: Using the Rule of Three when one quantity increases while the other decreases.

📖 Textbook Page 172

📖 Page 172Activity 2

Sachet 6 mL ₹2, small 180 mL ₹154, medium 340 mL ₹276, large 1000 mL ₹540. Is price proportional to volume?

1Volumes 6 : 180 = 1 : 30. Prices 2 : 154 = 1 : 77 → not proportional.

2Price per mL: sachet ₹0.33, small ₹0.86, medium ₹0.81, large ₹0.54.

3Packaging, transport and brand costs are not proportional to volume.

✅ Answer: Not proportional. The large bottle is cheaper per mL than the small and medium bottles; sachets are cheapest per mL but create the most plastic waste. A big bottle (or refills) is better for the planet.

📖 Page 172Activity 3

Share 12 counters: equally; if your partner gets 5; in the ratio 3 : 1.

1Equally: 6 : 6 = 1 : 1.

2Partner 5, you 7 → 5 : 7.

33 : 1 → take 3 and 1 again and again: 3 rounds → 9 and 3.

✅ Answer: 1 : 1; 5 : 7; 9 and 3.

📖 Textbook Page 173

📖 Page 173Think

Share 42 counters in the ratio 4 : 3. What is the size of each group?

1Total groups = 4 + 3 = 7.

2Size of each group = 42 ÷ 7 = 6.

3Partner: 4 × 6 = 24; you: 3 × 6 = 18.

42 split into 7 equal groups of 6: 4 groups and 3 groups.

✅ Answer: 24 and 18 (groups of 6).

General rule: x in the ratio m : n gives \(m \times \tfrac{x}{m+n}\) and \(n \times \tfrac{x}{m+n}\).

📖 Textbook Page 174

📖 Page 174Example 11

Prashanti invested ₹75,000 and Bhuvan ₹25,000. Share a profit of ₹4,000 in the ratio of investment.

175000 : 25000 = 3 : 1

2Groups = 4; 4000 ÷ 4 = 1000

3Prashanti 3 × 1000, Bhuvan 1 × 1000

✅ Answer: Prashanti ₹3,000, Bhuvan ₹1,000

📖 Page 174Example 12

40 kg of sand and cement are in the ratio 3 : 1. How much cement must be added to make it 5 : 2?

1Sand = \(\tfrac{3}{4} \times 40 = 30\) kg, cement = 10 kg.

2Sand stays 30 kg. 5 : 2 :: 30 : ? → \(\tfrac{2}{5} \times 30 = 12\) kg cement needed.

3Add 12 − 10 = 2 kg.

✅ Answer: Add 2 kg of cement.

📖 Textbook Page 175

📖 Page 175Figure it Out · Q1

Divide ₹4,500 in the ratio 2 : 3.

1Groups = 5; 4500 ÷ 5 = 900

22 × 900 = 1800; 3 × 900 = 2700

5 groups of ₹900.

✅ Answer: ₹1,800 and ₹2,700

📖 Page 175Figure it Out · Q2

Acid : water = 1 : 5. How much of each is in 240 mL of solution?

1Groups = 6; 240 ÷ 6 = 40

2Acid 1 × 40, water 5 × 40

✅ Answer: Acid 40 mL, water 200 mL

📖 Page 175Figure it Out · Q3

Blue : yellow = 3 : 5 for green paint. How much of each for 40 mL? If 20 mL more yellow is added, what is the new ratio?

1Groups = 8; 40 ÷ 8 = 5 → blue 15 mL, yellow 25 mL.

2New yellow = 25 + 20 = 45 mL.

315 : 45 = 1 : 3

✅ Answer: Blue 15 mL, yellow 25 mL; new ratio 1 : 3.

📖 Page 175Figure it Out · Q4

Rice : urad dal = 2 : 1 for idlis. How many cups of each for 6 cups of mixture?

1Groups = 3; 6 ÷ 3 = 2

2Rice 2 × 2, dal 1 × 2

✅ Answer: Rice 4 cups, urad dal 2 cups

📖 Page 175Figure it Out · Q5

A bucket of orange paint has red : yellow = 3 : 5. One more (same-sized) bucket of yellow is added. What is the new red : yellow ratio?

1Think of one bucket as 8 parts: red 3, yellow 5.

2Another full bucket of yellow = 8 more parts of yellow.

3Red 3, yellow 5 + 8 = 13.

✅ Answer: 3 : 13

💡 Tip: We assume both buckets hold the same amount of paint.

📖 Textbook Page 176

📖 Page 176Unit conversions

Check: 25 °C is 77 °F. Also convert 100 °F to °C.

1°F = \(\tfrac{9}{5}\) × °C + 32 = \(\tfrac{9}{5} \times 25 + 32 = 45 + 32 = 77\) ✔

2°C = \(\tfrac{5}{9}\)(°F − 32) = \(\tfrac{5}{9} \times 68 \approx 37.8\)

✅ Answer: 25 °C = 77 °F; 100 °F ≈ 37.8 °C.

Unit	Equals
1 metre	3.281 feet
1 square metre	10.764 square feet
1 acre	43,560 square feet
1 hectare	10,000 m² = 2.471 acres
1 mL	1 cc
1 litre	1,000 mL

📖 Page 176Figure it Out · Q1

Anagh mixes 600 mL orange juice with 900 mL apple juice. Ratio of orange to apple in simplest form?

1HCF of 600 and 900 is 300.

2600 ÷ 300 = 2, 900 ÷ 300 = 3

✅ Answer: 2 : 3

📖 Page 176Figure it Out · Q2

Last year 3 full buses carried 162 people. This year there are 204. How many buses? Will all be full?

1One bus holds 162 ÷ 3 = 54.

2204 ÷ 54 = 3 remainder 42 → need 4 buses.

34 buses hold 216 → 216 − 204 = 12 empty seats.

✅ Answer: 4 buses; not all full — 12 seats will be empty (the 4th bus has 42 people).

📖 Page 176Figure it Out · Q3

Delhi: 1,484 sq km, 30 million people. Mumbai: 550 sq km, 20 million. Which is more crowded?

1Compare people per sq km (same unit).

2Delhi: 30000000 ÷ 1484 ≈ 20,216

3Mumbai: 20000000 ÷ 550 ≈ 36,364

✅ Answer: Mumbai — about 36,364 people per sq km against Delhi's 20,216.

⚠️ Common mistake: Saying Delhi because it has more people. Crowding means people per unit area.

📖 Page 176Figure it Out · Q4

A crane's neck : rest of body = 4 : 6. If your body had the same ratio, how tall would your neck be?

1Groups = 4 + 6 = 10, so neck = \(\tfrac{4}{10}\) of total height.

2Crane: \(\tfrac{4}{10} \times 155 = 62\) cm.

3If you are 150 cm: neck = 0.4 × 150 = 60 cm!

✅ Answer: Neck = \(\tfrac{4}{10}\) × your height (e.g. 60 cm for 150 cm).

📖 Page 176Figure it Out · Q5 (Līlāvatī)

'If \(2\tfrac{1}{2}\) palas of saffron cost \(\tfrac{3}{7}\) niskas, how much saffron for 9 niskas?'

1niskas : palas → \(\tfrac{3}{7} : \tfrac{5}{2} :: 9 : x\)

2\(x = \dfrac{\tfrac{5}{2} \times 9}{\tfrac{3}{7}} = \tfrac{45}{2} \times \tfrac{7}{3} = \tfrac{105}{2}\)

✅ Answer: \(52\tfrac{1}{2}\) palas

📖 Textbook Page 177

📖 Page 177Figure it Out · Q6

Harmain is 1, her brother is 5. When will the ratio of their ages be 1 : 2?

1Difference is always 4 years.

2In 1 : 2, the difference is 1 part = 4 years.

3Harmain = 4, brother = 8 (check 4 : 8 = 1 : 2).

✅ Answer: When Harmain is 4 years old (3 years from now).

📖 Page 177Figure it Out · Q7

Equal volumes of gold and water have masses in the ratio 37 : 2. 1 L of water is 1 kg. Mass of 1 L of gold?

137 : 2 :: x : 1

2\(x = \tfrac{37}{2} = 18.5\)

✅ Answer: 18.5 kg

📖 Page 177Figure it Out · Q8

10 tonnes of manure per acre. How much for a 200 ft × 500 ft plot?

1Area = 100,000 sq ft.

2In acres: 100000 ÷ 43560 ≈ 2.296 acres.

3Manure = 10 × 2.296 ≈ 22.96 t

✅ Answer: About 23 tonnes (≈ 22.96 t).

📖 Page 177Figure it Out · Q9

A tap fills a 500 mL mug in 15 s. How long for a 10 L bucket?

110 L = 10,000 mL = 20 mugs.

220 × 15 = 300 s

✅ Answer: 300 seconds = 5 minutes

⚠️ Common mistake: Forgetting to change litres to mL.

📖 Page 177Figure it Out · Q10

1 acre costs ₹15,00,000. What is the cost of 2,400 sq ft?

143560 : 1500000 :: 2400 : x

2\(x = \tfrac{1500000 \times 2400}{43560} \approx 82645\)

✅ Answer: About ₹82,645

📖 Page 177Figure it Out · Q11

Oxen take 6 h per acre; a tractor is 4 times faster. Time for a 20-acre field with oxen? With a tractor?

1Oxen: 6 × 20 = 120 hours.

2Tractor is 4 times faster → takes ¼ of the time: 120 ÷ 4 = 30 hours.

✅ Answer: Oxen 120 h; tractor 30 h

⚠️ Common mistake: Multiplying by 4 — faster means LESS time.

📖 Page 177Figure it Out · Q12

A ₹10 coin (7.74 g) is copper : nickel = 3 : 1. Copper ₹906/kg, nickel ₹1,341/kg. Cost of the metals?

1Copper = \(\tfrac{3}{4} \times 7.74 = 5.805\) g; nickel = 1.935 g.

2Copper cost = 906 × 5.805 ÷ 1000 ≈ ₹5.26

3Nickel cost = 1341 × 1.935 ÷ 1000 ≈ ₹2.59

4Total ≈ ₹7.85

✅ Answer: About ₹7.85 (less than ₹10!)

📖 Textbook Page 178

📖 Page 178Puzzle: Binairo 1

Solve Binairo puzzle 1: each row and column has 3 horizontal and 3 vertical lines, no more than two of the same next to each other, and every row (and column) is different.

1Look for two equal lines side by side → the cells on both ends must be the other symbol.

2Look for a gap between two equal lines (H _ H) → the middle must be the other symbol.

3When a row or column already has 3 of one symbol, fill the rest with the other.

4Finally check that no two rows or columns are the same.

Puzzle 1: bright cells were given; thin lines are the cells we filled.

✅ Answer: See the solution grid.

📖 Page 178Puzzle: Binairo 2

Solve Binairo puzzle 2: each row and column has 3 horizontal and 3 vertical lines, no more than two of the same next to each other, and every row (and column) is different.

1Look for two equal lines side by side → the cells on both ends must be the other symbol.

2Look for a gap between two equal lines (H _ H) → the middle must be the other symbol.

3When a row or column already has 3 of one symbol, fill the rest with the other.

4Finally check that no two rows or columns are the same.

Puzzle 2: bright cells were given; thin lines are the cells we filled.

✅ Answer: See the solution grid.

📖 Page 178Puzzle: Binairo 3

Solve Binairo puzzle 3: each row and column has 3 horizontal and 3 vertical lines, no more than two of the same next to each other, and every row (and column) is different.

1Look for two equal lines side by side → the cells on both ends must be the other symbol.

2Look for a gap between two equal lines (H _ H) → the middle must be the other symbol.

3When a row or column already has 3 of one symbol, fill the rest with the other.

4Finally check that no two rows or columns are the same.

Puzzle 3: bright cells were given; thin lines are the cells we filled.

✅ Answer: See the solution grid.

🌍 Where do we use this?

• Cooking for more people, mixing paint, cement and medicines, maps and scale drawings, sharing profits, comparing prices.

🚀 Link to higher classes

• Part 2: Proportional Reasoning-2 (inverse proportion). Class 9–10: similar triangles, trigonometry. Science: speed, density, concentration.

⬆ Top 🏠 Class 8 Hub

🔑 keytoenjoylearningmaths.blogspot.com · Solutions written in our own words, based on NCERT Ganita Prakash Class 8 (Part 1)