CBSE Class 8 Mathematics Proportional Reasoning 2 MCQs Set 11

Answer: (b) a × d = b × c

Explanation:
1. Two ratios are proportional when their cross products are equal.
2. For a : b and c : d the cross products are a × d and b × c, so the condition is a × d = b × c, which is the same as \( \frac{a}{b} \) = \( \frac{c}{d} \).
3. Example: 6 : 3 and 4 : 2 give 6 × 2 = 12 and 3 × 4 = 12, so they are proportional.

Teacher's Note:
1. Multiply the outer terms and the inner terms; equal products mean a proportion.
2. Adding or subtracting terms is never the test for a ratio.

Answer: (c) Representative Fraction

Explanation:
1. RF is short for Representative Fraction.
2. It is the ratio printed on a map (usually in the lower right corner) that compares a distance on the map with the matching actual distance on the ground.

Teacher's Note:
1. The RF tells us by what factor the ground has been scaled down.
2. It is written like 1 : 60,00,000.

Answer: (a) 2 km

Explanation:
1. 1 cm on the map stands for 2,00,000 cm on the ground.
2. 1 km = 1,00,000 cm, so 2,00,000 cm = 2,00,000 ÷ 1,00,000 = 2 km.

Teacher's Note:
1. Divide centimetres by 1,00,000 to get kilometres.
2. Count the zeros carefully in the Indian place-value system.

Answer: (a) Only Statement I is true

Explanation:
1. 1 cm stands for 50,000 cm = 500 m (since 1 m = 100 cm). Statement I is true.
2. A map scale gives the geographical (straight-line) distance. The road distance is longer because roads bend. Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. 50,000 cm ÷ 100 = 500 m.
2. Scale distances are always straight-line distances.

Answer: (b) 4 red chillies

Explanation:
1. In 8 : 4 : 2 : 1, the last term 1 belongs to fenugreek seeds and the second term 4 belongs to red chillies.
2. So with 1 spoon of fenugreek seeds there are 4 red chillies.

Teacher's Note:
1. Match each term with its ingredient in the same order.
2. 8 would be the coriander seeds, and 2 the toor dal.

Answer: (c) 2 : 3 : 5

Explanation:
1. The HCF of 6, 9 and 15 is 3.
2. Dividing every term by 3: 6 ÷ 3 = 2, 9 ÷ 3 = 3, 15 ÷ 3 = 5.
3. So the simplest form is 2 : 3 : 5.

Teacher's Note:
1. Divide all the terms by the same number, their HCF.
2. A three-term ratio is simplified exactly like a two-term ratio.

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

Explanation:
1. Sum of the terms = 2 + 3 = 5, so one part = 600 ÷ 5 = ₹120.
2. The shares are 2 × 120 = ₹240 and 3 × 120 = ₹360. The smaller share is ₹240, not ₹200, so A is false.
3. R correctly describes the method, so R is true. Answer: A is false but R is true.

Teacher's Note:
1. Always work out the shares before judging the assertion.
2. Check: ₹240 + ₹360 = ₹600.

Answer: (c) 8 equal parts

Explanation:
1. Add the terms of the ratio: 3 + 5 = 8.
2. The whole is thought of as 8 equal parts; one share gets 3 parts and the other gets 5 parts.

Teacher's Note:
1. The number of equal parts is the sum of the terms, not their product.
2. 15 = 3 × 5 is a common wrong choice.

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

Explanation:
1. Angle of a slice = share × 360°.
2. \( \frac{1}{2} \) × 360° = 180° (s); \( \frac{1}{3} \) × 360° = 120° (r); \( \frac{1}{4} \) × 360° = 90° (q); \( \frac{1}{8} \) × 360° = 45° (p).
3. So the matching is (i)-(s), (ii)-(r), (iii)-(q), (iv)-(p).

Teacher's Note:
1. The whole circle is 360°, so multiply each fraction by 360°.
2. A bigger share always gives a bigger angle.

Answer: (a) the share of that category in the whole

Explanation:
1. A pie chart divides the 360° at the centre in the ratio of the data.
2. So each slice's angle is proportional to that category's share of the whole.
3. The radius and the number of categories do not decide the angle.

Teacher's Note:
1. Angle = share × 360°.
2. Any radius can be used; only the angles carry the data.

Answer: (b) 10

Explanation:
1. The bus slice is 100° out of 360°.
2. Number of students = \( \frac{100}{360} \) × 36 = 10.

Teacher's Note:
1. Here 360° stands for 36 students, so each student is 10°.
2. 100° ÷ 10° = 10 students.

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

Explanation:
1. More taps fill the same tank in less time, and (number of taps) × (time) stays the same, so A is true.
2. R is the correct definition of inverse proportion, so R is true.
3. R is exactly the test that shows A is true: both true and R explains A.

Teacher's Note:
1. The constant product is the work needed to fill the tank.
2. Inverse proportion means a constant product, not a constant ratio.

Answer: (d) 14 : 35

Explanation:
1. 6 : 15 in simplest form is 2 : 5.
2. Cross-multiply 6 : 15 with 14 : 35: 6 × 35 = 210 and 15 × 14 = 210. They are equal, so 14 : 35 is proportional.
3. 9 : 20, 10 : 24 = 5 : 12 and 12 : 32 = 3 : 8 do not reduce to 2 : 5.

Teacher's Note:
1. Reduce every ratio to its simplest form and compare.
2. Cross-multiplication gives a quick check.

Answer: (b) 150 km

Explanation:
1. 1 cm stands for 25,00,000 cm = 25 km.
2. So 6 cm stands for 6 × 25 = 150 km.

Teacher's Note:
1. First find what 1 cm stands for in km.
2. Then multiply by the map distance.

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

Explanation:
1. 1 cm stands for 5,00,000 cm = 5 km, so 3 cm stands for 3 × 5 = 15 km, not 150 km. A is false.
2. R correctly says what an RF is, so R is true.
3. Answer: A is false but R is true.

Teacher's Note:
1. 5,00,000 cm is 5 km, not 50 km.
2. An extra zero makes the answer ten times too big.

Answer: (b) \( \frac{a}{p} = \frac{b}{q} = \frac{c}{r} \)

Explanation:
1. In a proportion with several terms, each term of the first ratio is the same multiple of the matching term of the second ratio.
2. So \( \frac{a}{p} = \frac{b}{q} = \frac{c}{r} \).
3. The sums of the terms need not be equal, and the products are not the test.

Teacher's Note:
1. Compare matching terms: first with first, second with second, third with third.
2. Example: 8 : 4 : 2 :: 4 : 2 : 1 gives 8/4 = 4/2 = 2/1 = 2.

Answer: (b) 48 L

Explanation:
1. White is 5 parts and equals 20 L, so 1 part = 20 ÷ 5 = 4 L.
2. Total parts = 3 + 4 + 5 = 12.
3. Total paint = 12 × 4 = 48 L (yellow 12 L, blue 16 L, white 20 L).

Teacher's Note:
1. Find the value of one part from the known quantity.
2. Then multiply by the total number of parts.

Answer: (a) Only Statement I is true

Explanation:
1. Sum of the terms = 1 + 1.5 + 3 = 5.5, so one part = 55 ÷ 5.5 = 10 units.
2. Cement = 1 × 10 = 10 units, so Statement I is true.
3. Sand = 1.5 × 10 = 15 units, not 20, so Statement II is false. Answer: only Statement I is true.

Teacher's Note:
1. Decimal terms are added just like whole-number terms.
2. Gravel would be 30 units, and 10 + 15 + 30 = 55.

Answer: (d) 80°

Explanation:
1. The angles of a triangle add up to 180°.
2. Sum of the terms = 2 + 3 + 4 = 9, so one part = 180° ÷ 9 = 20°.
3. The angles are 40°, 60° and 80°, so the largest is 80°.

Teacher's Note:
1. The whole to be divided here is the angle sum 180°.
2. Check: 40° + 60° + 80° = 180°.

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

Explanation:
1. 72 families share 360°, so one family is worth 360° ÷ 72 = 5°.
2. 12 → 60° (q), 30 → 150° (s), 6 → 30° (p), 24 → 120° (r).
3. So the answer is (i)-(q), (ii)-(s), (iii)-(p), (iv)-(r).

Teacher's Note:
1. Find the angle for one item first.
2. Check: 60° + 150° + 30° + 120° = 360°.

Answer: (a) 40

Explanation:
1. Difference in the angles = 140° − 60° = 80°.
2. Number of students = \( \frac{80}{360} \) × 180 = 40. (Cricket 70 students, badminton 30 students.)

Teacher's Note:
1. Here 1° stands for half a student, since 360° = 180 students.
2. Subtracting the angles first saves one step.

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

Explanation:
1. Sum of the terms = 7 + 5 + 4 + 3 + 1 = 20, so one part = 360° ÷ 20 = 18°.
2. Largest slice = 7 × 18° = 126°, so A is true.
3. The slices of a pie chart add up to 360°, not 180°, so R is false. Answer: A is true but R is false.

Teacher's Note:
1. The angle at the centre of a circle is 360°.
2. 180° is the angle sum of a triangle, not of a circle.

Answer: (c) 60

Explanation:
1. Machines and days are in inverse proportion, so their product stays the same.
2. Constant = 36 × 45 = 1620 machine-days.
3. Machines needed = 1620 ÷ 27 = 60.

Teacher's Note:
1. Fewer days need more machines.
2. Check: 60 × 27 = 1620.

Answer: (b) Only Statement I is true

Explanation:
1. If xy is constant and x is tripled, y must become one-third. Statement I is true.
2. At a fixed rate per metre, more cloth costs more money; the ratio money : length stays constant. This is direct proportion, so Statement II is false.
3. Answer: only Statement I is true.

Teacher's Note:
1. Inverse: one goes up, the other goes down by the same factor.
2. A fixed price per unit always gives a direct proportion.

Answer: (c) 1.8 cm

Explanation:
1. On the first map, 1 cm stands for 40,00,000 cm = 40 km, so P and Q are 4.5 × 40 = 180 km apart.
2. On the second map, 1 cm stands for 1,00,00,000 cm = 100 km, so the gap is 180 ÷ 100 = 1.8 cm.

Teacher's Note:
1. Find the real distance first; it does not change between maps.
2. A smaller-scale map shows the same distance as a shorter length.

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

Explanation:
1. 1 : 2 : 3 → largest = \( \frac{3}{6} \) × 1800 = ₹900 (r).
2. 1 : 1 : 4 → \( \frac{4}{6} \) × 1800 = ₹1200 (p). 1 : 2 : 2 → \( \frac{2}{5} \) × 1800 = ₹720 (q). 2 : 3 : 4 → \( \frac{4}{9} \) × 1800 = ₹800 (s).
3. So the answer is (i)-(r), (ii)-(p), (iii)-(q), (iv)-(s).

Teacher's Note:
1. Largest share = (largest term ÷ sum of terms) × whole.
2. In 1 : 2 : 2 two children get the same largest share.

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

Explanation:
1. 6 × 40 = 240 = 10 × t, so t = 240 ÷ 10 = 24 minutes. A is true.
2. R is true, and it is exactly the rule used to get 24 minutes.
3. Answer: both true and R explains A.

Teacher's Note:
1. More taps, less time.
2. The constant 240 is measured in tap-minutes.

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

Explanation:
1. Share = \( \frac{24}{120} \) = \( \frac{1}{5} \), so the slice = \( \frac{1}{5} \) × 360° = 72°, not 60°. A is false.
2. R states the correct rule for the angle of a slice, so R is true.
3. Answer: A is false but R is true.

Teacher's Note:
1. Simplify the share before multiplying by 360°.
2. A 60° slice would stand for 20 people.

Answer: (a) Only Statement I is true

Explanation:
1. Students and days are in inverse proportion: 150 × 20 = 3000, and 3000 ÷ 200 = 15 days. Statement I is true.
2. Doubling the number of students halves the number of days, so Statement II is false.
3. Answer: only Statement I is true.

Teacher's Note:
1. The constant 3000 is in student-days.
2. More mouths, fewer days.

Answer: (d) \( 1\frac{1}{5} \) hours

Explanation:
1. In one hour Ravi does \( \frac{1}{2} \) of the job and Meera does \( \frac{1}{3} \).
2. Together in one hour: \( \frac{1}{2} \) + \( \frac{1}{3} \) = \( \frac{5}{6} \) of the job.
3. Whole job takes \( \frac{6}{5} \) = \( 1\frac{1}{5} \) hours, that is 1 hour 12 minutes.

Teacher's Note:
1. Add the work done per hour, not the times.
2. Together they must be faster than the faster person alone (2 hours).