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According to the question,

∵ 25 m of cloth cost = ₹ 337.50

∴ 1m of cloth cost = ₹ 337.50/25

a) Cost of 40 m of the same type of cloth = 337.50/25 x 40 = 13500/25 = ₹540

b) The length of cloth bought for ₹ 810 = 810/337.50 x 25 = 20250/337.50 = 60 cm

Correct option is (A) xy = k

\(x\propto\frac1y\)

\(\Rightarrow\) \(x=\frac ky\)

\(\Rightarrow\) xy = k

Correct option is  A) xy = k

More the dolls, more will be the cost. So it is a direct proportion.

Let no. of dolls be x, \(\frac{18}{630}=\frac{\text{x}}{455}\)

⇒ 630 × x = 18 × 455

⇒ x = \(\frac{18\,\times\,455}{630}\) = 13

False

When the distance is kept fixed, speed and time vary indirectly/inversely with each other. Since, if we increase speed, then taken time will less and vice-versa.

More the amount of sugar, more will be the cost. So it is a direct proportion.

Let the amount of sugar be x kg, \(\frac{9}{238.50}=\frac{\text{x}}{371}\)

⇒ 238.50 × x = 9 × 371

⇒ x = \(\frac{9\,\times\,371}{238.50}\) = 14kg

False

Length of a side of an equilateral triangle and its perimeter vary directly with each other, e.g. Let a be the side of an equilateral triangle. So, perimeter = 3 x (Side) = 3 x a = 3a .

So, if we increase the length of side of the equilateral triangle, then their perimeter will also increases.

⇒ 15 × 6 = 9 × y

⇒ y = \(\frac{15\,\times\,6}{9}\) = 10

False

Length of a side of a square and its area does not vary directly with each other, e.g. Let a be length of each side of a square.

So, area of the square = side² = a²

So, if we increase the length of the side of a square, then their area increases but not directly.

More the amount of oranges, more will be the cost. So it is a direct proportion.

Let the amount be x, \(\frac{8}{52} = \frac{x}{169}\)

x =  \(\frac{8\,\times\,169}{52}\) = Rs 26

We use the relation \(\frac{\text{x}}{y}=\frac{\text{x}}{Y}\) Here x₁ = 3, y₁ = 36 and y₂ = 96 

Here, \(\frac{3}{36}=\frac{\text{x}{_1}}{96}\)

⇒ x₁ × 36 = 3 × 96

⇒ x₁ = \(\frac{3\,\times\,96}{36}\) = 8

(a) If land to be uniformly fertile, then the area of land and the yield on it vary directly with each other.

Hence, option (a) is correct.

Note:

Two quantities x and y are said to be in direct proportion, if they increase or decrease together in such a manner that the ratio of their corresponding values remains constant.

Option (a)

Explanation: 

In option (a), the values of x is directly proportional to values of y, such as;

y = 4x

If we put the values of x = 0.5, 2, 8 and 32, we get the values of y as 2, 8, 32 and 128 respectively.

If two quantities x and y vary directly with each other, then x/y = k = constant.

Since, in direct proportion, both x and y increases or decreases together such a manner that the ratio of their corresponding value remains constant.

Hence, option (a) is correct

If two quantities p and q vary inversely with each other then product of their corresponding values remains constant.

If on increasing a, b decreases in such a manner that ab remains constant and positive, then a and b are said to vary inversely with each other.

(d) 45 and 1.3

Explanation: 

Since x and y vary inversely, so

xy = k (constant)

Putting the value of x and y, we get;

10×6 = 60

Now if we observe the options available;

Option (a) – 12 and 5 – 12×5 = 60

Option (b) – 15 and 4 – 15×4 = 60

Option (c) – 25 and 2.4 – 25×2.4 = 60

Option (d) – 45 and 1.3 = 45×1.3 = 58.3

Therefore, option (d) is not a possible pair of corresponding values of x and y

If two quantities x and y vary directly with each other, then ratio of their corresponding values remains constant.

If distance increases the fare also increases therefore it’s directly proportional.

Let us consider the required fare as ₹x, 2550/150 = x/124

150 × x = 2550 × 124

150x = 316200

x = 316200/150

x = ₹2108

∴ ₹2108 is the charge for journey of 124 km

Fare increases as the distance of the journey increases. So it is a direct proportion.

Let required fare be Rs x, \(\frac{2550}{150}=\frac{\text{x}}{124}\)

⇒ 50 × x = 2550 × 124

⇒ x = \(\frac{2550\,\times\,124}{150}\) = Rs. 2108

Rice : Cost :: Rice : Cost

7 kg : Rs.1120 :: x kg : Rs.3680

x = (7 x 3680/1120) = 23 kg

Notebooks : Cost :: Notebooks : Cost

6 : Rs.156 :: 54 : Rs. x

x = (156 x 54/6) = Rs. 1404

Let’s consider number of men required be ‘x’ men

(27/4)/11 = 27/x

By cross-multiplying

x27/4 = 27(11)

x27/4 = 297

x = (297 × 4)/27

= 44

∴ To dig 27 m long trench of same type in one day requires 44 men.

Arc length = 15 cm 

Angle subtend = 45°

= \(\frac{45\timesπ}{180°}\) = \(\frac{π}{4}\) radius

radius of circle = \(\frac{arc\,length}{angle\,subtend\,at\,centre}\)

= \(\frac{15\times4}{π}\) = \(\frac{60°}{π}\) cm

Given,

Length of arc = 20π cm

And. θ = Angle subtended at the centre of circle = 144°

We know that,

Length of arc = θ/360 ∗ 2πr cm

θ/360 ∗ 2πr cm = 144/360 ∗ 2πr cm = 4π/5 ∗ r cm

From the question, we can equate

20π cm = 4π/5 ∗ r cm

r = 25 cm.

Thus, the radius of the circle is 25 cm.

Given,

Radius of small quadrant, r = 2 cm

Radius of big quadrant, R = 3.5 cm

(i) Area of quadrant OACB = 1/4 πR²

= 1/4 (22/7)(3.5)²

= 269.5/28 = 9.625 cm²

(ii) Area of shaded region = Area of big quadrant – Area of small quadrant

= 1/4 π(R² – r²)

= 1/4 (22/7)(3.5² – 2²)

= 1/4 (22/7)(12.25 – 4)

= 1/4(22/7)(8.25)

= 6.482 cm²

Arc length = \(\frac{5π}3\) cm

Radius of circle = 5 cm 

Formula: 

Arc length = r × q  

r = radius of circle 

q = angle subtended by arc at the center

= \(\frac{5π}3\) = 5 x q

q = \(\frac{5π}{3\times5}\) = \(\frac{π}3\) = \(\frac{180}3\) = 60°

(c) 35

Cost of 1 Orange = 568 = ₹ 7

Cost of 5 Orange = 7 × 5 = ₹ 35

In my purchase cost of 6 apples = ₹ 90

Cost of 1 apple = 90/6 = ₹ 15

In my friend’s purchase cost of 5 apples = ₹ 70

Cost of 1 apple = 70/5 = ₹ 14

My friend buy for lower prize.

My friend’s purchase is a better buy.

(i) 3 : 5

(ii) 3 : 2

Hint: 3m = 300 cm

(iii) 9 : 10

Hint: 5km 400 m = 5400m and 6 km = 6000 m

(iv) 3 : 8

Hint: ₹ 2 = 200 paise

Here the extremes are 3 and 20 and the means are 2 and 30.

Product of extremes, ad = 3 × 20 = 60.

Product of means, bc = 2 × 30 = 60.

Thus by proportionality law, we find ad = bc and hence 3 : 2 and 30 : 20 are in proportion.

(i) False

(ii) True

(d) 20/28

Hint: 16/24 = (8×2)/(8×3) = 2/3

(i) 78 : 130 :: 12 : 20

Extreme terms are 78 and 20.

Mean terms are 130 and 12.

Product of Extremes = 78 × 20 = 1560

Product of Means = 130 × 12 = 1560

Product of Extremes = Product of means

It is in proportion.

(ii) 400 : 50 :: 25 : 625

Product of extremes = 400 × 625 = 250,000

Product of means = 50 × 25 = 1250

Here product of extremes ≠ product of means

400 : 50 and 25 : 625 are not in proportion.

A : B = 2 : 1

B : C = 2 : 1

They are in proportion.

Total length = 63 cm Ratio = 3 : 4

Sum of the ratio = 3 + 4 = 7

7 parts = 63 cm

1 part = 63/7 = 9 cm

3 parts = 3 × 9 cm = 27 cm

4 parts = 4 × 9 cm = 36 cm

∴ 63 cm can be divided into the parts as 27 cm and 36 cm.

(d) 8

5x = 2 × 20 

⇒ x = 8

(c) 2 : 5, 10 : 25

(d) 12 : 21

An equivalent ratio of 4 : 7 is 12 : 21.

(c) 35

If 7 : 5 is in proportion to x : 25, then ‘x’ is 35.

(i) False

(ii) False

(iii) False

(a) 6

Hint: 2/3 = (2×2)/(3×2) = 4/6

(c) 7/49

The equivalent fraction of 17 is 7/49.

Continents are very large land masses that are surrounded by vast water bodies called oceans on all sides. They are the primary divisions of land. There are seven continents in the world. Asia is the largest continent. It occupies about one-third of the land area of our planet. It is joined to the land mass of Europe and, thus, we use the term Eurasia for this combined land mass. The Ural Mountains, the Black Sea and the Caspian Sea separate the two continents. We find oceans on three sides of this land mass. To its north lies the Arctic Ocean, to its east is the Pacific Ocean and to its south is the Indian Ocean. Africa is the second largest continent after Asia. It is surrounded by water bodies on all sides. In the north, the Mediterranean Sea separates it from Europe, and on its east is the Indian Ocean. The Atlantic Ocean in the west separates it from the two Americas. In Africa lies the largest desert of the world, the Sahara Desert.

The major landforms on earth are :

• Mountains — Most of the rivers originate from mountains. They are rich in minerals and metals deposits and support variety of flora and fauna.
•  Plateau — They are storehouses of minerals such as gold, silver, iron, copper, manganese, etc.
• Valley — They are low-lying lands between hills formed by river flowing down the mountains or due to movement of earth plates.
•  Plains — They are the most fertile landforms which support easy habitation.

The Isthmus of Panama joins North America and South America with the Pacific Ocean on one side and the Atlantic Ocean on the other. The Isthmus of Suez joins Africa to Asia and separates the Mediterranean Sea and the Red Sea.

Importance of Mountains:

1. Mountains are a storehouse of water. Many rivers originate in the glaciers in the mountains. 

2. Reservoirs are made and the water from the mountains is used for irrigation and generation of hydro-electricity. 

3. The river valleys and terraces are most suitable for farming and cultivation of crops as the land is very fertile. 

4. Mountains support a rich variety of flora and fauna. 

5. Mountains also affect the climate of an area. For example, the Himalayas cause rainfall in India by blocking the south-west monsoon winds. They also protect us from the cold winds of Central Asia in winter. 

6. Mountains are also rich in mineral and metal deposits which are essential for industries. 

7. According to the United Nations Development Programme, mountains provide home for around 720 million people. 

8. The forests in the mountainous regions provide fuel, fodder, shelter and other products like gum, wild fruits, mushrooms, resins, etc. mountains provide an ideal holiday for tourists. They visit the for their natural scenic beauty and relaxation. 

9. Many recreation, sporting and tourism activities takes place in the mountains. Paragliding, hang gliding, river rafting and skiing are popular sports in the mountains.

Population of India may be classified into rural and urban population depending on where the people live. 

Rural population — They live in villages. They generally earn their livelihood from agriculture, livestock rearing and other primary activities. In India, more people live in rural areas (about 69 per cent). In recent years, a large number of people have migrated from rural to urban areas in search of job, good education and better living conditions. Urban population — They live in cities and towns. They earn their livelihood from jobs in offices, factories, trade, transport and services. In India, about 31 per cent people live in urban areas. Migration from rural to urban areas have had an impact on the population of urban areas.