Chapter 7 exercise answers: Proportional Reasoning-1

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(i) 4 : 7 :: 12 : 21

1. Cross-multiply: 4 × 21 = 84.
2. Cross-multiply the other way: 7 × 12 = 84.
3. 84 = 84, so the products match.

AnswerTrue

(ii) 8 : 3 :: 24 : 6

1. Cross-multiply: 8 × 6 = 48.
2. Cross-multiply the other way: 3 × 24 = 72.
3. 48 ≠ 72, so the products don't match.

AnswerFalse

(iii) 7 : 12 :: 12 : 7

1. Cross-multiply: 7 × 7 = 49.
2. Cross-multiply the other way: 12 × 12 = 144.
3. 49 ≠ 144.

AnswerFalse

(iv) 21 : 6 :: 35 : 10

1. Cross-multiply: 21 × 10 = 210.
2. Cross-multiply the other way: 6 × 35 = 210.
3. 210 = 210, so the products match.

AnswerTrue

(v) 12 : 18 :: 28 : 12

1. Cross-multiply: 12 × 12 = 144.
2. Cross-multiply the other way: 18 × 28 = 504.
3. 144 ≠ 504.

AnswerFalse

(vi) 24 : 8 :: 9 : 3

1. Cross-multiply: 24 × 3 = 72.
2. Cross-multiply the other way: 8 × 9 = 72.
3. 72 = 72, so the products match.

AnswerTrue

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Open NCERT p. 165Checked by computerAnswers can differ: one example

1. Multiply both terms of 4 : 9 by the same number each time.
2. ×2 gives 8 : 18.
3. ×3 gives 12 : 27.
4. ×4 gives 16 : 36.

Answer8 : 18, 12 : 27, and 16 : 36 (any three different multiples of 4 : 9 work).

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1. Simplify 18 : 24 to its simplest form: divide both terms by 6, giving 3 : 4.
2. Every ratio proportional to 18 : 24 must equal 3 : 4.
3. For first term 3: second term = 3 × 4⁄3 = 4.
4. For first term 12: second term = 12 × 4⁄3 = 16.
5. For first term 20: second term = 20 × 4⁄3 = 80/3.
6. For first term 27: second term = 27 × 4⁄3 = 36.

Answer4, 16, 80/3, and 36.

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Open NCERT p. 166Checked by computer

1. Measure each rectangle's longer side and shorter side with a scale.
2. Divide the longer side by the shorter side for each rectangle.
3. A and D both come out close to 3 (the long side is about three times the short side).
4. C and E both come out close to 5⁄2 = 2.5.
5. B comes out close to 3⁄2 = 1.5, which matches none of the others.
6. Rectangles with the same long-to-short ratio are similar, whatever their actual size.

AnswerA and D are similar to each other, and C and E are similar to each other; B is not similar to any of them.

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Open NCERT p. 166One way to think about it

1. Measure the given rectangle with a ruler. It is about 35 mm wide and 21 mm tall (your copy may differ slightly, so use your own measurements).
2. Its width to height ratio is 35 : 21. Divide both terms by 7 to get 5 : 3.
3. Smaller rectangle: multiply both sides by the same number less than 1, for example 3/7. You get 15 mm wide and 9 mm tall, and 15 : 9 = 5 : 3.
4. Bigger rectangle: multiply both sides by the same number greater than 1, for example 2. You get 70 mm wide and 42 mm tall, and 70 : 42 = 5 : 3.
5. Your classmates' rectangles will probably be different sizes, because each person may pick a different multiplying number. They are not wrong. Every rectangle whose width : height is 5 : 3 has the same shape.
6. A drawing is wrong only if its two sides were changed by different factors. For example, adding 10 mm to both sides gives 45 mm by 31 mm, and 45 : 31 is not 5 : 3.

In shortThere are many correct answers. One example: the given rectangle is about 35 mm by 21 mm (5 : 3), so a smaller one is 15 mm by 9 mm and a bigger one is 70 mm by 42 mm. Drawings of different sizes are all correct as long as width : height stays 5 : 3.

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(a)

1. The wall carries on beyond the picture, so count one block of the pattern that repeats. Do not count the whole picture, because it stops wherever the picture happens to stop.
2. Along every row the pattern repeats every 5 bricks. So take a block 5 bricks long and 3 rows deep, which is 15 bricks.
3. In that block, the top row has 2 grey and 3 coloured, the middle row has 3 grey and 2 coloured, and the bottom row has 4 grey and 1 coloured.
4. Grey = 2 + 3 + 4 = 9 and coloured = 3 + 2 + 1 = 6.
5. Grey : coloured = 9 : 6. Divide both terms by 3 to get 3 : 2.

Answer3 : 2

(b)

1. Along every row this pattern repeats every 4 bricks. So take a block 4 bricks long and 7 rows deep, which is 28 bricks.
2. In that block, the top and bottom rows each have 3 grey and 1 coloured. Each of the 5 middle rows has 2 grey and 2 coloured.
3. Grey = 3 + 3 + 5 × 2 = 16 and coloured = 1 + 1 + 5 × 2 = 12.
4. Grey : coloured = 16 : 12. Divide both terms by 4 to get 4 : 3.

Answer4 : 3

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Open NCERT p. 167One way to think about it

1. Measure your friend with a measuring tape. As an example only, suppose the head is 20 cm, the torso 50 cm, each arm 60 cm and each leg 80 cm. Your friend's numbers will be different.
2. Write the ratios. Head : torso = 20 : 50 = 2 : 5. Torso : arms = 50 : 60 = 5 : 6. Torso : legs = 50 : 80 = 5 : 8.
3. To fit the figure in your notebook, multiply every length by the same number, for example 1/10. That gives a head of 2 cm, a torso of 5 cm, arms of 6 cm and legs of 8 cm.
4. Check: 2 : 5, 5 : 6 and 5 : 8 are the same ratios as before, so the drawing is a proportional copy of your friend.
5. If you change the parts by different factors, for example a 5 cm torso with 12 cm legs (5 : 12 instead of 5 : 8), the figure looks stretched, like a photo that has been resized in only one direction.

In shortYes. The drawing looks more realistic when its ratios are the same as your friend's, because every part has been changed by the same factor and so the body keeps its shape. If different parts are scaled by different factors, the figure looks distorted. (Cartoons change the ratios on purpose, for example by drawing a very big head, and that is exactly why they do not look real.) Everyone's measurements will be different, so many answers are correct.

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1. Take a year as 52 weeks. It is really 52 weeks and 1 day, but 940 million km is itself only an approximate figure, so this is close enough.
2. One week is 1/52 of a year, so the Earth travels 1/52 of the year's distance.
3. 940,000,000 ÷ 52 = 235000000/13 ≈ 1,80,76,923 km.

AnswerAbout 1,80,76,923 km in a week, which is roughly 18 million km (exactly 235000000/13 km when a year is taken as 52 weeks).

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Open NCERT p. 170Checked by computerReads two ways: both answers shown

1. Bricks for 1 ft of wall = 1450 ÷ 10 = 145.
2. The top of the house is 9 + 15 = 24 ft long. The small part below is 6 ft wide and 9 ft deep, and starts under the left end of the 15 ft room.
3. Outer walls, going round the outside: 24 + 12 + 9 + 9 + 6 + 9 + 9 + 12 = 90 ft. (The bottom edge beside the small part is 24 − 15 = 9 ft on the right and 9 ft on the left.)
4. The inner wall between the 9 ft room and the 15 ft room is 12 ft long.
5. Reading 1 (the printed words: the outer walls and the inner wall that separates two rooms). We lead with this because it follows the question's words; the book gives no answer key. Wall = 90 + 12 = 102 ft, so bricks = 102 × 145 = 14790.
6. Reading 2 (the drawing also shows a 6 ft wall between the 15 ft room and the small part below it): wall = 102 + 6 = 108 ft, so bricks = 108 × 145 = 15660.

AnswerRead as the outer walls plus the one inner wall: about 14790 bricks. Read with the drawn 6 ft wall as well: about 15660 bricks.

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1. Add the ratio terms: 2 + 3 = 5 groups.
2. One group = ₹4,500 ÷ 5 = ₹900.
3. Smaller part = 2 groups = 2 × ₹900 = ₹1,800.
4. Larger part = 3 groups = 3 × ₹900 = ₹2,700.

Answer₹1,800 and ₹2,700.

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1. Add the ratio terms: 1 + 5 = 6 groups.
2. One group = 240 mL ÷ 6 = 40 mL.
3. Acid = 1 group = 40 mL.
4. Water = 5 groups = 5 × 40 = 200 mL.

Answer40 mL of acid and 200 mL of water.

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1. Add the ratio terms: 3 + 5 = 8 groups.
2. One group = 40 mL ÷ 8 = 5 mL.
3. Blue = 3 groups = 15 mL.
4. Yellow = 5 groups = 25 mL.
5. After adding 20 mL more yellow: new yellow = 25 + 20 = 45 mL; blue stays at 15 mL.
6. New ratio = 15 : 45 = 1 : 3.

Answer15 mL blue and 25 mL yellow to start with; after adding the extra yellow, the new ratio of blue to yellow is 1 : 3.

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1. Add the ratio terms: 2 + 1 = 3 groups.
2. One group = 6 cups ÷ 3 = 2 cups.
3. Rice = 2 groups = 4 cups.
4. Urad dal = 1 group = 2 cups.

Answer4 cups of rice and 2 cups of urad dal.

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1. In the original bucket, red : yellow = 3 : 5, so it holds 3 + 5 = 8 parts in all.
2. Assume the second bucket of yellow is the same size as the first bucket, i.e. 8 parts of pure yellow.
3. Red stays at 3 parts; yellow becomes 5 + 8 = 13 parts.
4. New ratio of red to yellow = 3 : 13.

Answer3 : 13 (assuming the added bucket is the same size as the original bucket of orange paint).

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1. Write the ratio: 600 : 900.
2. Divide both terms by their highest common factor, 300.
3. 600 ÷ 300 = 2 and 900 ÷ 300 = 3.

Answer2 : 3.

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1. Last year, 162 people exactly filled 3 buses, so one bus seats 162 ÷ 3 = 54 people.
2. This year, 204 ÷ 54 = 3.77…, which rounds UP to 4 buses — 3 buses would leave some students behind.
3. 4 buses can seat 4 × 54 = 216 people, but only 204 are travelling.
4. 216 is more than 204, so the buses will not all be full.

Answer4 buses; no, they will not all be full (12 seats will be empty).

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1. Crowding means people per square km, not the total number of people.
2. Delhi: 30,000,000 ÷ 1,484 ≈ 20,216 people per sq km.
3. Mumbai: 20,000,000 ÷ 550 ≈ 36,364 people per sq km.
4. Mumbai's people-per-area is bigger, even though its total population is smaller.

AnswerMumbai is more crowded — about 36,364 people per sq km against Delhi's about 20,216, even though Delhi has more people overall.

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1. Neck : whole body = 4 : (4 + 6) = 4 : 10.
2. For your own height h, neck length = 4⁄10 of h.
3. 4⁄10 simplifies to 2⁄5.

AnswerNeck length = (2/5) × h, i.e. two-fifths of your height.

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1. Saffron bought is directly proportional to money spent: 2½ palas for 3/7 niska.
2. Find the scaling factor from 3/7 niska up to 9 niskas: 9 ÷ (3/7) = 21.
3. Multiply the saffron amount by that same factor: 2½ × 21 = 52½ palas.

Answer52½ palas (105/2 palas).

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1. The gap between their ages never changes: 5 − 1 = 4 years, always.
2. When Harmain's age : brother's age = 1 : 2, the brother is exactly twice as old, so the gap equals Harmain's own age.
3. So Harmain's age = 4 years (and her brother will then be 8).

Answer4 years old.

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1. Equal volumes of gold and water have masses in the ratio 37 : 2.
2. 1 litre of water has mass 1 kg, matching the '2' part of the ratio.
3. So 1 litre of gold (the '37' part) has mass 37/2 = 18.5 kg.

Answer18.5 kg (37/2 kg).

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1. Plot area = 200 × 500 = 1,00,000 sq ft.
2. 1 acre = 43,560 sq ft, so the plot is 1,00,000 ÷ 43,560 = 2500/1089 acres.
3. Manure needed = 10 × 2500/1089 = 25000/1089 ≈ 22.96 tonnes.

Answer25000/1089 tonnes, about 22.96 tonnes.

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Open NCERT p. 177Matches NCERT’s answer

1. 10 litres = 10,000 mL (1 litre = 1,000 mL).
2. The bucket holds 10,000 ÷ 500 = 20 times as much as the mug.
3. So it takes 20 times as long: 15 × 20 = 300 seconds, i.e. 5 minutes.

Answer300 seconds, i.e. 5 minutes.

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1. 1 acre = 43,560 sq ft.
2. 2,400 sq ft is 2400/43560 of an acre.
3. Cost = ₹15,00,000 × 2400/43560 = ₹10000000/121 ≈ ₹82,644.63.

Answer₹10000000/121, which is about ₹82,644.63.

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1. Oxen take 6 hours per acre, so for 20 acres: 6 × 20 = 120 hours.
2. The tractor is 4 times as fast, so it takes 1/4 of the oxen's time: 120 ÷ 4 = 30 hours.

AnswerThe oxen would take 120 hours; the tractor would take 30 hours.

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1. Split 7.74 g in the ratio 3 : 1: total groups = 4, one group = 7.74 ÷ 4 = 1.935 g.
2. Copper mass = 3 groups = 5.805 g = 0.005805 kg.
3. Nickel mass = 1 group = 1.935 g = 0.001935 kg.
4. Cost of copper = 0.005805 × ₹906 = ₹5.25933.
5. Cost of nickel = 0.001935 × ₹1,341 = ₹2.594835.

AnswerCopper costs about ₹5.26 and nickel costs about ₹2.59 (exactly ₹5.25933 and ₹2.594835).

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