CBSE Class 8 Mathematics Proportional Reasoning 1 MCQs Set 11

Answer: (b) terms

Explanation:
1. The two numbers written on the two sides of the ratio sign are called the terms of the ratio.
2. In a : b, a is the first term and b is the second term.

Teacher's Note:
1. Terms is the special name for the two numbers of a ratio.
2. Factors, multiples and divisors are words from the topic of divisibility.

Answer: (d) ::

Explanation:
1. The double colon :: is read as "is proportional to".
2. So a : b :: c : d means the ratios a : b and c : d are proportional.
3. A single colon only separates the two terms of one ratio.

Teacher's Note:
1. One colon makes a ratio; two colons join two ratios in a proportion.
2. Read a : b :: c : d aloud as "a is to b as c is to d".

Answer: (d) 3 : 4

Explanation:
1. HCF of 36 and 48 = 12.
2. 36 ÷ 12 = 3 and 48 ÷ 12 = 4, so 36 : 48 = 3 : 4.
3. 6 : 8 and 9 : 12 are equivalent ratios but can still be reduced; 4 : 3 has the terms the wrong way round.

Teacher's Note:
1. Divide by the HCF to reach the simplest form in one step.
2. Keep the order of the terms unchanged.

Answer: (c) 12 laddoos

Explanation:
1. Number of equal groups = 3 + 2 = 5.
2. Each group = 20 ÷ 5 = 4 laddoos.
3. Larger share = 3 × 4 = 12 laddoos (smaller share = 2 × 4 = 8 laddoos).

Teacher's Note:
1. Add the terms of the ratio to get the number of equal groups.
2. Check: 12 + 8 = 20.

Answer: (c) 7 : 9

Explanation:
1. HCF of 7 and 9 is 1, so 7 : 9 cannot be reduced further.
2. The others reduce: 8 : 12 = 2 : 3, 9 : 15 = 3 : 5 and 14 : 21 = 2 : 3.

Teacher's Note:
1. A ratio is in simplest form when the HCF of its terms is 1.
2. Look for a common factor in each pair before choosing.

Answer: (b) 10,000 square metres

Explanation:
1. From the conversion table, 1 hectare = 10,000 square metres.
2. 43,560 is the number of square feet in one acre, and 2.471 is the number of acres in a hectare.

Teacher's Note:
1. A hectare is the area of a square of side 100 m.
2. Do not mix up the numbers for acres and hectares.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. HCF of 25 and 15 is 5; 25 ÷ 5 = 5 and 15 ÷ 5 = 3, so 25 : 15 = 5 : 3. A is true.
2. R states the rule that was used to get this simplest form, so R is true and correctly explains A.

Teacher's Note:
1. When the reason is the very method used in the assertion, choose (a).
2. Always check the HCF before dividing.

Answer: (c) Both Statement I and Statement II are true.

Explanation:
1. 1 litre = 1,000 mL, so Statement I is true.
2. 1 mL = 1 cubic centimetre (1 cc), so Statement II is true.
3. Hence both statements are true.

Teacher's Note:
1. Volume conversions: 1 L = 1,000 mL = 1,000 cc.
2. Medicine labels often use cc in place of mL.

Answer: (a) 6 kg

Explanation:
1. Total groups = 2 + 1 = 3.
2. Each group = 9 ÷ 3 = 3 kg.
3. Sand = 2 groups = 2 × 3 = 6 kg.

Teacher's Note:
1. Sand gets 2 of every 3 parts of the mixture.
2. Cement is then 3 kg, and 6 + 3 = 9 kg checks out.

Answer: (b) 3.281 feet

Explanation:
1. The conversion table gives 1 m ≈ 3.281 feet.
2. 10.764 is the number of square feet in a square metre, and 2.471 is the number of acres in a hectare.

Teacher's Note:
1. A metre is a little more than 3 feet.
2. Length and area factors are different numbers.

Answer: (d) (i)-(r), (ii)-(p), (iii)-(q), (iv)-(s)

Explanation:
1. 18 : 24 = 3 : 4 (HCF 6), so (i)-(r).
2. 40 : 20 = 2 : 1 (HCF 20), so (ii)-(p).
3. 60 : 60 = 1 : 1 (HCF 60), so (iii)-(q).
4. 90 : 60 = 3 : 2 (HCF 30), so (iv)-(s).

Teacher's Note:
1. Reduce each ratio by its HCF and match.
2. Equal terms always give 1 : 1.

Answer: (a) 1.5

Explanation:
1. Width: 6 units becomes 9 units, factor = \( \frac{9}{6} \) = 1.5.
2. Height: 4 units becomes 6 units, factor = \( \frac{6}{4} \) = 1.5.
3. Both measures are multiplied by 1.5, so Q is similar to P.

Teacher's Note:
1. Divide the new measure by the old one to get the factor.
2. The same factor for width and height means the shape is kept.

Answer: (a) 9 : 15 and 21 : 35

Explanation:
1. 9 : 15 = 3 : 5 and 21 : 35 = 3 : 5. Same simplest form, so they are proportional.
2. Other pairs differ: 3 : 5 and 5 : 8; 2 : 3 and 5 : 7; 4 : 7 and 3 : 5.

Teacher's Note:
1. Two ratios are proportional when their simplest forms match.
2. Cross multiplication also works: 9 × 35 = 315 = 15 × 21.

Answer: (d) A is false, but R is true.

Explanation:
1. A is false: 3 : 30 = 1 : 10, but adding 9 to both terms gives 12 : 39 = 4 : 13, a different ratio.
2. R is true: proportional ratios come from multiplying (or dividing) both terms by the same number.

Teacher's Note:
1. Adding changes a ratio; multiplying keeps it.
2. One counter-example is enough to show a statement with "always" is false.

Answer: (d) ad = bc

Explanation:
1. a : b :: c : d means \( \frac{a}{b} = \frac{c}{d} \).
2. Multiplying both sides by bd gives ad = bc.
3. The product of the extreme terms equals the product of the middle terms.

Teacher's Note:
1. Extremes are the outer terms a and d; means are the inner terms b and c.
2. This is the quickest test for proportion.

Answer: (b) their widths and their heights are multiplied by the same factor

Explanation:
1. A picture keeps its shape only when the width and the height are multiplied by the same factor.
2. Equal differences, equal perimeters or equal heights do not keep the width-to-height ratio the same.

Teacher's Note:
1. Similar shapes have the same width-to-height ratio.
2. Subtracting 20 mm from both sides squashes a picture.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. HCF of 75,000 and 25,000 is 25,000, so the ratio is 3 : 1. A is true.
2. R is a true fact about proportional ratios.
3. But A is found by dividing by the HCF, not by cross multiplication, so R does not explain A.

Teacher's Note:
1. Both statements can be true and still be unrelated.
2. Ask: is R the reason A happens?

Answer: (c) A is true, but R is false.

Explanation:
1. 15 ÷ 5 = 3 and 35 ÷ 5 = 7, so A is true.
2. R is false: dividing both terms by the same non-zero number gives an equivalent ratio, which is proportional to the original.

Teacher's Note:
1. Dividing both terms by the same number never changes a ratio.
2. Watch for the word "not" in a reason.

Answer: (a) Both Statement I and Statement II are true.

Explanation:
1. In 15 : 35, decoction is \( \frac{15}{50} \) of the cup; in 10 : 40 it is \( \frac{10}{50} \). So the first coffee is stronger. Statement I is true.
2. 15 : 35 = 3 : 7 (HCF 5) and 10 : 40 = 1 : 4 (HCF 10). Statement II is true.
3. Hence both statements are true.

Teacher's Note:
1. Strength depends on the fraction of decoction in the whole cup.
2. Both cups here hold 50 parts, which makes the comparison easy.

Answer: (a) Only Statement I is true.

Explanation:
1. 1 m² = 10.764 ft² is in the conversion table, so Statement I is true.
2. 1 hectare = 10,000 m², not 1,000 m², so Statement II is false.
3. Hence only Statement I is true.

Teacher's Note:
1. 1 hectare = 100 m × 100 m = 10,000 m².
2. Count the zeros carefully in conversions.

Answer: (c) 192 km

Explanation:
1. Use the same unit: 2 hours = 120 minutes.
2. 45 : 72 :: 120 : x gives 45x = 72 × 120 = 8640.
3. x = 8640 ÷ 45 = 192 km.

Teacher's Note:
1. Change hours to minutes before writing the proportion.
2. Taking 2 hours as 2 would give a silly answer.

Answer: (b) ₹1,400

Explanation:
1. Number of groups = 5 + 3 = 8; one group = 5600 ÷ 8 = ₹700.
2. Shares: 5 × 700 = ₹3,500 and 3 × 700 = ₹2,100.
3. Difference = 3500 − 2100 = ₹1,400.

Teacher's Note:
1. The difference is (5 − 3) groups = 2 × 700.
2. Option (c) is the smaller share, not the difference.

Answer: (c) (i)-(q), (ii)-(p), (iii)-(s), (iv)-(r)

Explanation:
1. 2 : 1 → 3 groups of ₹300 → ₹600 and ₹300, so (i)-(q).
2. 3 : 2 → 5 groups of ₹180 → ₹540 and ₹360, so (ii)-(p).
3. 5 : 4 → 9 groups of ₹100 → ₹500 and ₹400, so (iii)-(s).
4. 3 : 1 → 4 groups of ₹225 → ₹675 and ₹225, so (iv)-(r).

Teacher's Note:
1. Find the size of one group first, then multiply.
2. Each pair must add up to ₹900.

Answer: (d) S

Explanation:
1. Price per kg: P = 180 ÷ 2 = ₹90, Q = 450 ÷ 5 = ₹90, R = 720 ÷ 8 = ₹90.
2. S = 950 ÷ 10 = ₹95 per kg.
3. So packet S is not proportional to the others.

Teacher's Note:
1. Equal price per kg means proportional mass-to-price ratios.
2. The odd one out has a different unit rate.

Answer: (c) 4 kg

Explanation:
1. 36 kg in 6 groups of 6 kg: sand = 5 × 6 = 30 kg, cement = 6 kg.
2. For 3 : 1 with 30 kg of sand, cement must be 30 ÷ 3 = 10 kg.
3. Cement to add = 10 − 6 = 4 kg.

Teacher's Note:
1. Keep the unchanged quantity fixed and rebuild the ratio around it.
2. Check: 30 : 10 = 3 : 1.

Answer: (d) A is false, but R is true.

Explanation:
1. The proportion 50 : 2 :: 75 : x would give x = 3 hours, a longer time at a higher speed, which is wrong. So A is false.
2. R is true: over the same distance, more speed means less time (here 100 ÷ 75 = 1 h 20 min).

Teacher's Note:
1. Not every pair of quantities is in direct proportion.
2. Speed and time for a fixed distance move in opposite directions.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. Now: 2 : 6 = 1 : 3. After 4 years: 6 : 10 = 3 : 5. So A is true.
2. R is true, and it is exactly why the ratio of ages changes: 4 is added to both ages.
3. So R correctly explains A.

Teacher's Note:
1. Both people age by the same number of years, so the ratio changes.
2. The difference in ages stays the same; the ratio does not.

Answer: (a) Only Statement I is true.

Explanation:
1. City A: 72000 ÷ 18 = 4,000 people per km².
2. City B: 95000 ÷ 25 = 3,800 people per km².
3. A is more crowded, so Statement I is true; 4,000 ≠ 3,800, so Statement II is false.
4. Hence only Statement I is true.

Teacher's Note:
1. Crowding is measured as people per square kilometre.
2. Bigger population alone does not mean more crowded.

Answer: (d) 18

Explanation:
1. 8 kg costs ₹340, so 1 kg costs 340 ÷ 8 = ₹42.50.
2. Mass for ₹765 = 765 ÷ 42.50 = 18 kg.
3. Check: 8 × 765 = 6120 = 340 × 18, so 8 : 340 :: 18 : 765.

Teacher's Note:
1. Values that sit one above the other on a double number line form equivalent ratios.
2. Find the unit rate first, then scale.

Answer: (b) (i)-(q), (ii)-(p), (iii)-(r), (iv)-(s)

Explanation:
1. Use \( F = \frac{9}{5} \times C + 32 \) and \( C = \frac{5}{9} \times (F - 32) \).
2. 25°C = 45 + 32 = 77°F, so (i)-(q). 50°F: \( \frac{5}{9} \) × 18 = 10°C, so (ii)-(p).
3. 0°C = 32°F, so (iii)-(r). 212°F: \( \frac{5}{9} \) × 180 = 100°C, so (iv)-(s).

Teacher's Note:
1. Water freezes at 0°C = 32°F and boils at 100°C = 212°F.
2. Subtract 32 first when changing °F to °C.