CBSE Class 8 Mathematics Proportional Reasoning 1 Very Short Answer Questions with Answers

Very Short Answer Questions

Question 1. What are the two numbers of a ratio a : b called?
Answer:
They are called the terms of the ratio; a is the first term and b is the second term.

Teacher's Note:
1. Learn the word "terms" for the two numbers of a ratio.
2. Order matters: a : b is not the same as b : a.

Question 2. Write the ratio 45 : 30 in its simplest form.
Answer:
HCF of 45 and 30 is 15. 45 ÷ 15 = 3 and 30 ÷ 15 = 2, so 45 : 30 = 3 : 2.

Teacher's Note:
1. Divide both terms by the HCF.
2. Check that 3 and 2 have no common factor.

Question 3. Which symbol is written between two ratios to show that they are proportional?
Answer:
The double colon ::, as in a : b :: c : d, read as "a : b is proportional to c : d".

Teacher's Note:
1. One colon inside a ratio, two colons between ratios.
2. The = sign between two fractions says the same thing.

Question 4. A photograph is 80 mm wide and 50 mm high. Write the ratio of its width to its height in the simplest form.
Answer:
Width : height = 80 : 50. Dividing both terms by the HCF 10 gives 8 : 5.

Teacher's Note:
1. Write width first because the question asks for width to height.
2. Both terms are in mm, so no unit change is needed.

Question 5. State the condition under which an enlarged image still looks similar to the original image.
Answer:
The image looks similar only when its width and height are both multiplied by the same factor, so that the width-to-height ratio stays the same.

Teacher's Note:
1. Same factor, same shape.
2. Adding or subtracting the same amount distorts the picture.

Question 6. Write any one ratio that is proportional to 5 : 8.
Answer:
Multiplying both terms by 2 gives 10 : 16, which is proportional to 5 : 8.
(Others such as 15 : 24 or 20 : 32 are also correct.)

Teacher's Note:
1. Multiply both terms by the same number to get an equivalent ratio.
2. Many answers are correct here.

Question 7. How many millilitres are there in 3.5 litres?
Answer:
1 litre = 1,000 mL, so 3.5 L = 3.5 × 1000 = 3,500 mL.

Teacher's Note:
1. Multiply by 1,000 to change litres to millilitres.
2. Moving the decimal point 3 places right does the same job.

Question 8. How many square metres make one hectare?
Answer:
One hectare = 10,000 square metres.

Teacher's Note:
1. Picture a square field of side 100 m.
2. 1 hectare is about 2.471 acres.

Question 9. Divide 35 marbles between two friends in the ratio 4 : 3.
Answer:
Groups = 4 + 3 = 7; each group = 35 ÷ 7 = 5 marbles.
Shares: 4 × 5 = 20 marbles and 3 × 5 = 15 marbles.

Teacher's Note:
1. Add the terms to find the number of groups.
2. Check: 20 + 15 = 35.

Question 10. Write the number of square feet in one acre.
Answer:
One acre = 43,560 square feet.

Teacher's Note:
1. This is a standard conversion used in land records.
2. Do not confuse it with 10.764 ft² in a square metre.

Question 11. If a : b :: c : d, write the relation obtained by cross multiplying the terms.
Answer:
ad = bc, that is, the product of the extreme terms equals the product of the middle terms.

Teacher's Note:
1. Extremes: a and d; means: b and c.
2. This relation is used to find a missing term.

Question 12. A mixture of 24 kg contains sand and cement in the ratio 5 : 1. Find the mass of cement in it.
Answer:
Groups = 5 + 1 = 6; each group = 24 ÷ 6 = 4 kg.
Cement = 1 × 4 = 4 kg.

Teacher's Note:
1. Cement is 1 part out of 6.
2. Sand is then 20 kg, and 20 + 4 = 24.

Question 13. Check whether the ratios 14 : 21 and 22 : 33 are proportional.
Answer:
14 : 21 = 2 : 3 (HCF 7) and 22 : 33 = 2 : 3 (HCF 11).
Same simplest form, so yes, they are proportional. (Check: 14 × 33 = 462 = 21 × 22.)

Teacher's Note:
1. Compare simplest forms, or cross multiply.
2. Both methods must give the same verdict.

Question 14. Fill in the missing term: 18 : 24 :: 27 : ____.
Answer:
The first term changes from 18 to 27, a factor of \( \frac{27}{18} \) = \( \frac{3}{2} \).
Missing term = 24 × \( \frac{3}{2} \) = 36.

Teacher's Note:
1. Both terms must change by the same factor.
2. Check: 18 × 36 = 648 = 24 × 27.

Question 15. Fill in the missing term: 20 : 35 :: 8 : ____.
Answer:
The factor of change is \( \frac{8}{20} \) = \( \frac{2}{5} \).
Missing term = 35 × \( \frac{2}{5} \) = 14.

Teacher's Note:
1. A factor less than 1 makes the terms smaller.
2. Check: 20 × 14 = 280 = 35 × 8.

Question 16. A cook prepares 12 kg of rice for 96 students. How much rice should he prepare for 72 students?
Answer:
The factor of change is \( \frac{72}{96} \) = \( \frac{3}{4} \).
Rice needed = 12 × \( \frac{3}{4} \) = 9 kg.

Teacher's Note:
1. Rice and number of students are in direct proportion.
2. Fewer students need less rice, as expected.

Question 17. A tap fills a 400 mL glass in 12 seconds. How long will the same tap take to fill a 6-litre jug?
Answer:
6 litres = 6,000 mL, which is 6000 ÷ 400 = 15 glassfuls.
Time = 15 × 12 = 180 seconds = 3 minutes.

Teacher's Note:
1. Change litres to millilitres first.
2. Counting glassfuls is a simple way to find the factor.

Question 18. Share ₹6,300 between two workers in the ratio 4 : 5.
Answer:
Groups = 4 + 5 = 9; each group = 6300 ÷ 9 = ₹700.
Shares: 4 × 700 = ₹2,800 and 5 × 700 = ₹3,500.

Teacher's Note:
1. Find one group first.
2. Check: 2800 + 3500 = 6300.

Question 19. Convert 35°C into degrees Fahrenheit.
Answer:
F = \( \frac{9}{5} \) × 35 + 32 = 63 + 32 = 95, so 35°C = 95°F.

Teacher's Note:
1. Multiply by \( \frac{9}{5} \) first, then add 32.
2. Order of operations matters here.

Question 20. Convert 68°F into degrees Celsius.
Answer:
C = \( \frac{5}{9} \) × (68 − 32) = \( \frac{5}{9} \) × 36 = 20, so 68°F = 20°C.

Teacher's Note:
1. Subtract 32 first, then multiply by \( \frac{5}{9} \).
2. 20°C is a pleasant room temperature.

Question 21. One shop sells 150 g of a spice for ₹90 and another sells 250 g of the same spice for ₹150. Are the mass-to-price ratios proportional?
Answer:
150 : 90 = 5 : 3 and 250 : 150 = 5 : 3.
Yes, the ratios are proportional; the spice costs the same rate in both shops.

Teacher's Note:
1. Same simplest form means the same rate.
2. Both work out to ₹0.60 per gram.

Question 22. Image A measures 60 mm × 40 mm and image B measures 40 mm × 20 mm. Give one reason why B does not look similar to A.
Answer:
The width changes by the factor \( \frac{40}{60} \) = \( \frac{2}{3} \), but the height changes by \( \frac{20}{40} \) = \( \frac{1}{2} \).
The factors are different, so B looks stretched and is not similar to A.

Teacher's Note:
1. Subtracting 20 mm from both sides does not keep the shape.
2. Compare factors, not differences.

Question 23. A farmer owns 2 acres of land. Express this area in square feet.
Answer:
1 acre = 43,560 ft², so 2 acres = 2 × 43560 = 87,120 square feet.

Teacher's Note:
1. Multiply by the conversion factor.
2. The land stays the same; only the unit changes.

Question 24. In a class of 54 students, the ratio of girls to boys is 5 : 4. Find the number of girls.
Answer:
Groups = 5 + 4 = 9; each group = 54 ÷ 9 = 6 students.
Girls = 5 × 6 = 30.

Teacher's Note:
1. Girls take 5 of every 9 parts.
2. Boys are then 24, and 30 + 24 = 54.

Question 25. A 50 kg mixture has sand and cement in the ratio 4 : 1. How much cement must be added so that the ratio becomes 5 : 2?
Answer:
Sand = 40 kg and cement = 10 kg.
For 5 : 2 with the same sand, cement = \( \frac{2}{5} \) × 40 = 16 kg.
Cement to add = 16 − 10 = 6 kg.

Teacher's Note:
1. The sand does not change, so build the new ratio around 40 kg.
2. Check: 40 : 16 = 5 : 2.

Question 26. A boy is 3 years old and his sister is 11 years old. After how many years will the ratio of their ages be 1 : 2?
Answer:
After x years the ages are 3 + x and 11 + x.
2(3 + x) = 11 + x gives 6 + 2x = 11 + x, so x = 5.
After 5 years the ages are 8 and 16, which is 1 : 2.

Teacher's Note:
1. Both ages grow by the same number of years.
2. Always check the answer by putting it back.

Question 27. A pair of oxen ploughs one acre in 6 hours, and a tractor ploughs the same area 4 times faster. How long will the tractor take to plough a 12-acre field?
Answer:
Oxen need 12 × 6 = 72 hours for 12 acres.
The tractor is 4 times faster, so it takes 72 ÷ 4 = 18 hours.

Teacher's Note:
1. Faster means less time, so divide by 4.
2. Or: the tractor ploughs 1 acre in 1.5 hours, and 12 × 1.5 = 18.

Question 28. One acre of land costs ₹18,00,000. Find the cost of 3,630 square feet of the same land.
Answer:
3,630 ft² is \( \frac{3630}{43560} \) = \( \frac{1}{12} \) of an acre.
Cost = 1800000 ÷ 12 = ₹1,50,000.

Teacher's Note:
1. Change the plot area to a fraction of an acre.
2. 43560 ÷ 3630 = 12 exactly.

Question 29. The masses of equal volumes of silver and water are in the ratio 21 : 2. If 1 litre of water has a mass of 1 kg, find the mass of 2 litres of silver.
Answer:
2 litres of water has a mass of 2 kg.
From 21 : 2 :: x : 2, we get x = \( \frac{21 \times 2}{2} \) = 21.
So 2 litres of silver has a mass of 21 kg.

Teacher's Note:
1. Compare equal volumes: 2 L of silver with 2 L of water.
2. Silver is 10.5 times as heavy as water.

Question 30. A recipe for 8 people uses 600 g of flour. Rohan has 975 g of flour. For how many people can he cook the same dish?
Answer:
Flour per person = 600 ÷ 8 = 75 g.
People = 975 ÷ 75 = 13 people.

Teacher's Note:
1. The unitary method finds the amount for one person first.
2. Check: 13 × 75 = 975.