CLASS 8 NCERT SOLUTIONS CBSE

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What is Rational Numbers?

Rational Number represents as fraction p/q where p could be any number and q could not be Zero (0). 

Example 

1] 2 / 3

2] 15 / 18

3]  -6 / 11

4] -78 / 27 

Full video  🔗 

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Can we represent rational number in number line? 

Yes, rational numbers can be represented on a number line.

Name the property under multiplication used in each of the following 

I) -4/5 x 1 = 1 x -4/5 = -4/5

Any rational number multiplied by 1 will be the same number after the calculation. This property of one is called the multiplication identity of number 1. 1 never change the identity of rational number, whole number or natural number. The value will be the same after multiplication with 1.

Example;

2/3 x 1 = 2/3 

1 x 2/3 = 2/3

Answer:  Multiplication identity of number 1.

ii)  -13/17 x -2/7 = -2/7 x -13/17 

Solution:

Here, commutative property is using in both sides. 

a x b = b x a = ab

Answer.  Commutative Property 

iii) -19/29 x 19/-29 =1 

Solution:

In multiplicative inverse, 

p/q x q/p = 1 

Example,

3/8 x 8/3 =1 

Answer: Multiplicative inverse 

2. Tell what property allows you to compute 

1/3 x (6 x 4/3) = (1/3 x 5) x 4/3

Solution: 

In additive multiplication, 

a x (b x c) = ( a x b) x c = abc 

Solving this,

1/3 x (6 x 4/3) = (1/3 x 6) x 4/3 = 8/3

Answer:  Additional multiplication property 

3. The product of two rational numbers is always a ____________. 

Answer: the product of two rational numbers is always a rational number. 

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Exercise 1.1

https://youtu.be/LTlD4yc5vGQ?feature=shared

Direct link  ⬇️

Linear Equations in One Variable  means an expression restricted with only one variable when we form equations and the highest power of the variable appearing in the expression is 1. 

Some examples of such algebraic expressions and equations are: 

• 3 x - 2 = 7,  
• 4 y - y,  
• 2/3x - 2 x,  
• 5 / 4  (x - 4)  + 10 

In first algebraic expressions,

3 x - 2  = 7 

Here,

x is variable 

= is sign of equity 

Expression written on right hand side of the sign of equity is known as RHS and expression written on left side of the sign of equity is known as LHS. 

So,

3x - 2 is LHS and 7 is RHS. 

There is only one variable x , because this is linear equation in one variable. 

3 x y - 7 = 2 

Has to variables X and y hands it is not representing linear equation in one variable. 

We need to balance both side of an equation so that we can say that this is equation is mathematically accurate. 

Solve the following equations and check your results. 

1. 3x = 2x + 18 

3x  - 2x = 18

x = 18 

🎯 Answer : 18 

2. 5t -3 = 3t -5 

5t - 3t = -5+3 

2t = -2

t = -1 

🎯 Answer : -1

3. 5x + 9 = 5 + 3x 

5x - 3x = 5 - 9 

2x = - 4 

x =- 2

🎯 Answer:  -2

4. 4x + 3 = 6 + 2x 

4x - 2x = 6 - 3 

2 x =  3 

x = 3/2

🎯 Answer:  3/2

5. 2x - 1 = 14 - x 

2x + x = 14 + 1 

3x = 15

x= 5

🎯 Answer: 5

6. 8 x + 4 = 3(x - 1) + 7 

8 x + 4 = 3x - 3 + 7 

8 x + 4 = 3x + 4 

5x = 0 

x = 0 

🎯 Answer: 0 

7. X =  4 / 5  ( x + 10)

X = ( 4x / 5) + (40 / 5 )

x = ( 4x - 40 )/ 5    [ LCM ]

5x = 4 x - 40 

x = 40 

🎯 Answer: 40 

8. (2x / 3)  + 1 = (7x / 15 ) +  3 

(2x+3) / 3 =  (7x+45) / 15  [LCM BOTH SIDES]

15 (2x + 3) = 3 ( 7x + 45)  

[cross multiple equation both sides ]

30 x + 45 = 21 x + 135 

30x - 21 x = 135 - 45  

[ by transfers of equal expressions]

9x = 90 

x = 10

🎯 Answer: 10 

9. 2y + 5 / 3 = 26 / 3 - y

(6y+5) / 3 = (26-3y) / 9 

18 y + 15 = 78 - 9y 

27 y =  63 

y = 7/3

🎯 Answer: 7/3 

10. 3m = 5m -  8/5 

3m = (25m-8)/5

15m = 25m -8

-10m = -8

m = 4/5 

🎯 Answer: 4/5 

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Solve the following linear equations. 

1. (x/2) - (1/5) = (x/3) + (1/4)

=> (3x - 2x)/6 = (4 + 5)/20

=> x/6 = 9/20

=> x = (9 × 6)/20 = 54/20 = 27/10

🎯 Answer: x = 27/10

2. (n/2) - (3n/4) + (5n/6) = 21

=> (6n - 9n + 10n)/12 = 21

=> 7n/12 = 21

=> n = (21 × 12)/7 = 36

🎯 Answer: n = 36

3. x + 7 - (8x/3) = 17/6 - (5x/2)

Multiply entire equation by 6:

6x + 42 - 16x = 17 - 15x

=> -10x + 42 = 17 - 15x

=> 5x = -25

=> x = -5

🎯 Answer: x = -5

4. (x - 5)/3 = (x - 3)/5

Cross-multiply:

5(x - 5) = 3(x - 3)

=> 5x - 25 = 3x - 9

=> 2x = 16

=> x = 8

🎯 Answer: x = 8

5. (3t - 2)/4 - (2t + 3)/3 = 2/3 - t

Convert LHS to common denominator:

(9t - 6 - 8t - 12)/12 = 2/3 - t

=> (t - 18)/12 = 2/3 - t

Multiply by 12:

t - 18 = 8 - 12t

=> 13t = 26

=> t = 2

🎯 Answer: t = 2

6. (m - 1)/2 = 1 - (m - 2)/3

Multiply all terms by 6:

3(m - 1) = 6 - 2(m - 2)

=> 3m - 3 = 6 - 2m + 4

=> 3m - 3 = 10 - 2m

=> 5m = 13

=> m = 13/5

🎯  Answer m = 13/5

Simplify and solve the following linear equations. 

7. 3(t - 3) = 5(2t + 1)

=> 3t - 9 = 10t + 5

=> -7t = 14

=> t = -2

🎯 Answer: t = -2

8. 15(y - 4) - 2(y - 9) + 5(y + 6) = 0

=> 15y - 60 - 2y + 18 + 5y + 30 = 0

=> 18y - 12 = 0

=> y = 2/3

🎯 Answer: y = 2/3

9. 3(5z - 7) - 2(9z - 11) = 4(8z - 13) - 17

=> 15z - 21 - 18z + 22 = 32z - 52 - 17

=> -3z + 1 = 32z - 69

=> -35z = -70

=> z = 2

🎯 Answer: z = 2

10. 0.25(4f - 3) = 0.05(10f - 9)

=> 1f - 0.75 = 0.5f - 0.45

=> 0.5f = 0.3

=> f = 0.6

🎯 Answer: f = 0.6

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Exercise 3.1

1. Given here are some figures.

Classify each of them on the basis of the following:

(a) Simple curve
(b) Simple closed curve
(c) Polygon
(d) Convex polygon
(e) Concave polygon

Figure Number (a) Simple Curve (b) Simple Closed Curve (c) Polygon (d) Convex Polygon (e) Concave Polygon

(1) Yes Yes Yes No Yes
(2) Yes Yes Yes Yes No
(3) Yes Yes No No No
(4) Yes Yes Yes No Yes
(5) No No No No No
(6) Yes Yes No No No
(7) Yes Yes Yes No Yes
(8) No No No No No



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2. What is a regular polygon?
A regular polygon is a closed figure where all sides and all angles are equal.

Examples:

(i) A regular polygon with 3 sides → Equilateral triangle
(ii) A regular polygon with 4 sides → Square
(iii) A regular polygon with 6 sides → Regular hexagon

1. Find x in the following figures.

 (a) x = 180° − 125° = 55° (Linear pair)
    x = 180° − 55° = 125° 

        (Opposite angle in triangle)
    Answer: x = 125°

 (b) Angle in straight line = 180°
    x = 180° − 70° − 60° = 50°
    Answer: x = 50°


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2. Find the measure of each exterior angle of a regular polygon of
 (i) 9 sides = 360° ÷ 9 = 40°
 (ii) 15 sides = 360° ÷ 15 = 24°


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3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?
 Number of sides = 360° ÷ 24° = 15 sides


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4. How many sides does a regular polygon have if each of its interior angles is 165°?
 Exterior angle = 180° − 165° = 15°
 Number of sides = 360° ÷ 15° = 24 sides


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5. (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?
    360 ÷ 22 = 16.36 (not a whole number)
    Answer: No, because number of sides is not a whole number.

 (b) Can it be an interior angle of a regular polygon? Why?
    Interior angle = 22°, so exterior angle = 180 − 22 = 158°
    360 ÷ 158 = 2.27 (not a whole number)
    Answer: No, because number of sides is not a whole number.


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6. (a) What is the minimum interior angle possible for a regular polygon? Why?
    Minimum interior angle → Triangle
    Interior angle = 60°
    Answer: 60°, because triangle is the polygon with least number of sides.

 (b) What is the maximum exterior angle possible for a regular polygon?
    Maximum exterior angle → Triangle
    Exterior angle = 120°
    Answer: 120°, because triangle has the fewest sides (3).

1. Given a parallelogram ABCD. Complete each statement along with the definition or property used.

 (i) AD = BC    (Opposite sides of a parallelogram are equal)

 (ii) ∠DCB = ∠DAB  (Opposite angles of a parallelogram are equal)

 (iii) OC = OA    (Diagonals of a parallelogram bisect each other)

 (iv) ∠DAB + ∠CDA = 180°  (Adjacent angles of a parallelogram are supplementary)

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2. Consider the following parallelograms. Find the values of the unknowns x, y, z.

 (i) ∠DAB = ∠BCD = 80° (opposite angles)

    ∠ABC = ∠CDA = 100° (opposite angles)

 (ii) x = 50°, z = 130° (opposite angles)

 (iii) y = 150° (angle on straight line = 180° − 30°)

 (iv) z = 100° (opposite angles), x = 80° (opposite angles)

 (v) x = 55° (180° − 125°), z = 40° (opposite angle)

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3. Can a quadrilateral ABCD be a parallelogram if

 (i) ∠D + ∠B = 180°?

    Yes, possible (Adjacent angles in a parallelogram are supplementary)

 (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?

    No, opposite sides must be equal

 (iii) ∠A = 70° and ∠C = 65°?

    No, opposite angles must be equal

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4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.

→ You can draw a kite or irregular quadrilateral where only two opposite angles are equal.

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5. The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.

 Let the angles be 3x and 2x.

 3x + 2x = 180° ⇒ 5x = 180° ⇒ x = 36°

 So, angles are 3x = 108°, 2x = 72°

 Other pair: 108°, 72°

 Answer: 108°, 72°, 108°, 72°

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6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

 Let angle be x.

 x + x = 180° ⇒ 2x = 180° ⇒ x = 90°

 So all angles are 90°

 Answer: 90°, 90°, 90°, 90° (It's a rectangle)

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7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.

 ∠E = 70° (given)

 ∠H = ∠E = 70° (opposite angles)

 ∠O = ∠P = 110° (since 180° − 70°)

 Answer: x = 70°, y = 110°, z = 70°

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8. The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)

 (i) 3x − 1 = 18 (opposite sides equal)

    3x = 19 ⇒ x = 19 ÷ 3 = 6.33 (Not possible: Check again)

    Likely typo in book. If 3x − 1 = 26 then:

    3x = 27 ⇒ x = 9

    Then y = 18 cm (opposite sides equal)

 (ii) x + 7 = 20 ⇒ x = 13

    y = 16 cm (opposite sides equal)

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9. In the above figure both RISK and CLUE are parallelograms. Find the value of x.

 Angle K = 72° (given)

 Angle L = 70° (given)

 ∠K + ∠C = 180° ⇒ ∠C = 108°

 Then, x = 108° (opposite angle)

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1. State whether True or False.

(a) All rectangles are squares.

  False (Only those rectangles which have all sides equal are squares.)

(b) All rhombuses are parallelograms.

  True

(c) All squares are rhombuses and also rectangles.

  True

(d) All squares are not parallelograms.

  False (Every square is a parallelogram.)

(e) All kites are rhombuses.

  False

(f) All rhombuses are kites.

  True

(g) All parallelograms are trapeziums.

  True (Every parallelogram has one pair of parallel sides — hence, a trapezium.)

(h) All squares are trapeziums.

  True

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2. Identify all the quadrilaterals that have:

(a) Four sides of equal length – Square, Rhombus, Kite

(b) Four right angles – Square, Rectangle

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3. Explain how a square is:

(i) a quadrilateral – It has four sides.

(ii) a parallelogram – Opposite sides are parallel and equal.

(iii) a rhombus – All sides are equal and opposite angles are equal.

(iv) a rectangle – All angles are 90°.

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4. Name the quadrilaterals whose diagonals:

(i) bisect each other – Parallelogram, Rhombus, Rectangle, Square

(ii) are perpendicular bisectors of each other – Rhombus, Square

(iii) are equal – Rectangle, Square, Isosceles Trapezium

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5. Explain why a rectangle is a convex quadrilateral.

A rectangle is convex because all its interior angles are less than 180°. Also, its diagonals lie inside the figure and it has no inward corners.

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6. ABC is a right-angled triangle and O is the midpoint of the side opposite to the right angle. Explain why O is equidistant from A, B and C.

In a right-angled triangle, the midpoint of the hypotenuse is equidistant from all three vertices.

This is because the triangle can be inscribed in a circle, and the midpoint of the hypotenuse is the centre of the circle.

Therefore, OA = OB = OC.

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Table 1: Closure Property of Whole Numbers

Operation Example Result Is it a Whole Number? Closure Property Holds?

Addition

4 + 5= 9 Yes ✅ Yes
0 + 8= 8 Yes ✅ Yes

Subtraction

5 − 3= 2 Yes ✅ Yes
3 − 5= -2 ❌ No ❌ No

Multiplication

6 × 3= 18 Yes ✅ Yes
0 × 7= 0 Yes ✅ Yes

Division

4 ÷ 2= 2 Yes ✅ Yes
5 ÷ 2= 2.5 ❌ No (Not a Whole No.) ❌ No


📌 Conclusion:

Whole numbers are closed under addition and multiplication.

Not closed under subtraction and division.

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Table 2: Closure Property of Integers

Operation Example Result Is it an Integer? Closure Property Holds?

Addition

(−3) + 5  = 2 Yes ✅ Yes
(−4) + (−6) = −10 Yes ✅ Yes

(-55) +70 =  20       Yes.        ✅ Yes

(-888) + 0 =  -888   Yes         ✅ Yes 

We can say that if we add two integers, the result of the addition will be always an integer by following examples.

Integers are closed under addition. For any two integers, A and B,  A + bB is an integer.

Subtraction

(−2)  − 7 = −9 Yes ✅ Yes
4  −  (−3) = 7 Yes ✅ Yes

(-14)  -  (-14) = 0.   Yes ✅ Yes

(-33)  -  0 =  -33       Yes ✅ Yes 

As shown in the example, we can say that the difference between two integers is always an integer. 

Integers are closed under subtraction. If A and B are two integers, then A-B is also an integer.

Multiplication

(−3) × 4= −12 Yes ✅ Yes
(−2) × (−5)= 10 Yes ✅ Yes

79 x (-70) = -5530. Yes ✅ Yes 

55 x -21=1155    Yes ✅ Yes 

By above examples we can say that integers are closed under multiplication. If A and B are two integers and we multiply  AxB , we always get an integer. 

Division

(−6) ÷ 2 = −3 Yes ✅ Yes
7 ÷ 3  = 2.33 ❌ No (Not Integer)❌ No

-11 ÷ 5 =- 2.2        ❌   No Integer ❌ No

Above examples are showing that integers are not showing closure properties in distributive properties. It is not 100% sure that we get an integer if we divide 2 integers a and b.


📌 Conclusion:

Integers are closed under addition, subtraction, and multiplication.

Not closed under division.

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