Step 1: HCF of 42 and 56 is 14.

Step 2: Divide both terms by 14:

Answer: 3 : 4

Multiply both terms of 5 : 7 by 4:

Answer: (c) 20 : 28

Use cross multiplication:

Answer: x = 30

For two proportional ratios, the cross products are equal.

Answer: (b) ad = bc

The scale factor is 20 ÷ 8 = 2.5. The height must change by the same factor.

Answer: 30 cm

False. For example, 2 : 3 is not proportional to 3 : 4 because

Equivalent ratios are obtained by multiplying or dividing both terms by the same non-zero factor, not generally by adding the same number.

HCF of 600 and 900 is 300.

Answer: 2 : 3

By the standard volume conversion used in the chapter:

Answer: 1000 mL

Answer: 43,560 square feet

Use F = (9/5)C + 32.

Answer: 77°F

HCF(72, 96) = 24.

Answer: 3 : 4

21 : 27 simplifies by dividing by 3:

So the ratios are proportional.

Answer: Yes

Divide both terms by 5:

Answer: 1 : 34

Total parts = 3 + 1 = 4. Value of one part = ₹280 ÷ 4 = ₹70.

Answer: ₹210

Total parts = 3 + 5 = 8. One part = 64 ÷ 8 = 8 L.

Answer: 24 L

HCF(15, 35) = 5.

Answer: 3 : 7

Regular coffee has decoction fraction 15/(15+35) = 15/50 = 0.30.

The second cup has decoction fraction 20/(20+30) = 20/50 = 0.40.

Since 0.40 > 0.30, the second cup contains a greater proportion of decoction.

Answer: Stronger

Regular decoction fraction = 15/50 = 0.30.

New decoction fraction = 10/50 = 0.20.

Since 0.20 < 0.30, it has less decoction per cup.

Answer: Lighter

Tea A: 200 g costs ₹200, so 1000 g costs 5 × ₹200 = ₹1000.

Tea B: 1000 g costs ₹800.

Answer: Tea B is cheaper per kilogram.

40 is one-third of 120, so the rice must also be one-third.

Answer: 5 kg

Since ad = bc, divide both sides by a:

Answer: d = bc/a

Answer: 3.281 feet

Answer: 10.764 square feet

Answer: 10,000 square metres

Answer: 2.471 acres

Initially: 3 : 30 = 1 : 10.

Nine years later: 12 : 39 = 4 : 13.

Since 1 : 10 ≠ 4 : 13, the ratio changes.

Answer: No

HCF(60, 40) = 20.

Answer: 3 : 2

Distance per minute = 90/150 = 0.6 km.

Answer: 36 km

Check cross products:

The cross products are equal.

Answer: True

Both ratios simplify to 3 : 1.

Answer: True

(i) 21 becomes 42 by multiplying by 2. Therefore 14 must also be multiplied by 2.

(ii) 14 becomes 6 by multiplying by 6/14 = 3/7. Apply the same factor to 21.

Answer: x = 28, y = 9

Simplify each ratio.

Since their simplest forms are identical, the ratios are proportional.

Answer: Yes, they are proportional.

First rectangle:

Second rectangle:

The corresponding side ratios match.

Answer: Yes, the rectangles are similar.

Total parts = 2 + 3 = 5.

Answer: ₹1,800 and ₹2,700

Total parts = 1 + 5 = 6.

Answer: 40 mL acid and 200 mL water

Convert 2 kg to 2000 g. The quantity factor is 2000 ÷ 500 = 4.

Answer: ₹1,680

Convert 2.5 L to 2500 mL.

2500 mL is 2500 ÷ 500 = 5 times 500 mL.

Answer: 75 seconds

Use C = (5/9)(F − 32).

Answer: 20°C

Cost of one notebook:

Cost of 12 notebooks:

Answer: ₹540

Capacity of one bus = 162 ÷ 3 = 54 people.

For 204 people:

So 3 buses are not enough; a fourth bus is required. The fourth bus will carry only 42 people.

Answer: 4 buses; no, all four will not be full.

Glasses and sugar are directly proportional when sweetness remains the same.

24 is 4 times 6, so sugar must also be multiplied by 4.

Cross-check: 6 × 40 = 240 and 10 × 24 = 240.

Answer: 40 spoons

Compare length : cement.

Both use one bag for every 20 ft of wall. Therefore the cement usage per unit length is the same.

Answer: No. Their length-to-cement ratios are proportional.

Total parts = 5 + 3 = 8.

Check: ₹1050 + ₹630 = ₹1680.

Answer: ₹1,050 and ₹630

Investment ratio:

Total parts = 4, so one part = ₹12,000 ÷ 4 = ₹3,000.

Answer: A = ₹9,000; B = ₹3,000

First find the original amounts. Total parts = 5 + 3 = 8.

Sand remains 30 kg. For the new ratio 3 : 2:

Cement to add = 20 − 18 = 2 kg.

Answer: 2 kg of cement

From the first pair:

The second pair confirms it: 30/4 = 7.5.

Therefore y = 7.5x.

Answer: 52.5, 75; constant y/x = 7.5

Find the cost per kilogram:

Then for 5 kg:

Equivalently, 2.4 : 168 :: 5 : x gives x = (5 × 168)/2.4 = 350.

Answer: ₹350

City A density:

City B density:

Since 4000 > 3000, City B has more people per square kilometre.

Answer: City B is more crowded.

Half an acre is one-half of 43,560 ft².

Answer: 21,780 square feet

Set up a direct proportion between money and quantity:

The money factor is:

Therefore the saffron quantity is also 21 times:

Answer: 52.5 palas

Step 1: Find the scale factor.

Step 2: Scale every ingredient by the same factor.

Step 3: Verify the rice : dal ratio.

The same factor was applied to both quantities, so the ratio is preserved.

Answer: 2.25 kg rice, 750 g dal, 1.875 kg vegetables; rice : dal remains 3 : 1.

Step 1: Add all wall lengths.

Step 2: Set up the proportion.

Step 3: Cross multiply.

Check by unit rate: 1450 ÷ 10 = 145 bricks/ft; 145 × 108 = 15,660.

Answer: Approximately 15,660 bricks

For equal volumes:

(i) For 1 L: water mass = 1 kg.

(ii) For 3 L of gold:

(iii) 3 L of water has mass 3 kg.

Answer: (i) 18.5 kg, (ii) 55.5 kg, (iii) 52.5 kg heavier

Step 1: Find the plot area.

Step 2: Convert the plot area to acres.

Step 3: Apply 10 tonnes per acre.

Step 4: Convert tonnes to kilograms.

Answer: About 22.96 tonnes, or 22,956.8 kg

Step 1: Split the mass in the ratio 3 : 1. Total parts = 4.

Step 2: Convert grams to kilograms.

Step 3: Find metal costs.

Answer: 5.805 g copper, 1.935 g nickel; total metal cost ≈ ₹7.85.

First convert/identify a common rate.

(i) 4 hours = 240 minutes.

(ii) Time for 216 km:

Answer: (i) 144 km, (ii) 360 minutes = 6 hours

(i) Original mixture: Total parts = 3 + 5 = 8.

(ii) After adding 20 mL yellow:

(iii) Restore 3 : 5. With 45 mL yellow, let required blue be x.

Blue already present = 15 mL, so blue to add = 27 − 15 = 12 mL.

Answer: (i) 15 mL blue, 25 mL yellow; (ii) 1 : 3; (iii) add 12 mL blue.

Step 1: Simplify the investment ratio.

Step 2: Find distributable profit.

Step 3: Share ₹28,000 in the ratio 3 : 2.

Check: ₹16,800 + ₹11,200 = ₹28,000.

Answer: A = ₹16,800; B = ₹11,200

Step 1: Convert volumes to mL.

(i) 10 L bucket:

(ii) 18 L tank:

Why proportional? The flow rate is constant, so multiplying the volume by a factor multiplies the filling time by the same factor.

Answer: (i) 5 min, (ii) 9 min

(a) Direct proportion. Price per pen is constant.

(b) Not a direct proportion over time.

Adding the same number to both ages does not preserve the ratio.

(c) Not direct proportion. For a fixed distance, increasing speed reduces time.

Thus speed changes 40 → 60 (factor 1.5), while time changes 3 → 2 (it decreases). The direct Rule of Three model must not be used as if both increased together.

Answer: (a) Direct, ₹270; (b) not direct; (c) not direct, 2 h.

6 : 9 = 2 : 3 and 14 : 21 = 2 : 3. Therefore both A and R are true, and R correctly explains A.

Answer: Option A

The assertion is false. Example: 2 : 3 becomes 7 : 8 after adding 5, and 2 : 3 ≠ 7 : 8.

The reason is true: equivalent ratios can be formed by multiplying both terms by the same non-zero factor.

Answer: Option D

The assertion is true because proportional ratios have equal cross products.

The reason is false; equality of sums is not a condition for proportion.

Answer: Option C

The assertion is false because the simplified ratios are different.

The reason is true:

Answer: Option D

Both cross products equal 60, so the ratios are proportional. The reason gives the exact test used to prove the assertion.

Answer: Option A

The assertion is false.

The reason is true and shows why the ratios are not proportional.

Answer: Option D

The assertion is true because:

The reason is also true, but it is unrelated to temperature conversion and does not explain the assertion.

Answer: Option B

60 : 40 = 3 : 2 and 90 : 60 = 3 : 2. Both statements are true, and the reason explains why the rectangles have the same width-to-height proportion.

Answer: Option A

The assertion is false. Example: ages 10 and 30 are 1 : 3, but after 10 years they are 20 : 40 = 1 : 2.

The reason is true.

Answer: Option D

Both statements are true. Add the two shares:

Thus R correctly explains A.

Answer: Option A

(i) Regular ratio:

Regular coffee has decoction fraction 3/(3+7) = 3/10 = 0.30.

(ii) Classification:

• A: 300 : 600 = 1 : 2. Decoction fraction = 1/3 ≈ 0.333, so stronger.
• B: 150 : 500 = 3 : 10. Decoction fraction = 3/13 ≈ 0.231, so lighter.
• C: 200 : 400 = 1 : 2. Decoction fraction = 1/3 ≈ 0.333, so stronger.
• D: 24 : 56 = 3 : 7, so regular.
• E: 100 : 300 = 1 : 3. Decoction fraction = 1/4 = 0.25, so lighter.

(iii) For regular coffee, 3 : 7 :: 60 : x.

Answer: Regular = 3 : 7; A strong, B light, C strong, D regular, E light; milk = 140 mL.

Original width : height:

(i)

(ii) For S:

Width doubles, so height must also double.

(iii) Proportional resizing requires multiplication by the same factor. Subtracting, for example, 20 mm gives 40 × 20, whose ratio is 2 : 1, not 3 : 2.

Answer: P and Q preserve shape; R does not; S height = 80 mm.

The quantities scale with the number of students.

(i) 120 → 80 has factor 80/120 = 2/3.

(ii) 120 → 200 has factor 200/120 = 5/3.

(iii) Verify ratio:

Answer: 80 students: 10 kg rice, 4 kg dal; 200 students: 25 kg rice, 10 kg dal; ratio remains 5 : 2.

(i) Compare ratios.

Since 1 : 30 ≠ 1 : 77, price is not proportional to volume for these packs.

(ii) Price per mL among bottles:

So the large bottle has the lowest unit price among the bottles.

(iii) Direct-proportion price based on small bottle:

Answer: Not proportional; large bottle is cheapest per mL among bottles; proportional 1000 mL price ≈ ₹855.56.

(i) Total parts = 8, so one part = 40/8 = 5 mL.

(ii) After adding 20 mL yellow:

(iii) Let final blue amount be x while yellow stays 45 mL.

Blue to add = 27 − 15 = 12 mL.

Answer: 15 mL blue, 25 mL yellow; new ratio 1 : 3; add 12 mL blue.

(i)

(ii)

So 4 buses are needed; the last has 42 people, so not all are full.

(iii)

Exactly 5 buses, all full.

(iv) Six buses hold 6 × 54 = 324 people, which is one short. Therefore 7 buses are required.

Answer: 54 per bus; 4 buses for 204 (not all full); 5 for 270; 7 for 325.

(i) Cost of 2400 ft²:

So the approximate cost is ₹82,645.

(ii) 0.25 acre:

(iii) With a constant price per square foot, multiplying area by a factor multiplies cost by the same factor, which is the defining pattern of direct proportion.

Answer: About ₹82,645; ₹3,75,000; cost is directly proportional to area at constant unit price.

(i) Delhi density:

Mumbai density:

(ii) Mumbai has the higher population per square kilometre, so it is more crowded by this measure.

(iii) Total population ignores land area. A city with fewer people can still be more crowded if those people occupy a much smaller area.

Answer: Delhi ≈ 20,216/km²; Mumbai ≈ 36,364/km²; Mumbai is more crowded.

(i)

(ii) Total ratio parts = 2 + 3 = 5. Convert 3 L = 3000 mL.

(iii)

Answer: 2 : 3; 1.2 L orange and 1.8 L apple; 77°F.

(i) Split 7.74 g in the ratio 3 : 1.

(ii) Convert to kilograms and price each metal.

(iii) For 1,000 coins:

Answer: 5.805 g Cu, 1.935 g Ni; ≈ ₹7.85 per coin; ≈ ₹7,854.17 for 1,000 coins.