Proportional Reasoning-1 — Mastery

Q1. The chapter opens with five photographs of a tiger labelled A, B, C, D and E. Why does it ask students to look at them before defining a ratio?

• A. To practise measuring with a ruler.
• B. To compare the colours of the photographs.
• C. To notice that A, C, D look similar but B and E look distorted, and to ask why.
• D. To count the legs of the tiger in each picture.

Q2. A Class 8 teacher must introduce 'proportional reasoning'. Four options are on the table. Which is the BEST first activity, given the chapter's design?

• A. Write the formula ad = bc on the board and ask students to memorise it.
• B. Drill students on HCF of pairs of numbers for ten minutes.
• C. Start with the Fahrenheit-to-Celsius conversion formula.
• D. Show printed images of the same picture in different sizes and ask which look similar and which look distorted, then ask 'why?'.

Q3. In a brick pattern, there are 24 grey bricks and 36 coloured bricks. Sara writes the ratio of grey to coloured bricks in simplest form. She gets

• A. 24 : 36
• B. 12 : 18
• C. 2 : 3
• D. 3 : 2

Q4. Image E (60 mm × 60 mm) looks square — a 'compressed' tiger. Comparing E to A (60 mm × 40 mm), which statement BEST explains why E looks distorted?

• A. Width and height are equal in E but unequal in A, so the simplest forms 1 : 1 and 3 : 2 differ.
• B. E has been printed on smaller paper than A.
• C. Width of E is greater than width of A.
• D. Height of E is the same as the width of A.

Q5. Aman is checking whether 12 : 18 :: 28 : 42 is true. He reduces both ratios to their simplest forms. Which conclusion is correct?

• A. 12 : 18 = 2 : 3 and 28 : 42 = 2 : 3, so the proportion is true.
• B. 12 : 18 = 2 : 3 and 28 : 42 = 4 : 6, so the proportion is false.
• C. 12 : 18 = 6 : 9 and 28 : 42 = 14 : 21, so the proportion is false.
• D. 12 : 18 = 3 : 2 and 28 : 42 = 3 : 2, so the proportion is true.

Q6. Are the ratios 24 : 8 and 9 : 3 proportional? Choose the correct reasoning.

• A. No, because 24 ≠ 9 and 8 ≠ 3.
• B. Yes, because both reduce to 3 : 1.
• C. Yes, because 24 × 3 = 9 × 8 = 72, so ad = bc.
• D. Both B and C are correct ways to see it.

Q7. Kesang's recipe uses 6 glasses : 10 spoons of sugar. She now wants to keep the same sweetness for 9 glasses. Which equation correctly uses proportional reasoning?

• A. 6 + 9 = 10 + s, so s = 5 spoons.
• B. 6 : 10 :: 9 : s, giving s = (10 × 9) ÷ 6 = 15 spoons.
• C. 6 × 9 = 10 × s, so s = 5.4 spoons.
• D. 10 − 6 = s − 9, so s = 13 spoons.

Q8. Nitin's wall: 60 ft using 3 bags of cement. Hari's wall: 40 ft using 2 bags. Why does the chapter say the two walls are equally strong, even though their lengths and bag counts are different?

• A. Because 60 and 40 share a common factor of 20.
• B. Because the ratio of length to bags simplifies to 20 : 1 in both cases, so the cement-per-foot is the same.
• C. Because 2 bags can build 40 ft of any wall.
• D. Because the chapter assumes all cement bags are the same size.

Q9. Anagh's drink mixes orange juice and apple juice in the ratio 2 : 3 (simplest form of 600 : 900 mL). He now has 1.2 L of orange juice and wants to keep the same ratio. Using the Rule of Three, how much apple juice does he need?

• A. 0.8 L
• B. 1.5 L
• C. 1.8 L
• D. 2.4 L

Q10. Harmain is 1 year old and her brother is 5. After how many years will the ratio of their ages become 1 : 2?

• A. 1 year — Harmain 2, brother 6, ratio 2 : 6 = 1 : 3.
• B. 2 years — Harmain 3, brother 7, ratio 3 : 7.
• C. 3 years — Harmain 4, brother 8, ratio 4 : 8 = 1 : 2.
• D. 4 years — Harmain 5, brother 9, ratio 5 : 9.

Q11. In a science lab, acid and water are mixed in the ratio 1 : 5 to make a solution. A bottle holds 240 mL of the solution. How much acid and how much water does it contain?

• A. 40 mL acid, 200 mL water
• B. 48 mL acid, 192 mL water
• C. 60 mL acid, 180 mL water
• D. 120 mL acid, 120 mL water

Q12. Green paint is made by mixing blue and yellow in the ratio 3 : 5. After making 40 mL of green paint, the artist adds 20 mL more yellow to lighten the colour. What is the new ratio of blue to yellow?

• A. 3 : 5
• B. 15 : 25
• C. 3 : 9, i.e., 1 : 3
• D. 3 : 10

Q13. Last year 3 buses carried 162 students with every seat full. This year 204 students must go. How many full buses are needed?

• A. 3 buses
• B. 4 buses, with some empty seats
• C. 5 buses
• D. Exactly 3.78 buses

Q14. Activity 2 lists shampoo prices: sachet 6 mL ₹2, small bottle 180 mL ₹154, medium bottle 340 mL ₹276. Hema wants to check whether the volume-to-price ratios of sachet and medium bottle are proportional. Which conclusion is correct?

• A. Both simplify to 3 : 1, so they are proportional.
• B. Sachet is 3 : 1; medium bottle is 85 : 69; they are not proportional.
• C. Sachet is 1 : 3; medium bottle is 1 : 3; they are proportional.
• D. They are proportional because both products are from the same company.

Q15. A teacher draws the table from — A: 60×40, B: 80×60, C: 30×20, D: 90×60, E: 60×60 — and asks Class 8: 'Which images are proportional to A?' Four students answer. Whose reasoning is the SOUNDEST?

• A. Riya: B, because 80 − 60 = 20 and 60 − 40 = 20, same difference.
• B. Sahil: only C, because C is exactly half of A.
• C. Meera: C and D, because both reduce to 3 : 2 — the same simplest form as A.
• D. Vivek: all four, because they all show a tiger.

Q16. To make soft idlis, rice and urad dal are mixed in the ratio 2 : 1. To make 6 cups of this mixture, how many cups of rice and how many of urad dal are needed?

• A. 4 cups rice, 2 cups urad dal
• B. 3 cups rice, 3 cups urad dal
• C. 2 cups rice, 4 cups urad dal
• D. 5 cups rice, 1 cup urad dal

Q17. A bucket of orange paint was made by mixing red and yellow paint in the ratio 3 : 5. Another bucket of pure yellow paint of the same size is added. What is the new ratio of red to yellow in the mixture?

• A. 3 : 5
• B. 3 : 10
• C. 3 : 13
• D. 3 : 6

Q18. The Earth travels approximately 940 million kilometres around the Sun in a year. Using proportional reasoning, about how many million kilometres does it travel in one week?

• A. About 18 million km
• B. About 78 million km
• C. About 134 million km
• D. About 2.6 million km

Q19. A mason uses about 1,450 bricks to build a 10-foot wall. Assuming all walls have the same height and thickness, about how many bricks are needed for a 25-foot wall?

• A. 2,900 bricks
• B. 3,625 bricks
• C. 4,350 bricks
• D. 1,450 bricks

Q20. Delhi has an area of 1,484 sq. km and a population of about 3 crore. Mumbai has an area of 550 sq. km and a population of about 2 crore. Which city is more crowded and why?

• A. Delhi, because its population is larger.
• B. Mumbai, because the people-to-area ratio (≈ 36,000 per sq. km) is greater than Delhi's (≈ 20,000 per sq. km).
• C. Both are equally crowded because area and population are both larger in Delhi.
• D. Delhi, because its area is larger.

Q21. A crane of total height 155 cm has its neck and the rest of its body in the ratio 4 : 6. If a student of total height 150 cm had her neck and the rest of her body in the same 4 : 6 ratio, how long would her neck be?

• A. 40 cm
• B. 60 cm
• C. 62 cm
• D. 100 cm

Q22. A classroom is 8 metres long. Using the conversion 1 metre = 3.281 feet the length of the room in feet is closest to

• A. 21.6 feet
• B. 26.2 feet
• C. 32.8 feet
• D. 11.3 feet

Q23. Using the formula Fahrenheit = (9/5) × Celsius + 32 from, the normal human body temperature of 37°C in degrees Fahrenheit is

• A. 66.6°F
• B. 98.6°F
• C. 94.5°F
• D. 100°F

Q24. 1 hectare = 2.471 acres. A farmer owns 5 hectares of land. About how many acres does he own?

• A. About 12.4 acres
• B. About 7.5 acres
• C. About 2.0 acres
• D. About 24.7 acres

Q25. Shampoo table shows: sachet 6 mL ₹2 and large bottle 1000 mL ₹540. Which conclusion correctly compares the per-millilitre cost?

• A. Sachet costs about ₹0.33 per mL; large bottle about ₹0.54 per mL, so the sachet is cheaper per mL.
• B. Sachet costs about ₹0.33 per mL; large bottle about ₹0.54 per mL, so the large bottle is cheaper per mL.
• C. Both cost the same per mL because both are shampoos of the same brand.
• D. The chapter does not allow this comparison.

Q26. A teacher is planning the chapter. Which sequence of teaching steps is most consistent with how the idea of a ratio and proportion is built up?

• A. Define ratio symbolically → state proportion symbol → drill HCF → solve missing-term problems.
• B. Show tiger images and ask 'which look similar?' → measure width and height → write ratios → reduce to simplest form → introduce '::' for equal simplest forms.
• C. Give the Aryabhata Trairasika rule → derive ad = bc → apply to lemonade and walls.
• D. Show the Fahrenheit-Celsius formula → derive proportions backwards → then explain ratios.

Q27. A Class 8 student writes: 'The ratio 3 : 5 is the same as 8 : 10 because I added 5 to both terms.' What is the most precise diagnosis of her error?

• A. She confused 'adding the same number to both terms' with 'multiplying both terms by the same factor', which is what preserves a ratio.
• B. She did not know how to find the HCF of 3 and 5.
• C. She forgot that ratios cannot use single-digit numbers.
• D. She wrote the second ratio in the wrong order.

Q28. Example 10 concludes that Himachal tea (200 g at ₹200) is more expensive than Meghalaya tea (1 kg at ₹800). Which line of reasoning BEST supports this conclusion using proportional reasoning?

• A. Himachal's price is ₹200; Meghalaya's is ₹800. Since ₹800 > ₹200, Meghalaya is more expensive.
• B. 1 kg of Himachal tea would cost 5 × ₹200 = ₹1,000, but 1 kg of Meghalaya tea costs only ₹800 — so per kilogram Himachal is more expensive.
• C. Both teas cost the same per gram because the simplest forms are 1 : 1 and 5 : 4.
• D. The 200-gram packet itself costs less, so Himachal is cheaper overall.

Q29. Using cross multiplication, which check confirms that image D (90 × 60) is proportional to image A (60 × 40)?

• A. 60 + 60 = 90 + 40
• B. 60 × 40 = 90 × 60
• C. 60 × 60 = 90 × 40, i.e., 3600 = 3600
• D. 60 ÷ 60 = 90 ÷ 40

Q30. Which one of the following everyday situations CANNOT be modelled as a direct proportion of the form a : b :: c : d?

• A. Sugar needed to keep the same sweetness in 6, 12, or 18 glasses of lemonade.
• B. The number of bricks required to build walls of different lengths (same height and thickness).
• C. The time taken to cover a fixed distance when the speed is doubled.
• D. Rice needed to feed 120, 80 or 240 students at the same per-head ration.