Gujarat BoardĀ GSEB Textbook Solutions Class 8 Maths Chapter 2 Linear Equations in One Variable Ex 2.2 Textbook Questions and Answers.

## Gujarat Board Textbook Solutions Class 8 Maths Chapter 2 Linear Equations in One Variable Ex 2.2

Question 1.

If you subtract \(\frac { 1 }{ 2 }\) from a number and multiply the result by \(\frac { 1 }{ 2 }\) you get \(\frac { 1 }{ 2 }\). What is the number?

Solution:

Let the required number be x.

ā“According to the condition, we have

(- \(\frac { 1 }{ 2 }\)) Ć \(\frac { 1 }{ 2 }\) = \(\frac { 1 }{ 8 }\) Ć 2

(Multipling both sides by 2)

x – \(\frac { 1 }{ 2 }\) = \(\frac { 1 }{ 4 }\)

Adding both sides \(\frac { 1 }{ 2 }\) we have

x – \(\frac { 1 }{ 2 }\) + \(\frac { 1 }{ 2 }\) = \(\frac { 1 }{ 4 }\) + \(\frac { 1 }{ 2 }\) or x = \(\frac { 1 + 2 }{ 4 }\) = \(\frac { 3 }{ 4 }\)

ā“The required number = \(\frac { 3 }{ 4 }\)

Question 2.

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:

Perimeter of the rectangular pool =154 m Let breadth = x metres

ā“Length = 2(breadth) + 2 metres.

= (2x + 2) metres.

Since, perimeter of a rectangle

= 2(length + breadth)

ā“2[(2x + 2) + x] = 154

or (2x + 2) + x = \(\frac { 154 }{ 2 }\) = 77

(Dividing both sides by 2)

or 2x + 2 + x = 77 or 3x + 2 = 77

Transposing 2 to RHS, we have

3x = 77 – 2 = 75

Dividing both sides by 3, we have

x = \(\frac { 75 }{ 3 }\) = 25

ā“Breadth = 25 m

Length = 2(25) + 2 = 52 m

Question 3.

The base of an isosceles triangle is \(\frac { 4 }{ 3 }\) cm.

The perimeter of the triangle is 4\(\frac { 2 }{ 15 }\) cm. What is the length of either of the remaining equal sides?

Solution:

Base of the isosceles triangle = \(\frac { 4 }{ 3 }\) cm.

Let the length of either equal sides = x cm.

Perimeter of the triangle = \(\frac { 4 }{ 3 }\) + x + x = \(\frac { 4 }{ 3 }\) + 2x

But the perimeter of the triangle = 4\(\frac { 2 }{ 15 }\) cm

ā“\(\frac { 4 }{ 3 }\) + 2x = 4\(\frac { 2 }{ 15 }\) = \(\frac { 62 }{ 15 }\)

Transposing \(\frac { 4 }{ 3 }\) to RHS, we have 2x = \(\frac { 62 }{ 15 }\) – \(\frac { 4 }{ 3 }\)

or 2x = \(\frac { 62 }{ 15 }\) – \(\frac { 4 }{ 3 }\) or 2x = \(\frac { 62 – 20 }{ 15 }\) = \(\frac { 42 }{ 15 }\)

Dividing both sides by 2, we have

x = \(\frac { 42 }{ 15 }\) Ć \(\frac { 1 }{ 2 }\) = \(\frac { 21 }{ 15 }\) = \(\frac { 7 }{ 5 }\)

or x = \(\frac { 7 }{ 5 }\) = 1 \(\frac { 2 }{ 5 }\) cm

Question 4.

Sum of two numbers is 95. If one exceeds the other by 15, find the numbers.

Solution:

Let the smaller number = x.

ā“ The other number = x + 15.

According to the condition, we have

x + [x + 15] = 95

2x + 15 = 95

or 2x = 95 – 15 = 80

Dividing both sides by 2, we have

x = 80 Ć· 2 = 40

ā“The smaller number = 40

The other number = 40 + 15 = 55

Question 5.

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. According to the condition, we have

5x – 3x = 18 or 2x = 18

Dividing both sides by 2, we have

\(\frac { 2x }{ 2 }\) = \(\frac { 18 }{ 2 }\) or x = 9

ā“5x = 5 x 9 = 45 and 3x = 3 x 9 = 27

ā“The required numbers are 45 and 27.

Question 6.

Three consecutive integers add up to 51. What are these integers?

Solution:

Let the three consecutive numbers be x, x + 1 and x + 2.

According to the condition, we have

x + [x + 1] + [x + 2] = 51

or x + x + 1 + x + 2 = 51

or 3x + 3 = 51

Transposing 3 to RHS, we have

3x = 51 – 3 = 48

Dividing both sides by 3, we have:

\(\frac { 3x }{ 3 }\) = \(\frac { 48 }{ 3 }\) or x = 16

Now x + 1 = 16 + 1 = 17

and x + 2=16 + 2 = 18

Thus, the required three consecutive numbers

are: 16, 17 and 18.

Question 7.

The sum of three consecutive multiples of 8 is 888. Find the multiples.

Solution:

Let the three multiples of 8 are: x, x + 8 and x+8+8=x+ 16.

According to the condition, we have

[x] + [x + 8] + [x + 16] = 888

or x + x + 8+ x + 16 = 888

or 3x + 24 = 888

Transposing 24 to RHS, we have

3x = 888 – 24 = 864

Dividing both sides by 3, we have

x = 864 + 3 = 288

ā“ x + 8 = 288 + 8 = 296

and x + 16 = 288 + 16 = 304

Thus, the required multiples of 8 are: 288, 296 and 304.

Question 8.

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 integers be x, (x + 1) and (x + 2).

ā“ According to the condition, we have

or 2x + 3x + 3 + 4x + 8 = 74

or 9x+11 = 63

Dividing both sides by 9, we have

x = 63 Ć· 9 = 7

The required integers are:

Question 9

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:

Since, ages of Rahul and Haroon are in the

ratio of 5 : 7.

Let their present ages are: 5x and 7x.

4 years later: Age of Rahul = (5x + 4) years

Age of Haroon = (7x + 4) years

According to the condition, we have

(5x + 4) + (7x + 4) = 56

or 5x + 4 + 7x + 4 = 56

or + 8 56

Transposing 8 to RHS, we have

12K = 56 – 8 – 48

Dividing both sides by 12, we have x = \(\frac { 48 }{ 12 }\) = 4

ā“Present age of Rahul = 5x 5 x 4 = 20 years

Present age Of Haroon 7x = 7 x 4 = 28 years

Question 10.

The number of boys and girls in a class are in the ratio 7 : 5. The number of boys is 8 more than the number of girls. What is the total class strength?

Solution:

Since, [number of boys] : [number of girls] = 7:5

Let the number of boys = lx and the number of girls = 5x

According to the condition, we have

7x = 5x + 8

Transposing 5x to LHS, we have

7x – 5x = 8 or 2x = 8

Dividing both sides by 2,

x = 8 = 2 = 4

ā“Number of boys = 7x = 7 x 4 = 28

Number of girls = 5x = 5 x 4 = 20

ā“Total class strength = 28 + 20 = 48 students

Question 11.

Baichungās father is 26 years younger than Baichungās 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 = x years

His fatherās age = (x + 29) years and

His grandfatherās age = (x + 29 + 26)

years = (x + 55) years

According to the condition, we have

[x] + [x + 29] + [x + 55]= 135 years

or x + x + 29 + x + 55 = 135

or 3x + 84 = 135

Transposing 84 to RHS, we have

3x = 135 – 84 or 3x = 51

Dividing both sides by 3, we have

x = 51 + 3 = 17

Now, Baichungās age = 17 years

His fatherās age = 17 + 29 = 46 years

His grandfatherās age = 46 + 26 = 72 years

Question 12.

Fifteen years from now Raviās age will be four times his present age. What is Ravi s present age?

Solution:

Let Raviās present age = x years

4 times the present age= 4 x x = 4x years

15 years from now, Raviās age = (x + 15) years

According to condition, we have

x + 15 = 4x

Transposing 15 and 4x, we have

x – 4x = -15 or -3x = -15

Dividing both sides by (-3), we have

x = (-15) Ć· (-3) = 5

ā“ Raviās present age = 5 years.

Question 13.

A rational number is such that when you multiply it by \(\frac { 5 }{ 2 }\) and add \(\frac { 2 }{ 3 }\) to the product, you 7 get . What is the number?

Solution:

Let the required rational number be x.

ā“ According to the condition, we have

Question 14.

Lakshmi is a cashier in a bank. She has currency notes of denominations ā¹ 100, ā¹ 50 and ā¹ 10, respectively. The ratio of the number of these notes is 2:3: 5. The total cash with Lakshmi is ā¹ 4,00,000. How many notes of each denomination does she have?

Solution:

Let the number of:

ā¹ 100-notes = 2x

ā¹ 50-notes = 3x

ā¹ 10-notes = 5x

Value of ā¹ 100-notes = 2x x ā¹ 100 = ā¹ 200x Value of ā¹ 50-notes = 3x Ć ā¹ 50 = ā¹ 150x Value of ā¹ 10 we have 10-notes = 5x Ć ā¹ 10 = ā¹ 50x

According to the condition, we have

ā¹ 200x + ā¹ 150x + ā¹ 50x = ā¹ 4,00,000

or 200x + 150x + 50x = 400000

or 400x = 400000

Dividing both sides by 400, we have

x = 400000 – Ć· 400 = 1000

2x = 2 x 1000 = 2000

3x = 3 x 1000 = 3000

5x = 5 x 1000 = 5000

Thus, the number of

ā¹ 100-notes = 2000

ā¹ 50-notes

ā¹ 10-notes

Question 15.

I have a total of ā¹ 300 in coins of denomination ā¹ 1, ā¹ 2 and ā¹ 5. The number of ā¹ 2 coins is 3 times the number off 5 coins. The total number of coins is 160. How many coins of each denomination are with me?

Solution:

Let the number of ā¹ 5 coins = x

The number of ā¹ 2 coins = 3x

Total number of coins = 160

Number of ā¹ 1 coins = 160 – 3x – x = (160 – 4x)

Now, value of:

ā¹ 5-coins = ā¹ 5 x x = ā¹ 5x

ā¹ 2-coins = ā¹ 2 x 3x = ā¹ 6x

ā¹ 1-coins = ā¹ 1 x (160 – 4x) = ā¹ (160 – 4x)

According to the condition, we have

5x + 6x + (160 – 4x) = 300

or 5x – 6x +160 – 4x = 300

or 7x + 160 = 300

Transposing 160 to RHS, we have

7x = 300 – 160 = 140

Dividing both sides by 7, we have

x = \(\frac { 140 }{ 7 }\) = 20

Number of: ā¹ 5-coins = 20

ā¹ 2-coins = 3 x 20 = 60

ā¹ 1-coins = 160 – (20 x 4) = 80

Question 16.

The organisers of an essay competition decide that a winner in the competition gets a prize of ā¹ 100 and a participant who does not win gets a prize of ā¹ 25. The total prize money distributed is ā¹ 3,000. Find the number of winners, if the total number of participants is 63.

Solution:

Let the number of winners = x

Number of participants who are not winners = (63 – x)

ā“ Prize money given to:

Winners = x Ć ā¹ 100 = ā¹ 100x

Non-winner participants = ā¹ 25 x (63 – x) = ā¹ 25 x 63 – ā¹ 25x = ā¹ 1575 – ā¹ 25x

According to the condition, we have

100x + 1575 – 25x = 3000

or 75x = 3000 – 1575

[Transposing 1575 to RHS]

or 75x = 1425

or x = 1425 – 75 = 19

Thus, the number of winners = 19