CBSE Class 8 Mathematics Proportional Reasoning 2 Very Short Answer Questions with Answers

Very Short Answer Questions

Question 1. When is the relationship between two or more quantities called a proportional relationship?
Answer:
When two or more related quantities change by the same factor, the relationship between them is called a proportional relationship. It is written in ratio notation, for example rice : urad dal = 2 : 1.

Teacher's Note:
1. The key words are "same factor".
2. Doubling one quantity must double the other as well.

Question 2. Write the condition that the ratios a : b and c : d must satisfy to be proportional.
Answer:
They are proportional when a × d = b × c, that is, when \( \frac{a}{b} \) = \( \frac{c}{d} \).

Teacher's Note:
1. This is the cross-multiplication test.
2. Outer product = inner product.

Question 3. What does RF written on a map stand for, and which two distances does it compare?
Answer:
RF stands for Representative Fraction. It compares a distance measured on the map with the matching actual (geographical) distance on the ground.

Teacher's Note:
1. Map distance comes first in the RF.
2. The RF is usually printed in the lower right corner of a map.

Question 4. A map is drawn to the scale 1 : 1,00,000. What ground distance does 1 cm on this map represent?
Answer:
1 cm on the map stands for 1,00,000 cm on the ground, which is 1 km.

Teacher's Note:
1. 1 km = 1,000 m = 1,00,000 cm.
2. This is a handy scale: every cm is one km.

Question 5. Express 25,00,000 cm in kilometres.
Answer:
25,00,000 ÷ 1,00,000 = 25, so 25,00,000 cm = 25 km.

Teacher's Note:
1. Divide by 1,00,000 to change cm to km.
2. Removing five zeros does the same job.

Question 6. A pickle masala is made with 9 spoons of chilli powder, 6 spoons of mustard and 3 spoons of salt. Write this as a ratio in its simplest form.
Answer:
Chilli powder : mustard : salt = 9 : 6 : 3. Dividing every term by the HCF 3 gives 3 : 2 : 1.

Teacher's Note:
1. Keep the order of the ingredients as given.
2. Divide all three terms by the same number.

Question 7. Write the ratio 10 : 15 : 25 in its simplest form.
Answer:
The HCF of 10, 15 and 25 is 5, so the simplest form is 2 : 3 : 5.

Teacher's Note:
1. Find the HCF of all the terms together.
2. 2, 3 and 5 have no common factor, so we stop.

Question 8. What is the first step in dividing a whole in a given ratio?
Answer:
Add the terms of the ratio to find how many equal parts the whole is to be split into; then find the value of one part.

Teacher's Note:
1. The sum of the terms gives the number of equal parts.
2. Each share is then a number of parts.

Question 9. Divide 36 in the ratio 1 : 2 : 3.
Answer:
Sum of the terms = 1 + 2 + 3 = 6, so one part = 36 ÷ 6 = 6.
The three shares are 6, 12 and 18.

Teacher's Note:
1. Check: 6 + 12 + 18 = 36.
2. Multiply one part by each term.

Question 10. What does a pie chart show, and what is the total angle at its centre?
Answer:
A pie chart shows how a whole is shared among its categories, each category being a slice. The total angle at its centre is 360°.

Teacher's Note:
1. The whole circle stands for the whole data.
2. All the slice angles together make 360°.

Question 11. In a pie chart, find the angle of the slice that stands for one-tenth of the whole.
Answer:
Angle = \( \frac{1}{10} \) × 360° = 36°.

Teacher's Note:
1. Angle = share × 360°.
2. Ten such slices fill the circle.

Question 12. Write the relation satisfied by two quantities x and y that are in inverse proportion.
Answer:
xy = k, where k is a constant; so x₁y₁ = x₂y₂ for any two pairs of values.

Teacher's Note:
1. The product stays the same.
2. In direct proportion it is the ratio that stays the same.

Question 13. Use cross-multiplication to check whether 8 : 12 and 14 : 21 are proportional.
Answer:
8 × 21 = 168 and 12 × 14 = 168.
The products are equal, so 8 : 12 and 14 : 21 are proportional.

Teacher's Note:
1. Both ratios also reduce to 2 : 3.
2. Equal cross products confirm a proportion.

Question 14. On a map with RF 1 : 5,00,000, two villages are 7 cm apart. Find the geographical distance between them.
Answer:
1 cm stands for 5,00,000 cm = 5 km, so 7 cm stands for 7 × 5 = 35 km.

Teacher's Note:
1. Convert the RF into km per cm first.
2. This is a straight-line distance.

Question 15. A classroom wall is 8 m long. How long will this wall be in a sketch of the room drawn to the scale 1 : 50?
Answer:
8 m = 800 cm, and the sketch makes every length 50 times smaller.
So the wall is drawn 800 ÷ 50 = 16 cm long.

Teacher's Note:
1. Change metres to centimetres before dividing.
2. Ground to sketch means divide by the scale.

Question 16. A spice mix has coriander seeds : red chillies : toor dal : fenugreek seeds = 8 : 4 : 2 : 1. Puneet has only 3 red chillies. How many spoons of coriander seeds should he grind with them?
Answer:
Puneet has 3 chillies instead of 4, which is \( \frac{3}{4} \) of the recipe.
Coriander = 8 × \( \frac{3}{4} \) = 6 spoons.

Teacher's Note:
1. Find the scaling factor from the ingredient you know.
2. Coriander is always twice the chillies here.

Question 17. Concrete is mixed as cement : sand : gravel = 1 : 1.5 : 3. How many bags of concrete mixture can be made using 4 bags of cement?
Answer:
Multiply every term by 4: cement : sand : gravel = 4 : 6 : 12.
Total = 4 + 6 + 12 = 22 bags.

Teacher's Note:
1. Cement is 1 part, so 1 part = 4 bags.
2. Total parts 5.5 × 4 = 22 gives the same answer.

Question 18. Divide ₹2400 among three friends in the ratio 3 : 4 : 5.
Answer:
Sum of the terms = 12, so one part = 2400 ÷ 12 = ₹200.
The shares are ₹600, ₹800 and ₹1000.

Teacher's Note:
1. Check: 600 + 800 + 1000 = 2400.
2. Multiply ₹200 by each term.

Question 19. The angles of a triangle are in the ratio 1 : 2 : 3. Find the three angles.
Answer:
Sum of the terms = 6 and the angle sum is 180°, so one part = 30°.
The angles are 30°, 60° and 90°.

Teacher's Note:
1. The whole here is 180°.
2. It is a right-angled triangle.

Question 20. A pie chart is drawn for a survey of 720 people. One slice of the chart measures 75°. How many people does this slice stand for?
Answer:
360° stands for 720 people, so 1° stands for 2 people.
The slice stands for 75 × 2 = 150 people.

Teacher's Note:
1. Find how many people one degree represents.
2. Check: 150/720 × 360° = 75°.

Question 21. Data in the ratio 9 : 6 : 5 : 4 is to be shown in a pie chart. Find the angle of the largest slice.
Answer:
Sum of the terms = 9 + 6 + 5 + 4 = 24, so one part = 360° ÷ 24 = 15°.
Largest slice = 9 × 15° = 135°.

Teacher's Note:
1. Divide 360° by the sum of the terms.
2. The largest term gives the largest slice.

Question 22. Eight pipes of the same kind fill a tank in 6 hours. How long will twelve such pipes take to fill it?
Answer:
Pipes and time are in inverse proportion: 8 × 6 = 12 × t.
t = 48 ÷ 12 = 4 hours.

Teacher's Note:
1. More pipes, less time.
2. 48 is the work in pipe-hours.

Question 23. The quantities x and y are in inverse proportion. If x₁ = 14, y₁ = 6 and x₂ = 21, find y₂.
Answer:
x₁y₁ = x₂y₂, so 14 × 6 = 21 × y₂.
y₂ = 84 ÷ 21 = 4.

Teacher's Note:
1. The product 84 stays the same.
2. x grew, so y must shrink.

Question 24. Are the number of workers and the time they take to finish a fixed piece of work directly or inversely proportional? Give the reason.
Answer:
They are inversely proportional, because more workers finish the same fixed work in fewer days, and (number of workers) × (days) stays constant.

Teacher's Note:
1. The fixed work is the constant product.
2. We assume all workers work at the same rate.

Question 25. A straight road is 24 km long on the ground and measures 3 cm on a map. Find the RF of this map.
Answer:
24 km = 24,00,000 cm, shown by 3 cm.
So 1 cm stands for 24,00,000 ÷ 3 = 8,00,000 cm, and the RF is 1 : 8,00,000.

Teacher's Note:
1. Both distances must be in the same unit.
2. The RF always has 1 as its first term.

Question 26. The three parts of a mixture are in the ratio 4 : 6 : 9. The middle part has a mass of 48 g. Find the total mass of the mixture.
Answer:
The middle term 6 stands for 48 g, so one part = 48 ÷ 6 = 8 g.
The parts are 32 g, 48 g and 72 g, and the total = 19 × 8 = 152 g.

Teacher's Note:
1. Use the known part to find one part.
2. Check: 32 + 48 + 72 = 152.

Question 27. A sum of money is shared among Asif, Beena and Charu in the ratio 2 : 3 : 5. Charu gets ₹900 more than Asif. Find the total sum shared.
Answer:
Charu has 5 parts and Asif 2 parts, so the difference is 3 parts = ₹900; one part = ₹300.
Total = 10 × 300 = ₹3000 (shares ₹600, ₹900 and ₹1500).

Teacher's Note:
1. A difference of shares is a difference of parts.
2. Check: 1500 − 600 = 900.

Question 28. A family earns ₹36,000 a month. In a pie chart of its spending, food takes 150°, rent 90° and travel 60°, and the remaining part is saved. How much money is saved?
Answer:
Savings slice = 360° − (150° + 90° + 60°) = 60°.
Money saved = \( \frac{60}{360} \) × 36,000 = ₹6000.

Teacher's Note:
1. The slices must add to 360°.
2. 60° is one-sixth of the circle.

Question 29. A hostel has food for 60 students for 20 days. After 5 days, 15 more students join. For how many more days will the remaining food last?
Answer:
Food = 60 × 20 = 1200 student-days; 60 × 5 = 300 are used in the first 5 days, leaving 900.
With 75 students, it lasts 900 ÷ 75 = 12 more days.

Teacher's Note:
1. Subtract the food already eaten first.
2. Then share the rest among the new number of students.

Question 30. Asha can mow a lawn in 4 hours and Bala can mow the same lawn in 6 hours. Working together, how long will they take to mow it?
Answer:
In one hour Asha mows \( \frac{1}{4} \) and Bala \( \frac{1}{6} \); together \( \frac{1}{4} \) + \( \frac{1}{6} \) = \( \frac{5}{12} \) of the lawn.
Whole lawn takes \( \frac{12}{5} \) = 2.4 hours = 2 hours 24 minutes.

Teacher's Note:
1. Add the parts done in one hour.
2. 0.4 hour = 0.4 × 60 = 24 minutes.