7. Proportional Reasoning-1

Its width and height have not changed by the same factor.

Compared to A (60 × 40), B (40 × 20) has height halved but width not halved. So B looks stretched.

Width: 90 ÷ 60 = 32. Height: 60 ÷ 40 = 32.

Yes, both change by the same factor 32. So D looks similar to A.

60 × 32 = 90 and 40 × 32 = 60. The factor is 32

Now: 3 : 30 = 1 : 10. After 9 years: 12 : 39 = 4 : 13.

No, the ratio does not remain the same. Adding the same number to both terms changes the ratio.

Each cup is 50 mL. In 20 : 30, there is more decoction (20 mL) than in regular coffee (15 mL). So it is stronger.

In 10 : 40, there is less decoction (10 mL). So it is lighter.

Regular coffee is 15 : 35 = 3 : 7, that is, 3 parts decoction in every 10 parts. A coffee with more than 3 parts decoction out of 10 is strong, and with less is light.

(i) 4 : 7 :: 12 : 21 → 12 : 21 = 4 : 7. True

(ii) 8 : 3 :: 24 : 6 → 24 : 6 = 4 : 1. False

(iii) 7 : 12 :: 12 : 7 → False

(iv) 21 : 6 :: 35 : 10 → both are 7 : 2. True

(v) 12 : 18 :: 28 : 12 → 2 : 3 and 7 : 3. False

(vi) 24 : 8 :: 9 : 3 → both are 3 : 1. True

8 : 18   12 : 27   16 : 36

18 : 24 = 3 : 4 in simplest form.

3 : 4   12 : 16   20 : 2623   27 : 36

For 20 : ?, the factor is 20 ÷ 3 = 203. So ? = 4 × 203 = 803 = 2623.

On measuring, the ratio of the longer side to the shorter side is about:

A = 3 : 1, B = 3 : 2, C = 5 : 2, D = 3 : 1, E = 5 : 2.

So A and D are similar, and C and E are similar. B is not similar to any other rectangle.

Measure the given rectangle, for example 3 cm × 2 cm (ratio 3 : 2). A smaller one can be 1.5 cm × 1 cm and a bigger one 6 cm × 4 cm.

Classmates’ rectangles may have different sizes. They are not wrong if the ratio of width to height is the same. They will all look similar.

(a) In one repeating block: grey = 2 + 3 + 4 = 9, red = 3 + 2 + 1 = 6. Ratio = 9 : 6 = 3 : 2

(b) In one repeating block: grey = 16, coloured = 12. Ratio = 16 : 12 = 4 : 3

Sample: head 22 cm, torso 55 cm, arms 66 cm, legs 88 cm.

head : torso = 22 : 55 = 2 : 5; torso : arms = 55 : 66 = 5 : 6; torso : legs = 55 : 88 = 5 : 8.

Yes, the drawing looks more realistic when the ratios are proportional, because the parts of a real body are in these ratios.

4 hours = 240 minutes. 150 : 90 :: 240 : x.

150 × x = 240 × 90, so x = 21600 ÷ 150 = 144 km

Another way: In 30 minutes it goes 18 km. 240 minutes has 8 such parts. 8 × 18 = 144 km.

Himachal: 200 g for ₹200, so 1 kg costs ₹1,000. Meghalaya: 1 kg costs ₹800.

So the Himachal Pradesh tea is more expensive. A small farmer grows less tea, often by hand, so the cost per kg is higher.

1 year ≈ 52 weeks. 940000000 : 52 :: x : 1.

x = 940000000 ÷ 52 ≈ 1,80,76,923 km (about 18 million km).

Total length of walls = 24 (top) + 12 + 12 (two sides) + 24 (bottom of big part) + 12 (inner wall) + 9 + 9 + 6 (small room) = 108 ft.

10 : 1450 :: 108 : x. So x = 1450 × 108 ÷ 10 = 15,660 bricks

Distance = 50 × 2 = 100 km. Time at 75 km/h = 100 ÷ 75 = 113 hours = 1 hour 20 minutes

It takes less time. We cannot write 50 : 2 :: 75 : __, because when speed increases, time decreases. They are not in direct proportion.

6 : 180 = 1 : 30, but 2 : 154 = 1 : 77. So they are not proportional

Price per mL: sachet ≈ ₹0.33, small ≈ ₹0.86, medium ≈ ₹0.81, large = ₹0.54. Companies set prices based on packing cost, demand and convenience, not only on quantity.

Large bottles use less plastic per mL. To reduce the ecological footprint, the company can offer refills, and customers can buy larger packs or refills instead of many sachets.

You get 12 − 5 = 7. The ratio is 5 : 7.

42 in the ratio 4 : 3: total parts = 7, each part = 42 ÷ 7 = 6. Shares = 4 × 6 = 24 and 3 × 6 = 18

Total parts = 5. One part = 4500 ÷ 5 = 900.

Parts = 2 × 900 = ₹1,800 and 3 × 900 = ₹2,700

Total parts = 6. One part = 240 ÷ 6 = 40 mL.

Acid = 40 mL, water = 5 × 40 = 200 mL

One part = 40 ÷ 8 = 5 mL. Blue = 15 mL, yellow = 25 mL

New yellow = 25 + 20 = 45 mL. New ratio = 15 : 45 = 1 : 3

One part = 6 ÷ 3 = 2 cups.

Rice = 4 cups, urad dal = 2 cups

Take one bucket as 8 parts. Red = 3 parts, yellow = 5 parts.

The added bucket is 8 more parts of yellow. Yellow = 5 + 8 = 13 parts.

New ratio = 3 : 13

600 : 900 = 2 : 3

One bus carries 162 ÷ 3 = 54 people.

204 ÷ 54 = 3 buses and 42 people left. So we need 4 buses

4 buses have 216 seats. So 12 seats will be empty; all buses will not be full.

Delhi: 3,00,00,000 ÷ 1484 ≈ 20,216 people per sq. km.

Mumbai: 2,00,00,000 ÷ 550 ≈ 36,364 people per sq. km.

Mumbai is more crowded because more people live in each square kilometre.

Neck = 410 of the height. (For the crane, neck = 155 × 410 = 62 cm.)

If your height is 150 cm, your neck would be 150 × 410 = 60 cm.

37 : 212 :: 9 : x.

x = 212 × 9 ÷ 37 = 52 × 9 × 73 = 3156 = 5212 palas

After x years: (1 + x) : (5 + x) = 1 : 2.

2(1 + x) = 5 + x, so x = 3.

Harmain will be 4 years old (brother 8).

37 : 2 :: x : 1. So x = 37 ÷ 2 = 18.5 kg

Area = 200 × 500 = 1,00,000 sq. ft. 1 acre = 43,560 sq. ft.

Manure = 10 × 1,00,000 ÷ 43,560 ≈ 22.96 tonnes (about 23 tonnes).

10 litres = 10,000 mL = 20 mugs.

Time = 20 × 15 = 300 seconds = 5 minutes

Cost = 15,00,000 × 2400 ÷ 43560 ≈ ₹82,645

Oxen: 20 × 6 = 120 hours

Tractor: 6 ÷ 4 = 1.5 hours per acre. 20 × 1.5 = 30 hours

Copper = 7.74 × 34 = 5.805 g. Nickel = 7.74 × 14 = 1.935 g.

Copper cost = 5.805 × 906 ÷ 1000 ≈ ₹5.26. Nickel cost = 1.935 × 1341 ÷ 1000 ≈ ₹2.59.

Total cost of metals ≈ ₹7.85

Each row and column has 3 horizontal and 3 vertical lines, no three same lines are next to each other, and all rows and columns are different.