Exercise 2.1 | Class 8th Mathematics

1.Solve the equation: x – 2 = 7.
Solution:
Given: x – 2 = 7
⇒ x – 2 + 2 = 7 + 2 (adding 2 on both sides)
⇒ x = 9 (Required solution)

2.Solve the equation: y + 3 = 10.
Given: y + 3 = 10
⇒ y + 3 – 3 = 10 – 3 (subtracting 3 from each side)
⇒ y = 7 (Required solution)

3.Solve the equation: 6 = z + 2
Solution:
We have 6 = z + 2
⇒ 6 – 2 = z + 2 – 2 (subtracting 2 from each side)
⇒ 4 = z
Thus, z = 4 is the required solution.

4.Solve the equation 6x = 12.
Solution:
We have 6x = 12
⇒ 6x ÷ 6 = 12 ÷ 6 (dividing each side by 6)
⇒ x = 2
Thus, x = 2 is the required solution.

5.Solve the equation t5 = 10.
Solution:
Given t5 = 10
⇒ t5 × 5 = 10 × 5 (multiplying both sides by 5)
⇒ t = 50
Thus, t = 50 is the required solution.

6.Solve the equation 2×3 = 18.
Solution:
We have 2×3 = 18
⇒ 2×3 × 3 = 18 × 3 (multiplying both sides by 3)
⇒ 2x = 54
⇒ 2x ÷ 2 = 54 ÷ 2 (dividing both sides by 2)
⇒ x = 27
Thus, x = 27 is the required solution.

7.Solve the equation 1.6 = y1.5
Solution:
Given: 1.6 = y1.5
⇒ 1.6 × 1.5 = y1.5 × 1.5 (multiplying both sides by 1.5)
⇒ 2.40 = y
Thus, y = 2.40 is the required solution.

8.Solve the equation 7x – 9 = 16.
Solution:
We have 7x – 9 = 16
⇒ 7x – 9 + 9 = 16 + 9 (adding 9 to both sides)
⇒ 7x = 25
⇒ 7x ÷ 7 = 25 ÷ 7 (dividing both sides by 7)
⇒ x = 257
Thus, x = 257 is the required solution.

9.Solve the equation 14y – 8 = 13.
Solution:
We have 14y – 8 = 13
⇒ 14y – 8 + 8 = 13 + 8 (adding 8 to both sides)
⇒ 14y = 21
⇒ 14y ÷ 14 = 21 ÷ 14 (dividing both sides by 14)
⇒ y = 2114
⇒ y = 32
Thus, y = 32 is the required solution.

10.Solve the equation 17 + 6p = 9.
Solution:
We have, 17 + 6p = 9
⇒ 17 – 17 + 6p = 9 – 17 (subtracting 17 from both sides)
⇒ 6p = -8
⇒ 6p ÷ 6 = -8 ÷ 6 (dividing both sides by 6)
⇒ p = −86
⇒ p = −43
Thus, p = −43 is the required solution.

 Exercise 2.2 | Class 8th Mathematics

1.The perimeter of a rectangular swimming pool is 154 m. Its length is 2 m more than twice its breadth. What are the length and the breadth of the pool?
Solution:
Let the breadth of the pool be x m.
Condition I: Length = (2x + 2) m.
Condition II: Perimeter = 154 m.
We know that Perimeter of rectangle = 2 × [length + breadth]
2 × [2x + 2 + x] = 154
⇒ 2 × [3x + 2] = 154
⇒ 6x + 4 = 154 (solving the bracket)
⇒ 6x = 154 – 4 [Transposing 4 from (+) to (-)]

⇒ 6x = 150
⇒ x = 150 ÷ 6 [Transposing 6 from (×) to (÷)]
⇒ x = 25
Thus, the required breadth = 25 m

and the length = 2 × 25 + 2 = 50 + 2 = 52 m.

2.Sum of two numbers be 95. If one exceeds the other by 15, find the numbers.
Solution:
Let one number be x
Other number = x + 15
As per the condition of the question, we get
x + (x + 15) = 95
⇒ x + x + 15 = 95
⇒ 2x + 15 = 95
⇒ 2x = 95 – 15 [transposing 15 from (+) to (-)]
⇒ 2x = 80
⇒ x = 802 [transposing 2 from (×) to (÷)]
⇒ x = 40
Other number = 95 – 40 = 55
Thus, the required numbers are 40 and 55.

3.Two numbers are in the ratio 5 : 3. If they differ by 18, what are the numbers?
Solution:
Let the two numbers be 5x and 3x.
As per the conditions, we get
5x – 3x = 18
⇒ 2x = 18
⇒ x = 18 ÷ 2 [Transposing 2 from (×) to (÷)]
⇒ x = 9.
Thus, the required numbers are 5 × 9 = 45 and 3 × 9 = 27

4.Three consecutive integers add up to 51. What are these integers?
Solution:
Let the three consecutive integers be x, x + 1 and x + 2.
As per the condition, we get
x + (x + 1) + (x + 2) = 51
⇒ x + x + 1 + x + 2 = 51
⇒ 3x + 3 = 51
⇒ 3x = 51 – 3 [transposing 3 to RHS]
⇒ 3x = 48
⇒ x = 48 ÷ 3 [transposing 3 to RHS]
⇒ x = 16
Thus, the required integers are 16, 16 + 1 = 17 and 16 + 2 = 18, i.e., 16, 17 and 18.

5.The sum of three consecutive multiples of 8 is 888. Find the multiples.
Solution:
Let the three consecutive multiples of 8 be 8x, 8x + 8 and 8x + 16.
As per the conditions, we get
8x + (8x + 8) + (8x + 16) = 888
⇒ 8x + 8x + 8 + 8x + 16 = 888
⇒ 24x + 24 = 888
⇒ 24x = 888 – 24 (transposing 24 to RHS)
⇒ 24x = 864
⇒ x = 864 ÷ 24 (transposing 24 to RHS)

⇒ x = 36
Thus, the required multiples are
36 × 8 = 288, 36 × 8 + 8 = 296 and 36 × 8 + 16 = 304,
i.e., 288, 296 and 304.

6.Three consecutive integers are such that when they are taken in increasing order and multiplied by 2, 3, and 4 respectively, they add up to 74. Find these numbers.
Solution:
Let the three consecutive integers be x, x + 1 and x + 2.
As per the condition, we have
2x + 3(x + 1) + 4(x + 2) = 74
⇒ 2x + 3x + 3 + 4x + 8 = 74
⇒ 9x + 11 = 74
⇒ 9x = 74 – 11 (transposing 11 to RHS)
⇒ 9x = 63
⇒ x = 63 ÷ 9
⇒ x = 7 (transposing 7 to RHS)
Thus, the required numbers are 7, 7 + 1 = 8 and 7 + 2 = 9, i.e., 7, 8 and 9.

7.The ages of Rahul and Haroon are in the ratio 5 : 7. Four years later the sum of their ages will be 56 years. What are their present ages?
Solution:
Let the present ages of Rahul and Haroon he 5x years and 7x years respectively.
4 years later, the age of Rahul will be (5x + 4) years.
4 years later, the age of Haroon will be (7x + 4) years.
As per the conditions, we get
(5x + 4) + (7x + 4) = 56
⇒ 5x + 4 + 7x + 4 = 56
⇒ 12x + 8 = 56
⇒ 12x = 56 – 8 (transposing 8 to RHS)
⇒ 12x = 48
⇒ x = 48 ÷ 12 = 4 (transposing 12 to RHS)
Hence, the required age of Rahul = 5 × 4 = 20 years.
and the required age of Haroon = 7 × 4 = 28 years.

8.The number of boys and girls in a class are in the ratio 7 : 5. The number of boys is 8 more than the numbers of girls. What is the total class strength?
Solution:
Let the number of boys be 7x
and the number of girls be 5x
As per the conditions, we get
7x – 5x = 8
⇒ 2x = 8
⇒ x = 8 ÷ 2 = 4 (transposing 2 to RHS)
the required number of boys = 7 × 4 = 28
and the number of girls = 5 × 4 = 20
Hence, total class strength = 28 + 20 = 48

9.Baichung’s father is 26 years younger than Baichung’s grandfather and 29 years older than Baichung. The sum of the ages of all the three is 135 years. What is the age of each one of them?
Solution:
Let the age of Baichung be x years.
The age of his father = x + 29 years,
and the age of his grandfather = x + 29 + 26 = (x + 55) years.
As per the conditions, we get
x + x + 29 + x + 55 = 135
⇒ 3x + 84 = 135
⇒ 3x = 135 – 84 (transposing 84 to RHS)
⇒ 3x = 51
⇒ x = 51 ÷ 3 (transposing 3 to RHS)
⇒ x = 17
Hence Baichung’s age = 17 years
Baichung’s father’s age = 17 + 29 = 46 years,
and grand father’s age = 46 + 26 = 72 years.

10.Fifteen years from now Ravi’s age will be four times his present age. What is Ravi’s present age?
Solution:
Let the present age of Ravi be x years.
After 15 years, his age will be = (x + 15) years
As per the conditions, we get
⇒ x + 15 = 4x
⇒ 15 = 4x – x (transposing x to RHS)
⇒ 15 = 3x
⇒ 15 ÷ 3 = x (transposing 3 to LHS)
⇒ x = 5
Hence, the present age of Ravi = 5 years.

Thus, the number of winners = 19

 Exercise 2.3 | Class 8th Mathematics

1.3x = 2x + 18
Solution:
We have 3x = 2x + 18
⇒ 3x – 2x = 18 (Transposing 2x to LHS)
⇒ x = 18
Hence, x = 18 is the required solution.
Check: 3x = 2x + 18
Putting x = 18, we have
LHS = 3 × 18 = 54
RHS = 2 × 18 + 18 = 36 + 18 = 54
LHS = RHS
Hence verified.

2.5t – 3 = 3t – 5
Solution:
We have 5t – 3 = 3t – 5
⇒ 5t – 3t – 3 = -5 (Transposing 3t to LHS)
⇒ 2t = -5 + 3 (Transposing -3 to RHS)
⇒ 2t = -2
⇒ t = -2 ÷ 2
⇒ t = -1
Hence t = -1 is the required solution.
Check: 5t – 3 = 3t – 5
Putting t = -1, we have
LHS = 5t – 3 = 5 × (-1)-3 = -5 – 3 = -8
RHS = 3t – 5 = 3 × (-1) – 5 = -3 – 5 = -8
LHS = RHS
Hence verified.

3. 5x + 9 = 5 + 3x
Solution:
We have 5x + 9 = 5 + 3x
⇒ 5x – 3x + 9 = 5 (Transposing 3x to LHS) => 2x + 9 = 5
⇒ 2x = 5 – 9 (Transposing 9 to RHS)
⇒ 2x = -4
⇒ x = -4 ÷ 2 = -2
Hence x = -2 is the required solution.
Check: 5x + 9 = 5 + 3x
Putting x = -2, we have
LHS = 5 × (-2) + 9 = -10 + 9 = -1
RHS = 5 + 3 × (-2) = 5 – 6 = -1
LHS = RHS
Hence verified.

4. 4z + 3 = 6 + 2z
Solution:
We have 4z + 3 = 6 + 2z
⇒ 4z – 2z + 3 = 6 (Transposing 2z to LHS)
⇒ 2z + 3 = 6
⇒ 2z = 6 – 3 (Transposing 3 to RHS)
⇒ 2z = 3
⇒ z = 32
Hence z = 32 is the required solution.
Check: 4z + 3 = 6 + 2z
Putting z = 32, we have
LHS = 4z + 3 = 4 × 32 + 3 = 6 + 3 = 9
RHS = 6 + 2z = 6 + 2 × 32 = 6 + 3 = 9
LHS = RHS
Hence verified.

5. 2x – 1 = 14 – x
Solution:
We have 2x – 1 = 14 – x
⇒ 2x + x = 14 + 1 (Transposing x to LHS and 1 to RHS)
⇒ 3x = 15
⇒ x = 15 ÷ 3 = 5
Hence x = 5 is the required solution.
Check: 2x – 1 = 14 – x
Putting x = 5
LHS we have 2x – 1 = 2 × 5 – 1 = 10 – 1 = 9
RHS = 14 – x = 14 – 5 = 9
LHS = RHS
Hence verified.

6. 8x + 4 = 3(x – 1) + 7
Solution:
We have 8x + 4 = 3(x – 1) + 7
⇒ 8x + 4 = 3x – 3 + 7 (Solving the bracket)
⇒ 8x + 4 = 3x + 4
⇒ 8x – 3x = 4 – 4 [Transposing 3x to LHS and 4 to RHS]
⇒ 5x = 0
⇒ x = 0 ÷ 5 [Transposing 5 to RHS]
or x = 0
Thus x = 0 is the required solution.
Check: 8x + 4 = 3(x – 1) + 7
Putting x = 0, we have
8 × 0 + 4 = 3(0 – 1) + 7
⇒ 0 + 4 = -3 + 7
⇒ 4 = 4
LHS = RHS
Hence verified.

7. x = 45 (x + 10)
Solution:
We have x = 45 (x + 10)
⇒ 5 × x = 4(x + 10) (Transposing 5 to LHS)
⇒ 5x = 4x + 40 (Solving the bracket)
⇒ 5x – 4x = 40 (Transposing 4x to LHS)
⇒ x = 40
Thus x = 40 is the required solution.
Check: x = 45 (x + 10)
Putting x = 40, we have
40 = 45 (40 + 10)
⇒ 40 = 45 × 50
⇒ 40 = 4 × 10
⇒ 40 = 40
LHS = RHS
Hence verified.

8. 2×3 + 1 = 7×15 + 3
Solution:
We have 2×3 + 1 = 7×15 + 3
15(2×3 + 1) = 15(7×15 + 3)
LCM of 3 and 15 is 15
2×3 × 15 + 1 × 15 = 7×15 × 15 + 3 × 15 [Multiplying both sides by 15]
⇒ 2x × 5 + 15 = 7x + 45
⇒ 10x + 15 = 7x + 45
⇒ 10x – 7x = 45 – 15 (Transposing 7x to LHS and 15 to RHS)
⇒ 3x = 30
⇒ x = 30 ÷ 3 = 10 (Transposing 3 to RHS)
Thus the required solution is x = 10

9. 2y + 53 = 263 – y
Solution:


10. 3m = 5m – 85
Solution:
We have

 Extra Questions Very Short Answer Type | Class 8th Mathematics

Question 1.
Identify the algebraic linear equations from the given expressions.
(a) x² + x = 2
(b) 3x + 5 = 11
(c) 5 + 7 = 12
(d) x + y² = 3
Solution:
(a) x² + x = 2 is not a linear equation.
(b) 3x + 5 = 11 is a linear equation.
(c) 5 + 7 = 12 is not a linear equation as it does not contain variable.
(d) x + y² = 3 is not a linear equation.

Question 2.
Check whether the linear equation 3x + 5 = 11 is true for x = 2.
Solution:
Given that 3x + 5 = 11
For x = 2, we get
LHS = 3 × 2 + 5 = 6 + 5 = 11
LHS = RHS = 11
Hence, the given equation is true for x = 2

Question 3.
Form a linear equation from the given statement: ‘When 5 is added to twice a number, it gives 11.’
Solution:
As per the given statement we have
2x + 5 = 11 which is the required linear equation.

Question 4.
If x = a, then which of the following is not always true for an integer k. (NCERT Exemplar)
(a) kx = ak
(b) xk = ak
(c) x – k = a – k
(d) x + k = a + k
Solution:
Correct answer is (b).

Question 5.
Solve the following linear equations:
(a) 4x + 5 = 9
(b) x + 32 = 2x
Solution:
(a) We have 4x + 5 = 9
⇒ 4x = 9 – 5 (Transposing 5 to RHS)
⇒ 4x = 4
⇒ x = 1 (Transposing 4 to RHS)
(b) We have x + 32 = 2x
⇒ 32 = 2x – x
⇒ x = 32

Question 6.
Solve the given equation 31x × 514 = 1712
Solution:
We have 31x × 514 = 1712

Question 7.
Verify that x = 2 is the solution of the equation 4.4x – 3.8 = 5.
Solution:
We have 4.4x – 3.8 = 5
Putting x = 2, we have
4.4 × 2 – 3.8 = 5
⇒ 8.8 – 3.8 = 5
⇒ 5 = 5
L.H.S. = R.H.S.
Hence verified.

Question 8.

Solution:

⇒ 3x × 3 – (2x + 5) × 4 = 5 × 6
⇒ 9x – 8x – 20 = 30 (Solving the bracket)
⇒ x – 20 = 30
⇒ x = 30 + 20 (Transposing 20 to RHS)
⇒ x = 50
Hence x = 50 is the required solution.

Question 9.
The angles of a triangle are in the ratio 2 : 3 : 4. Find the angles of the triangle.
Solution:
Let the angles of a given triangle be 2x°, 3x° and 4x°.
2x + 3x + 4x = 180 (∵ Sum of the angles of a triangle is 180°)
⇒ 9x = 180
⇒ x = 20 (Transposing 9 to RHS)
Angles of the given triangles are
2 × 20 = 40°
3 × 20 = 60°
4 × 20 = 80°

Question 10.
The sum of two numbers is 11 and their difference is 5. Find the numbers.
Solution:
Let one of the two numbers be x.
Other number = 11 – x.
As per the conditions, we have
x – (11 – x) = 5
⇒ x – 11 + x = 5 (Solving the bracket)
⇒ 2x – 11 = 5
⇒ 2x = 5 + 11 (Transposing 11 to RHS)
⇒ 2x = 16
⇒ x = 8
Hence the required numbers are 8 and 11 – 8 = 3.

Exercise 1.1 NCERT Solution Class 8th

Ex 1.1 Class 8 Maths Question 1.
Using appropriate properties find:

Solution:

Ex 1.1 Class 8 Maths Question 2.
Write the additive inverse of each of the following:
(i) 28
(ii) −59
(iii) −6−5
(iv) 2−9
(v) 19−6
Solution:

Ex 1.1 Class 8 Maths Question 3.
Verify that -(-x) = x for
(i) x = 115
(ii) x = −1317
Solution:

Ex 1.1 Class 8 Maths Question 4.
Find the multiplicative inverse of the following:

Solution:

Ex 1.1 Class 8 Maths Question 5.
Name the property under multiplication used in each of the following:

Solution:
(i) Commutative property of multiplication
(ii) Commutative property of multiplication
(iii) Multiplicative inverse property

Ex 1.1 Class 8 Maths Question 6.
Multiply 613 by the reciprocal of −716.
Solution:

Ex 1.1 Class 8 Maths Question 7.
Tell what property allows you to compute

Solution:
Since a × (b × c) = (a × b) × c shows the associative property of multiplications.

Ex 1.1 Class 8 Maths Question 8.
Is 89 the multiplicative inverse of -118? Why or Why not?
Solution:
Here -118 = −98.
Since multiplicative inverse of 89 is 98 but not −98
89 is not the multiplicative inverse of -118

Ex 1.1 Class 8 Maths Question 9.
If 0.3 the multiplicative inverse of 313? Why or why not?
Solution:

Multiplicative inverse of 0.3 or 310 is 103.
Thus, 0.3 is the multiplicative inverse of 313.

Ex 1.1 Class 8 Maths Question 10.
Write:
(i) The rational number that does not have a reciprocal.
(ii) The rational numbers that are equal to their reciprocals.
(iii) The rational number that is equal to its negative.
Solution:
(i) 0 is the rational number which does not have its reciprocal
[∵ 10 is not defined]
(ii) Reciprocal of 1 = 11 = 1
Reciprocal of -1 = 1−1 = -1
Thus, 1 and -1 are the required rational numbers.
(iii) 0 is the rational number which is equal to its negative.

Ex 1.1 Class 8 Maths Question 11.
Fill in the blanks.
(i) Zero has ……….. reciprocal.
(ii) The numbers ……….. and ……….. are their own reciprocals.
(iii) The reciprocal of -5 is ………
(iv) Reciprocal of 1x, where x ≠ 0 is ……….
(v) The product of two rational numbers is always a …………
(vi) The reciprocal of a positive rational number is ……….
Solution:
(i) no
(ii) -1 and 1
(iii) −15
(iv) x
(v) rational number
(vi) positive

Exercise 1.2 NCERT Solution Class 8th

Ex 1.2 Class 8 Maths Question 1.
Represent these numbers on the number line
(i) 74
(ii) −56
Solution:
(i) 74

Here, point A represents 74 on the number line.
(ii) −56

Here, point B represents −56 on the number line.

Ex 1.2 Class 8 Maths Question 2.
Represent −211, −511 , −911 on a number line.
Solution:
We have −211, −511 and −911

Here, point A represents −211 , point B represents −511, point C represents −911

Ex 1.2 Class 8 Maths Question 3.
Write five rational numbers which are smaller than 2.
Solution:
Required five rational numbers smaller than 2 are
1, 0, 12, 13 and 14

Ex 1.2 Class 8 Maths Question 4.
Find ten rational numbers between −25 and 12.
Solution:

Ex 1.2 Class 8 Maths Question 5.
Find five rational numbers between
(i) 23 and 45
(ii) −32 and 53
(iii) 14 and 12
Solution:

Ex 1.2 Class 8 Maths Question 6.
Write five rational numbers greater than -2.
Solution:
Required rational numbers greater than -2 are -1, 0, 12, 34 , 1.

Ex 1.2 Class 8 Maths Question 7.
Find ten rational numbers between 35 and 34.
Solution:
Given rational numbers are 35 and 34.

Rational Number Extra Questions Class 8th

Question 1.
Pick up the rational numbers from the following numbers.
67, −12, 0, 10, 1000
Solution:
Since rational numbers are in the form of ab where b ≠ 0.
Only 67, −12 and 0 are the rational numbers.

Question 2.
Find the reciprocal of the following rational numbers:
(a) −34
(b) 0
(c) 611
(d) 5−9
Solution:
(a) Reciprocal of −34 is −43
(b) Reciprocal of 0, i.e. 10 is not defined.
(c) Reciprocal of 611 is 116
(d) Reciprocal of 5−9 = −95

Question 3.
Write two such rational numbers whose multiplicative inverse is same as they are.
Solution:
Reciprocal of 1 = 11 = 1
Reciprocal of -1 = 1−1 = -1
Hence, the required rational numbers are -1 and 1.

Question 4.
What properties, the following expressions show?
(i) 23+45=45+23
(ii) 13×23=23×13
Solution:
(i) 23+45=45+23 shows the commutative property of addition of rational numbers.
(ii) 13×23=23×13 shows the commutative property of multiplication of rational numbers.

Question 5.
What is the multiplicative identity of rational numbers?
Solution:
1 is the multiplicating identity of rational numbers.

Question 6.
What is the additive identity of rational numbers?
Solution:
0 is the additive identity of rational numbers.

Question 7.
If a = 12, b = 34, verify the following:
(i) a × b = b × a
(ii) a + b = b + a
Solution:

Question 8.
Multiply 58 by the reciprocal of −38
Solution:

Question 9.
Find a rational number between 12 and 13.
Solution:
Rational number between

Question 10.
Write the additive inverse of the following:
(a) −67
(b) 101213
Solution:

Question 11.
Write any 5 rational numbers between −56 and 78. (NCERT Exemplar)
Solution:

Question 12.
Identify the rational number which is different from the other three : 23, −45, 12, 13. Explain your reasoning.
Solution:
−45 is the rational number which is different from the other three, as it lies on the left side of zero while others lie on the right side of zero on the number line.

Rational Numbers Class 8 Extra Questions Short Answer Type

Question 13.
Calculate the following:

Solution:

Question 14.
Represent the following rational numbers on number lines.
(a) −23
(b) 34
(c) 32
Solution:

Question 15.
Find 7 rational numbers between 13 and 12.
Solution:

Question 16.
Show that:

Solution:

Question 17.
If x = 12, y = −23 and z = 14, verify that x × (y × z) = (x × y) × z.
Solution:
We have x = 12, y = −23 and z = 14
LHS = x × (y × z)

Question 18.
If the cost of 412 litres of milk is ₹8912, find the cost of 1 litre of milk.
Solution:

Question 19.
The product of two rational numbers is 1556. If one of the numbers is −548, find the other.
Solution:
Product of two rational numbers = 1556
One number = −548
Other number = Product ÷ First number

Hence, the other number = −187

Question 20.
Let O, P and Z represent the numbers 0, 3 and -5 respectively on the number line. Points Q, R and S are between O and P such that OQ = QR = RS = SP. (NCERT Exemplar)
What are the rational numbers represented by the points Q, R and S. Next choose a point T between Z and 0 so that ZT = TO. Which rational number does T represent?
Solution:

As OQ = QR = RS = SP and OQ + QR + RS + SP = OP
therefore Q, R and S divide OP into four equal parts.

Question 21.
Let a, b, c be the three rational numbers where a = 23, b = 45 and c = −56 (NCERT Exemplar)
Verify:
(i) a + (b + c) = (a + b) + c (Associative property of addition)
(ii) a × (b × c) – (a × b) × c (Associative property of multiplication)
Solution:

Rational Numbers Class 8 Extra Questions Higher Order Thinking Skills (HOTS)

Question 22.
Rajni had a certain amount of money in her purse. She spent ₹ 1014 in the school canteen, bought a gift worth ₹ 2534 and gave ₹ 1612 to her friend. How much she have to begin with?
Solution:
Amount given to school canteen = ₹ 1014
Amount given to buy gift = ₹ 2534
Amount given to her friend = ₹ 1612
To begin with Rajni had

Question 23.
One-third of a group of people are men. If the number of women is 200 more than the men, find the total number of people.
Solution:
Number of men in the group = 13 of the group
Number of women = 1 – 13 = 23
Difference between the number of men and women = 23 – 13 = 13
If difference is 13, then total number of people = 1
If difference is 200, then total number of people
= 200 ÷ 13
= 200 × 3 = 600
Hence, the total number of people = 600

Question 24.
Fill in the blanks:
(a) Numbers of rational numbers between two rational numbers is ……….

Solution:
(a) Countless
(b) 611
(c) −32
(d) 35
(e) Commutative
(f) associative
(g) equivalent
(h) 311

Pythagoras Theorem Statement

Pythagoras theorem states that “In a right-angled triangle,  the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple.

Pythagoras Theorem Formula

Consider the triangle given above:

Where “a” is the perpendicular,

“b” is the base,

“c” is the hypotenuse.

According to the definition, the Pythagoras Theorem formula is given as:

Hypotenuse2 = Perpendicular2 + Base2c2 = a2 + b2

The side opposite to the right angle (90°)  is the longest side (known as Hypotenuse) because the side opposite to the greatest angle is the longest.

Pythagoras Theorem Proof

Given: A right-angled triangle ABC, right-angled at B.

To Prove- AC² = AB² + BC²

Construction: Draw a perpendicular BD meeting AC at D.

Proof:

We know, △ADB ~ △ABC

Therefore, 

 (corresponding sides of similar triangles)

Or, AB² = AD × AC ……………………………..……..(1)

Also, △BDC ~△ABC

Therefore,

(corresponding sides of similar triangles)

Or, BC²= CD × AC ……………………………………..(2)

Adding the equations (1) and (2) we get,

AB² + BC² = AD × AC + CD × AC

AB² + BC² = AC (AD + CD)

Since, AD + CD = AC

Therefore, AC² = AB² + BC²

Hence, the Pythagorean theorem is proved.

Applications of Pythagoras Theorem

• To know if the triangle is a right-angled triangle or not.
• In a right-angled triangle, we can calculate the length of any side if the other two sides are given.
• To find the diagonal of a square.

Useful For

Pythagoras theorem is useful to find the sides of a right-angled triangle. If we know the two sides of a right triangle, then we can find the third side.

Pythagorean Theorem Problems

Problem 1: The sides of a triangle are 5, 12 & 13 units. Check if it has a right angle or not.

Solution: From Pythagoras Theorem, we have;

Perpendicular² + Base² = Hypotenuse²

Let,

Perpendicular = 12 units

Base = 5 units

Hypotenuse = 13 units {since it is the longest side measure}

12² + 5² = 13²

⇒ 144 + 25 = 169

⇒ 169 = 169 

L.H.S. = R.H.S.

Therefore, the angle opposite to the 13 units side will be a right angle.

Problem 2: Given the side of a square to be 4 cm. Find the length of the diagonal.

Solution- Given;

Sides of a square = 4 cm

To Find- The length of diagonal ac.

Consider triangle abc (or can also be acd)

(ab)² +(bc)² = (ac)²

(4)² +(4)²= (ac)²

16 + 16 = (ac)²

32 = (ac)²

(ac)² = 32

ac = 4√2.

Thus, the length of the diagonal is 4√2 cm.

Probability is defined as the likelihood or chance that an event will occur. It is defined as the
numerical method of measuring uncertainty involved in a situation.

To understand and measure
the chance, we perform the experiments like tossing a coin, rolling a die and spinning the
spinner etc.

Terms related to Probability:

 Sample space: It is the set of all possible outcomes in an experiment.
 Trial: An action which results in one or several outcomes.
 Experiment: An experiment is defined as an action or process that results in well defined outcomes.
 Event: The collection of some outcomes of the experiment.

Experimental Probability:
 The fraction of times event E is expected to occur.
 The estimated probability ≅ empirical probability for large number of trials.
The empirical probability P(E) of an event E is given as:
P(E) = (Number of trials in which event E has occurred)/(Total number of trials)

Example

If we throw a dice then what is the probability that we will get a 5?

Solution

Favourable outcome = 1 (there is only one possibility of getting 5)

Total no. of possible outcomes = 6 (total six numbers are there on a dice)

Probability of getting 5 = 1/6

Triangle

Triangle is a closed curve made up of three line segments. It has three vertices, sides and angles.

Here, in ∆ABC,

• AB, BC and CA are the three sides.
• A, B and C are three vertices.
• ∠A, ∠B and ∠C are the three angles.

Types of Triangle on the basis of sides

Types of Triangle on the basis of angles

Medians of a Triangle

Median is the line segment which made by joining any vertex of the triangle with the midpoint of its opposite side. Median divides the side into two equal parts.

Every triangle has three medians like AE, CD and BF in the above triangle.

The point where all the three medians intersect each other is called Centroid.

Altitudes of a Triangle

Altitude is the line segment made by joining the vertex and the perpendicular to the opposite side. Altitude is the height if we take the opposite side as the base.

• The altitude form angle of 90°.
• There are three altitudes possible in a triangle.
• The point of intersection of all the three altitudes is called Orthocenter.

The Exterior Angle of a Triangle

If we extend any side of the triangle then we get an exterior angle.

• An exterior angle must form a linear pair with one of the interior angles of the triangle.
• There are only two exterior angles possible at each of the vertices.

Here ∠4 and ∠5 are the exterior angles of the vertex but ∠6 is not the exterior angle as it is not adjacent to any of the interior angles of the triangle.

Exterior Angle Property of the Triangle

An Exterior angle of a triangle will always be equal to the sum of the two opposite interior angles of the triangle.

Here, ∠d = ∠a + ∠b

This is called the Exterior angle property of a triangle.

Example

Find the value of “x”.

Solution

x is the exterior angle of the triangle and the two given angles are the opposite interior angles.

Hence,

x = 64°+ 45°

x = 109°

Angle Sum Property of a Triangle

This property says that the sum of all the interior angles of a triangle is 180°.

Example

Find the value of x and y in the given triangle.

Solution

x + 58° = 180° (linear pair)

x = 180° – 58°

x = 122°

We can find the value of y by two properties-

1. Angle sum property

60° + 58° + y = 180°

y = 180°- (60° + 58)

y = 62°

2. Exterior angle property

x = 60°+ y

122° = 60° + y

y = 122° – 60°

y = 62°

Two Special Triangles

1. Equilateral Triangle

It is a triangle in which all the three sides and angles are equal.

2. Isosceles Triangle

It is a triangle in which two sides are equal and the base angles opposite to the equal sides are also equal.

Sum of the length of the two sides of a triangle

Sum of the length of the two sides of a triangle will always be greater than the third side, whether it is an equilateral, isosceles or scalene triangle.

Example

Check whether it is possible to make a triangle using these measurements or not?

1. 3 cm, 4 cm, 7 cm

We have to check whether the sum of two sides is greater than the third side or not.

4 + 7 = 11

3 + 7 = 10

3 + 4 =7

Here the sum of the two sides is equal to the third side so the triangle is not possible with these measurements.

2. 2 cm, 5 cm, 6 cm

2 + 5 = 7

6 + 5 =11

6 + 2 = 8

Here the sum of the two sides is greater than the third side so the triangle could be made with these measurements.

Right Angled Triangle

A right-angled triangle is a triangle which has one of its angles as 90° and the side opposite to that angle is the largest leg of the triangle which is known as Hypotenuse .the other two sides are called Legs.

Pythagoras theorem

In a right angle triangle,

(Hypotenuse)² = (base)² + (height)²

The reverse of Pythagoras theorem is also applicable, i.e. if the Pythagoras property holds in a triangle then it must be a right-angled triangle.

Example

Find the value of x in the given triangle if the hypotenuse is 5 cm and height is 4 cm.

Solution

Given:

Hypotenuse = 5 cm

Height = 4 cm

Base = x cm

(Hypotenuse)² = (base)² + (height)²

5² = x² + 4²

x² = 5² – 4²

x² = 25 – 16

x = 9

x = 3 cm

Revision Notes on Weather, Climate and Adaptations of Animals to Climate

Weather

The weather of a place can be defined as the measure of its daily atmospheric conditions such as humidity, temperature, lightning events, rainfall Storms, snow and so on.

Different elements of weather are:

• Rainfall
• Temperature
• Humidity
• Snowfall
• Storms
• Winds etc.

Figure 1: Different types of Weather

A weather report generally contains the information about the weather of the day.

The government has a special department called the Meteorological Department that predict the weather of a place and prepare the weather report.

The weather report is generally published in newspapers, radio and television.

The weather forecast is important for people because many of our day-to-day activities are based on weather conditions. For Example, we can check the possibility of rainfall on a particular day and carry an umbrella with us accordingly.

Figure 2: Weather Forecast

The weather of a place is never constant. It can alter every day or even every hour. For instance, the weather might be sunny in the morning in an area but really in the evening.

The weather report of a place always includes the minimum and maximum temperatures of the day which are measured using a minimum-maximum thermometer. The minimum temperature can be experienced in the morning time while the maximum temperature is experienced in the afternoon.

To measure the rainfall of a place an instrument called the Rain Gauge is used. The rain gauge collects the rainwater of origin and has a measuring scale which determines the quantity of rainfall of that place.

Figure 3: Rain Gauge

How do changes in weather occur?

Any change in the weather of a place on the earth is because of the Sun which radiates large amounts of heat and light energy on the earth. The formation of winds, the phenomena of rainfall and the change in seasons, all occur because of the Sun.

Why days are shorter in the winter season?

• We know that the Earth spins on an axis around the sun.
• Hence, the amount of sunlight a place receives various throughout the year as its position with respect to the sun changes because of the rotation of the Earth.
• This also leads to change in the seasons of a place.
• In the summer season, the position of the place is closer to the sun and hence it receives sunlight for longer hours while in the winter season the position of a place is farther to the sun and hence, it receives sunlight for shorter hours.

Figure 4: Change in Seasons

Climate

• The climate of a place can be defined as the prevailing weather conditions of the place for a long period of time, for example, 25 years.
• For Example, the temperature of Rajasthan is generally high throughout the year and it does not receive much rainfall so we can say that Rajasthan is a hot and dry place.

Figure 5: Different Climates on Earth

Climate and Adaptation

• The climate of a place can affect the living organisms of that area.
• The animals living in a particular region adapt themselves so that they can survive the weather conditions of that place.
• The features and habits of the animals start to change as per the climate of the place.

Polar Region

Figure 6: The polar region on the earth

• The area of the earth that surrounds the North Pole and the South Pole is called the Polar Region.
• The climate of the polar region is extremely cold throughout the year and receives heavy snow.
• The sun does not rise for 6 months of the year in the polar region and then It stays up for the next 6 months.
• The temperature in the polar region can be as low as – 37^°C.
• Most common animals found in these regions are polar bears and penguins. Other animals that can be found in polar areas are fishes, birds, oxen, musk, reindeers, fox, whales and seals.
• They have adapted themselves so that they can survive easily in these places. 

Figure 7: Animals in Polar Region

The Polar Bear

• The white fur of the polar bear makes it easier for them to hide in the snow and therefore save them from predators.
• In the same way, it makes it easier for them to catch their prey.
• The polar bears have two layers of thick fur on them so that they can survive extremely cold conditions.
• The polar bears move slowly and rest a lot so that they do not get overheated because of their thick fur.
• The polar bears often swim on warm days to keep themselves cool.
• The paws of the polar bear are large and wide so that it can swim as well as walk easily in the snow.
• The polar bear can swim underwater as well because it can keep its nostrils closed for a long time.
• The strong sense of smell of polar bears makes it possible to locate its prey during such harsh weathers.

Figure 8: Adaptation of Polar Bear

The Penguins

• The penguins are also white in colour so that they can hide in the snow.
• They have thick skin with large fat content in their body so that they can survive the cold weather easily.
• The Penguins generally live in a crowd or nest closely so that they can stay warm.
• The Penguins have webbed feet which allow them to swim.

Figure 9: Adaptation of Penguin

Migratory Birds in the Polar Region

• The birds in order to protect themselves from cold weather of the winters in the polar region often migrate from these areas to warmer places.
• They then return back after the winter season.
• For example, The Siberian crane migrates to India in Rajasthan, Haryana and some North East regions during the winter season in Siberia.
• These birds that migrate to different places during a change in weather are often called migratory birds.
• They can travel used instances of 15000 km to protect themselves from the extremely cold environment.
• Such birds migrate to the same places every year.
• The migratory birds fly very high so that the heat generated by the flight wings can be disposed of in the cold conditions.
• The migratory birds have a sense of direction so that they can travel to the same place every time.
• The migratory birds also use landmarks or follow the direction of the sun and stars to migrate.
• Some birds also use the magnetic field of the earth and find direction.
• Apart from birds, fishes, insects and mammals also migrate.

Figure 10: The Siberian Crane

Tropical Rainforests

Figure 11: The tropical region on earth

• The tropical regions on the earth are the regions which are close to the equator and hence receive more amount of sunlight during the year.
• Because of this, these areas have a hot climate.
• The temperature in tropical regions can be as high as 40^o C and can drop until 15 ^o C only.
• The length of the day and night are almost equal in these regions.
• However, there is a lot of rainfall and so the tropical rainforests are found in this region.
• The Tropical rainforests are home to a wide variety of vegetation and animals.
• Due to large habitation, the animals often compete for food in these regions.
• Many animals have adapted themselves so that they can live on the trees and find their food easily. The skin colour of these animals is generally similar to that of the surroundings so that they can catch their prey easily and protect themselves from the predators. Also, many of these animals have a good eyesight and better sense of hearing.

Different animals found in the tropical rainforests and their features:

Animals	Adaptation
The red-eyed frog	It has a sticky feat so that it can climb up on the trees easily.	Figure 12: The red-eyed frog
The monkeys	They have long tails so that they can climb the trees easily. Even their hands and feet have a structure that helps them in holding the branches of the trees easily.	Figure 13: Monkey in Tropical Rainforest
Toucan Birds	The Toucan birds adapt themselves so that they can find food easily. The Toucan bird has a large and long beak with which it can reach the food that is found on weaker branches as well.	Figure 14: Toucan Bird
Lions and Tigers	They have thick skin, sharp eyesight and sensitive hearing.	Figure 15: Tiger in Tropical Rainforest
The lion-tailed macaque (beard ape)	The beard ape lives generally on the trees in the tropical forest because it can find its food easily on them such as insects, seeds, fruits, leaves, flowers and stems. The beard ape also has a silver-white mane that starts from its head to its cheek.	Figure 16: The lion-tailed macaque (beard ape)
Elephants	The elephant found in the tropical rainforest also have adapted themselves according to the climate. Its huge trunk gives it a nice sense of smell and helps in picking the food easily. The elephant also eats the bark of the trees. The tusks or large teeth of elephant allow it to clear the bark of the trees. The large ears of the elephant allow it to hear sensitive sounds that make it easier for it to protect itself from predators. The ears also protect the elephant from the hot and humid climate of the rainforests.	Figure 17: Elephant in Tropical Rainforest

Importance of Soil

• Soil allows the growth of plants. It supplies water and nutrients that are required in the growth of plants.
• The soil is the main part of agriculture. Different types of soils support different kinds of crops. Without agriculture, food, shelter and clothing are not possible.
• Many microorganisms live in the soil.
• Underground water is used for various purposes.

Figure 1: Importance of Soil

What pollutes the soil?

• Dumping non-biodegradable substances such as plastic bags and polythene causes soil pollution.
• Waste products from industries which contain chemicals can affect the soil adversely.
• Excess use of fertilizers and pesticides pollute the soil and decrease its fertility.

Therefore, before dumping anything waste into the soil it must be treated properly. Pesticides and fertilizers should be used in minimum quantity. Lastly, materials like plastic should be banned as we pollute the soil and affect the living organisms as well.

Figure 2: What Causes Soil Pollution?

Soil Profile

The soil consists of distinct layers which are also called Horizons of the Soil.

The Soil Profile is a vertical section of the soil which depicts all the layers of the soil. The layers of the soil can be seen if we dig deep through it like while creating a well or while laying the foundation of a building.

• Humus – The decaying matter in the soil is called Humus.
• Weathering – Soil is formed when rocks break down. This process is also called Weathering. The weathering of rocks takes place because of rains, flowing water, winds, temperature and climatic conditions of a place.
• Parent Rock – The nature of the soil that is its texture and availability of minerals depends upon the rock from which it is formed. This rock is often called as the Parent Rock.

Figure 3: Soil Profile

Layers of the Soil

Horizon A

• This layer is also called the topsoil. It is visible to us.
• It contains large amounts of humus and minerals which makes it dark in colour.
• The soil is rich in nutrients because of the presence of humus.
• The topsoil has a soft texture and can retain water easily. That is why plants roots grow in the topsoil region.
• The topsoil is a home to many living organisms as well like insects, worms, beetles, rodents and moles.

Horizon B or the Middle Layer

• It is the next layer of the soil which does not contain much humus.
• The minerals are found in large quantities in this layer.
• This layer has a hard texture, light colour and is more compact than the topsoil.

Horizon C or Third Layer

• The third layer of the soil consists of small rocks with cracks in them. These rocks are partly weathered.

Bedrock

• The last layer of the soil is called the Bedrock.
• It contains large pieces of rocks that are not weathered or exposed to any winds or water.
• Bedrock cannot be dug with the help of a spade. It is very hard in texture.

How is Soil Formed?

We know that soil is formed from weathering of the parent rock and the texture of the soil depends upon the parent rocks only. This process takes time, maybe a hundred years, and then the fine soil is formed.

• In the first stage of soil formation, the soil is generally non-porous in nature. Then it slowly turns into soil having air and water in the pores.
• We can define soil as a mixture of rock particles and humus. Based on the size of the particles and the textures of the soil it can be divided into various types.

Figure 4: Formation of Soil

Types of Soil

Figure 5: Types of Soil

Sandy Soil

• Sandy soil has big particles that have large spaces between them.
• The spaces between these particles are filled with air. Hence, sandy soils are called well-aerated soils.
• Because of large spaces, water can easily penetrate through the particles of sand. Sandy soils, however, cannot hold water.
• Hence, sandy soils are light aerated and dry in nature.
• Sandy soils lack much nutrients hence do not support the diverse growth of plants.

Clayey Soil

• It consists of fine particles which have less space between them.
• Since there is not much space between the particles clayey soils are not well-aerated like sandy soils.
• The tiny gaps between the particles although allow absorption of water in the clayey soils easily
• They are able to hold water hence are suitable for the growth of different kinds of plants.

Loamy Soils

• Loamy soil contains a similar amount of large and small particles in them.
• They are combination of sandy, clayey and silty soil.
• They also contain humus.
• They can hold water in appropriate amounts and therefore support the growth of plants.
• They are also called Agricultural Soils because of their fertility and appropriate texture.
• They contain good amounts of calcium and have a high pH level.

Silt Soil

• The silt soil particles are smaller than that of sandy soils but larger than clayey soils.
• Silt soil can hold water to some extent because of its fine quality.
• They are generally found near the water bodies like river banks and lakes.
• They are rich in nutrients, highly fertile and hence are suitable for agriculture.
• They are often mixed with other soils to improve the fertility of the soil. 

Figure 6: Particle Size in Sand, Silt and Clay

Properties of Soils

1. Percolation of water through the Soil

Percolation can be defined as the property of the soil by which it allows the flow of water through it. The rate at which water percolates or moves through soils may vary in different kind of soils. Some soils absorb water while others allow it to flow through them. The rate of percolation can be calculated by:

Figure 7: Percolation of Water

2. Moisture

Moisture is the amount of water that is present in the soil. Even a dry soil has some amount of moisture in the air. However, the clayey soil has the highest content of moisture.

Why air above farmland appears shimmering during the daytime?

We know that soil contains water. Due to sunlight, the water from the soil begins to evaporate and turns into water vapour. This water vapour when reflects the sunlight appears as if it is shining and hence the air above the soil makes the land look shimmery.

3. Absorption

Every soil has a water absorption capacity which depends upon how porous the soil is. Clayey and loamy soils are most porous hence can retain water in large quantities. That is why crops can grow over these soils. Sandy soils, on the other hand, do not absorb water and hence do not support much vegetation.

4. Texture

The texture is the size of particles of the soil. Different kinds of soils have a different texture.

5. Colour

Different soils have different colours as well. This is because of the minerals and nutrients present in the soil. For instance, some soils are black in colour because of the presence of humus and minerals while some soils are red in colour because they have iron in large quantities in them.

Figure 8: Soils have different Colors

6. pH of Soil

Soils can have different pH depending upon their acidic, basic or neutral nature. Based upon the pH different types of crops grow in the soil.

7. Air Content

Since soil is made up of particles of different sizes these particles can be loosely bound or tightly bound. The air often occupies the space in between these particles. This allows life to sustain in the soil such as microorganisms.

Figure 9: Different Properties of Soils

Soil and Crops

Different kinds of soils are found in different regions because of the following factors that decide the soil structure of that place:

• temperature
• humidity
• rainfall
• sunlight
• winds

The type of crops that will grow in the soil depends upon these factors as well as the properties of a soil.

Type of Soil	Crops Grown
Sandy	Potato, Lettuce, Corn, Peppers
Clayey	Sprouts, Broccoli, Kale, Beans, Cabbage
Loamy	Apples, Carrots, Tomatoes, Cucumber

What is soil erosion?

• When the top layer of soil gets removed it is called soil erosion.
• The soil erosion mainly occurs when the soil is left loose without vegetation or when deforestation occurs.
• In such a situation, strong winds and flowing water or rainwater takes away the topsoil and therefore decrease its quality.
• Also, this kills the organisms living inside the soil.
• The roots of the plants and trees keep the soil together and allow several microorganisms to grow and survive there. Therefore, it is always advised to plant more trees and avoid deforestation

What are Rocks?

Rocks are mineral aggregates with a combination of properties of all the mineral traces. Any unique combination of chemical composition, mineralogy, grain size, texture, or other distinguishing characteristics can describe rock types. Additionally, different classification systems exist for each major type of rock. There are different types of rocks existing in nature.

Rocks which are found in nature rarely show such simple characteristics and usually exhibit some variation in the set of properties as the measurement scale changes.

Types of RocksTypes of Rocks in India

Types of Rocks

There are three types of rocks:

• Igneous Rocks
• Sedimentary Rocks
• Metamorphic Rocks

Igneous Rock

Igneous rock is one of the three main rock types. Igneous rock is formed through the cooling and solidification of magma or lava. Igneous rock may form with or without crystallization, either below the surface as intrusive (plutonic) rocks or on the surface as extrusive (volcanic) rocks.

This magma can be derived from partial melts of existing rocks in either a planet’s mantle or crust. Typically, the melting is caused by one or more of three processes: an increase in temperature, a decrease in pressure, or a change in composition.

Types of Igneous Rock

Following are the two types of igneous rock:

1. Intrusive igneous rock: These rocks crystallize below the earth’s surface resulting in large crystals as the cooling takes place slowly. Diorite, granite, pegmatite are examples of intrusive igneous rocks.
2. Extrusive igneous rock: These rocks erupt onto the surface resulting in small crystals as the cooling takes place quickly. The cooling rate is for a few rocks is so quick that they form an amorphous glass. Basalt, tuff, pumice are examples of extrusive igneous rock.

Igneous Rock Examples

Basalt	Diorite
Granite	Mica and quartz

Sedimentary Rock

The sedimentary rocks are formed by the deposition and subsequent cementation of that material within bodies of water and at the surface of the earth. The process that causes various organic materials and minerals to settle in a place is termed as sedimentation.

The particles that form a sedimentary rock by accumulating are called sediment. Before being deposited, the sediment was formed by weathering and erosion from the source area and then transported to the place of deposition by water, wind, ice, mass movement or glaciers, which are called agents of denudation. Sedimentation may also occur as minerals precipitate from water solution or shells of aquatic creatures settle out of suspension.

Types of Sedimentary Rock

Following are the three types of sedimentary rock:

1. Clastic sedimentary rocks: These rocks are formed from the mechanical weathering debris. Sandstone, siltstone are examples of clastic sedimentary rocks.
2. Chemical sedimentary rocks: These rocks are formed from the dissolved materials that precipitate from the solution. Iron ore, limestones are examples of chemical sedimentary rocks.
3. Organic sedimentary rocks: These rocks are formed from the accumulation of plant and animal debris. Coal, some dolomites are examples of organic sedimentary rocks.

Sedimentary Rock Examples

Halite	Limestone
Sandstone	Siltstone

Metamorphic Rocks

The metamorphic rocks make up a large part of the Earth’s crust and are classified by texture and by chemical and mineral assemblage. They may be formed simply by being deep beneath the Earth’s surface, subjected to high temperatures and the great pressure of the rock layers above it.

Metamorphic rocks arise from the transformation of existing rock types, in a process called metamorphism, which means “change in form”. The original rock is subjected to heat with temperatures greater than 150 to 200°C and pressure around 1500 bars, causing profound physical and/or chemical change.

Types of Metamorphic Rock

Following are the two types of metamorphic rock:

1. Foliated metamorphic rocks: These rocks are produced by the exposure to heat and pressure which makes them appear layered. Phyllite, gneiss are examples of foliated metamorphic rocks.
2. Non-foliated metamorphic rocks: These rocks don’t have layers. Marble, quartzite are examples of non-foliated metamorphic rocks.

Metamorphic Rock Examples

Marble	Quartzite
Slate	Phyllite

Types of Rocks in India

Following are the classification of rocks in India:

1. Rocks of the Archaean system: These rocks get this name as they are formed from the hot molten earth and are the oldest and primary rocks. Gneiss is an example and is found in Karnataka, Andhra Pradesh, Tamil Nadu, Madhya Pradesh, Orissa and some parts of Jharkhand and Rajasthan.
2. Rock of Dhawar system: These are formed from the erosion and sedimentation of the Archaean system and are the oldest sedimentary rocks. These are mainly found in Karnataka.
3. Rocks of Cuddapah system: These are formed from the erosion and sedimentation of Dhawar system. Sandstone, limestone and marble asbestos are the examples and are mainly found in Rajasthan.
4. Rocks of the Vindhyan system: These are formed from the silt of river valleys and shallow oceans. Red sandstone is an example and is mainly found in Madhya Pradesh.
5. Rocks of Gondwana system: These are formed from the depressions in the basins. Coal is an example and is mainly found in Madhya Pradesh.
6. Rocks of Deccan trap: These are formed from the volcanic eruption. Dolorite and basalt are examples and are mainly found in Maharashtra and parts of Gujarat, Tamil Nadu and Madhya Pradesh.
7. Rocks of Tertiary system: These rocks are found mainly in the Himalayan regions.
8. Rocks of the Quarternary system: These rocks are found in the plains of the Indus and Ganga.

What is Global Warming?

Earth absorbs about 75 % of the total solar energy reaching its surface thereby increasing its temperature. Some of this energy is radiated back into the atmosphere. The gases present in the atmosphere for example ozone, methane, carbon dioxide, water vapour and chlorofluorocarbons are called greenhouse gases, they absorb some heat thereby restricting the heat to escape our atmosphere. These gases add to the heating of the atmosphere and result in global warming.

In places where the temperature is low, we use glass covered areas known as a greenhouse to grow flowers, fruits, and vegetables. It is very interesting to know that even we live in a greenhouse, but the difference is that we are not covered by the glass but by the blanket of air called the atmosphere. It is this atmosphere which has kept the earth’s temperature constant for centuries and helped in the survival of life. Atmosphere traps the heat around the earth and keeps it warm. This is called as natural greenhouse effect because it maintains the temperature and sustains life.

In a greenhouse, the solar energy enters through the glass, warms the soil and atmosphere and helps in the growth of plants. In return, the soil and plants emit infrared radiation, which is partly absorbed and partly reflected by the glass. This mechanism traps the sun’s energy in the greenhouse. Similarly, we have carbon dioxide which absorbs heat (as they are transparent to sunlight but not to infrared heat radiation) and is the major contributor to the global warming.

Other than carbon dioxide we have methane, ozone, CFC’s and nitrous oxides forming a major part of greenhouse gases. These chemicals either occur naturally or are man-made. The use of these compounds should be reduced; otherwise, the average temperature of the earth will rise. This will result in melting of polar ice caps and flooding of the coastal areas. Increase in the global temperature also increases the incidence of diseases like dengue, malaria, yellow fever etc.

What is the Greenhouse Effect?

To understand the nature of the greenhouse effect on climate change which leads to global warming, we must first know what the greenhouse effect is.

When you enter a car that has been in the sun for quite some time, what is the first thing you notice? It is a lot hotter inside than it is outside. The sun’s rays (UV radiation, thermal radiation, visible light) enter the car through the glass panes and all the rays entering the car do not leave owing to the build up of gases inside the car and the refractive properties of the glass itself. The weaker thermal radiation does not completely leave the car. This eventually heats up the inside of the car.

The same thing happens in a greenhouse. The plants in a greenhouse require a warm temperature to grow. Have you ever seen a greenhouse? It is made almost entirely out of glass. The heat is retained and the plants thrive. The temperature inside a greenhouse is always higher than the temperature outside.

A typical greenhouse used to regulate the climate for plant growth

This is called the greenhouse effect. What happens when the greenhouse effect occurs on a large scale in the world itself? This is where greenhouse gases come in.

Let’s have a little back-story here. Other planets in our solar system are either extremely cold or really hot. It’s the only planet earth that has a climate which is mild enough to support life. This is because of the presence of a thin layer of naturally occurring greenhouse gases like carbon dioxide, methane gas, water vapour and nitrous oxide. These gases are part of our atmosphere. The atmosphere here plays the role of the glass pane like in the greenhouse. They let the sun’s rays inside but not all of it is reflected back. The greenhouse gases even facilitate the absorption of the heat thereby warming up the earth and not letting it become extremely cold like it otherwise should have been.

The Greenhouse Effect

Since the beginning of the 18th century, the concentration of these gases in the atmosphere has started rising gradually. Since then, CO₂ levels have risen around 40%. And why is this happening? Different human activities, mainly industrial, have led to the production of these gases; the most common being carbon dioxide. The unnatural presence of increased greenhouse gases has led to a more pronounced greenhouse effect which has altered the temperature of the earth, leading to a phenomenon termed ‘Global Warming’.

Climate Change Due To Global Warming

A universal consensus of climate scientists is that there has definitely been a rise in the global temperature over the past century. The phenomenon of Global Warming if continued unchecked will have profound implications.

One of the main effects of global warming will be the rise in sea and ocean levels. Currently, this is already occurring around us. Melting of glaciers and polar ice caps will contribute to the rise in water levels all over the world. Apart from this, fresh water sources will also reduce. According to scientific bodies like NASA, other consequences of global warming are ocean acidification, extreme weather events and other natural and societal impacts.

Projected change in annual average precipitation for the 21st century based on the SRES A1B emissions scenario, and simulated by the GFDL CM 2.1 model

So can we control or check this from happening? There are some like Josef Werne, an associate professor at the department of geology & planetary science at the University of Pittsburgh, who believes that we have already crossed the point of no return. All we can do now is to adapt to the changing environment and the rising sea and ocean levels. We can still lessen the severity of climate change by aggressively enforcing policies that require different bodies to lessen CO₂ levels in the atmosphere. Still, others are even more optimistic believing that strong international agreements and actions can save the planet and it’s changing atmosphere.

Projected impact of climate change on agricultural yields by the 2080s, compared to 2003 levels (Cline,2007)

Greenhouse Gases

Many chemical compounds in the atmosphere act as greenhouse gases. These gases allow sunlight (short wave radiation) to freely pass through the Earth’s atmosphere and heat the land and oceans. The warmed Earth releases this heat in the form of infrared light (long wave radiation), invisible to human eyes. Some infrared light released by the Earth passes through the atmosphere back into space. However, greenhouse gases will not let all the infrared light pass through the atmosphere. They absorb some and radiate it back down to the Earth. This phenomenon, called the greenhouse effect, is naturally occurring and keeps the Earth’s surface warm. It is vital to our survival on Earth. Without the greenhouse effect, the Earth’s average surface temperature would be about 60° Fahrenheit colder, and our current way of life would be impossible.

We know that several gases in the atmosphere can absorb heat. These greenhouse gases are produced both by natural processes and by human activities. The primary ones are:

• Carbon dioxide (CO₂)
• Methane (CH₄)
• Nitrous oxide (N₂O)

Industrial Gases, including hydrofluorocarbons, per fluorocarbons, and sulphur hexafluoride.

Greenhouse Gas Conc. Chart in ppm

Greenhouse Gases List

Greenhouse gas	How it’s produced	100-year global warming potential	Average lifetime in the atmosphere
Methane	Released during the production and transport of natural gas, coal, and oil. It also results from agricultural practices, livestock and decay of organic waste in municipal solid waste landfills.	21	12 years
Carbon Dioxide	Released primarily when fossil fuels like natural gas, coal, and oil are burnt. Burning of trees, wood, and waste also release carbon dioxide. Changes in land use also play a role. Soil degradation and deforestation add carbon dioxide to the atmosphere, while forest regrowth takes it out of the atmosphere.	1
Nitrous oxide	Emitted during industrial and agricultural activities as well as during combustion of solid waste and fossil fuels	310	114 years
Fluorinated gases	A group of gases that includes per fluorocarbons, hydrofluorocarbons, and sulphur hexafluoride, among other chemicals. These gases are released from a variety of commercial and industrial processes and household uses and do not occur naturally. Sometimes used as substitutes for ozone-depleting substances such as chlorofluorocarbons (CFCs).	Varies	Few weeks to thousands of years