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Exercise 2.2 - Chapter 2 Linear Equations In One Variable class 8 ncert solutions Maths - SaraNextGen [2024]


Question 1:

If you subtract https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2888/Chapter%202_html_m5a4d85ce.gif from a number and multiply the result byhttps://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2888/Chapter%202_html_m5a4d85ce.gif , you gethttps://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2888/Chapter%202_html_4db079ef.gif . What is the number?

Answer:

Let the number be x. According to the Question,

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2888/Chapter%202_html_m69cdeb9a.gif

On multiplying both sides by 2, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2888/Chapter%202_html_m15a96f17.gif

On transposinghttps://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2888/Chapter%202_html_m5a4d85ce.gif  to R.H.S, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2888/Chapter%202_html_m2ba1dfb2.gif

Therefore, the number is https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2888/Chapter%202_html_1f30e4cd.gif .

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?

Answer:

Let the breadth be x m. The length will be (2x + 2) m.

Perimeter of swimming pool = 2(l + b) = 154 m

2(2x + 2 + x) = 154

2(3x + 2) = 154

Dividing both sides by 2, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2890/Chapter%202_html_2efe2166.gif

3+ 2 = 77

On transposing 2 to R.H.S, we obtain

3x = 77 − 2

3x = 75

On dividing both sides by 3, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2890/Chapter%202_html_5e0f40c0.gif

x = 25

2x + 2 = 2 × 25 + 2 = 52

Hence, the breadth and length of the pool are 25 m and 52 m respectively.

Question 3:

The base of an isosceles triangle is https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2893/Chapter%202_html_37cb7ba8.gif cm. The perimeter of the triangle is https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2893/Chapter%202_html_79cfff04.gif cm. What is the length of either of the remaining equal sides?

Answer:

Let the length of equal sides be x cm.

Perimeter = cm + x cm + Base = https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2893/Chapter%202_html_79cfff04.gif cm

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2893/Chapter%202_html_m786ab428.gif

On transposinghttps://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2893/Chapter%202_html_37cb7ba8.gif  to R.H.S, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2893/Chapter%202_html_m339f40c2.gif

On dividing both sides by 2, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2893/Chapter%202_html_m4c94a630.gif

Therefore, the length of equal sides is https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2893/Chapter%202_html_m23306187.gif cm.

Question 4:

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

Answer:

Let one number be x. Therefore, the other number will be x + 15.

According to the Question,

x + x + 15 = 95

2x + 15 = 95

On transposing 15 to R.H.S, we obtain

2x = 95 − 15

2x = 80

On dividing both sides by 2, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2898/Chapter%202_html_5f2a07e6.gif

x = 40

x + 15 = 40 + 15 = 55

Hence, the numbers are 40 and 55.

Question 5:

Two numbers are in the ratio 5:3. If they differ by 18, what are the numbers?

Answer:

Let the common ratio between these numbers be x. Therefore, the numbers will be 5x and 3x respectively.

Difference between these numbers = 18

5x − 3x = 18

2x = 18

Dividing both sides by 2,

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2900/Chapter%202_html_m6c709b7b.gif

x = 9

First number = 5x = 5 × 9 = 45

Second number = 3x = 3 × 9 = 27

Question 6:

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

Answer:

Let three consecutive integers be xx + 1, and x + 2.

Sum of these numbers = x+ x + 1 + x + 2 = 51

3x + 3 = 51

On transposing 3 to R.H.S, we obtain

3x = 51 − 3

3x = 48

On dividing both sides by 3, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2904/Chapter%202_html_d7907cf.gif

x = 16

x + 1 = 17

x + 2 = 18

Hence, the consecutive integers are 16, 17, and 18.

Question 7:

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

Answer:

Let the three consecutive multiples of 8 be 8x, 8(x + 1), 8(x + 2).

Sum of these numbers = 8+ 8(x + 1) + 8(x + 2) = 888

8(x + x + 1 + x + 2) = 888

8(3x + 3) = 888

On dividing both sides by 8, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2906/Chapter%202_html_176a88ba.gif

3x + 3 = 111

On transposing 3 to R.H.S, we obtain

3x = 111 − 3

3x = 108

On dividing both sides by 3, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2906/Chapter%202_html_m59b5c99e.gif

x = 36

First multiple = 8x = 8 × 36 = 288

Second multiple = 8(x + 1) = 8 × (36 + 1) = 8 × 37 = 296

Third multiple = 8(x + 2) = 8 × (36 + 2) = 8 × 38 = 304

Hence, the required numbers 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.

Answer:

Let three consecutive integers be xx + 1, x + 2. According to the Question,

2+ 3(x + 1) + 4(x + 2) = 74

2x + 3x + 3 + 4x + 8 = 74

9x + 11 = 74

On transposing 11 to R.H.S, we obtain

9x = 74 − 11

9x = 63

On dividing both sides by 9, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2914/Chapter%202_html_5a4e9ba8.gif

x = 7

x + 1 = 7 + 1 = 8

x + 2 = 7 + 2 = 9

Hence, the numbers are 7, 8, and 9.

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?

Answer:

Let common ratio between Rahul’s age and Haroon’s age be x.

Therefore, age of Rahul and Haroon will be 5x years and 7x years respectively. After 4 years, the age of Rahul and Haroon will be (5+ 4) years and (7+ 4) years respectively.

According to the given Question, after 4 years, the sum of the ages of Rahul and Haroon is 56 years.

∴ (5+ 4 + 7+ 4) = 56

12+ 8 = 56

On transposing 8 to R.H.S, we obtain

12x = 56 − 8

12x = 48

On dividing both sides by 12, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2918/Chapter%202_html_m27cecd10.gif

x = 4

Rahul’s age = 5x years = (5 × 4) years = 20 years

Haroon’s age = 7x years = (7 × 4) years = 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?

Answer:

Let the common ratio between the number of boys and numbers of girls be x.

Number of boys = 7x

Number of girls = 5x

According to the given Question,

Number of boys = Number of girls + 8

∴ 7x = 5x + 8

On transposing 5x to L.H.S, we obtain

7x − 5x = 8

2x = 8

On dividing both sides by 2, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2922/Chapter%202_html_709e9b7.gif

x = 4

Number of boys = 7x = 7 × 4 = 28

Number of girls = 5x = 5 × 4 = 20

Hence, total class strength = 28 + 20 = 48 students

Question 11:

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?

Answer:

Let Baichung’s father’s age be x years. Therefore, Baichung’s age and Baichung’s grandfather’s age will be (x − 29) years and (x + 26) years respectively.

According to the given Question, the sum of the ages of these 3 people is 135 years.

∴ x + x − 29 + x + 26 = 135

3x − 3 = 135

On transposing 3 to R.H.S, we obtain

3x = 135 + 3

3x = 138

On dividing both sides by 3, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2940/Chapter%202_html_12dbbdad.gif

x = 46

Baichung’s father’s age = x years = 46 years

Baichung’s age = (x − 29) years = (46 − 29) years = 17 years

Baichung’s grandfather’s age = (x + 26) years = (46 + 26) years = 72 years

Question 12:

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

Answer:

Let Ravi’s present age be x years.

Fifteen years later, Ravi’s age = 4 × His present age

x + 15 = 4x

On transposing x to R.H.S, we obtain

15 = 4x − x

15 = 3x

On dividing both sides by 3, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2942/Chapter%202_html_m499e5a7d.gif

5 = x

Hence, Ravi’s present age = 5 years

Question 13:

A rational number is such that when you multiply it by https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2945/Chapter%202_html_21c6157c.gif and add https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2945/Chapter%202_html_m64d8549d.gif to the product, you gethttps://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2945/Chapter%202_html_m75d0a82f.gif . What is the number?

Answer:

Let the number be x.

According to the given Question,

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2945/Chapter%202_html_58833be6.gif

On transposing https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2945/Chapter%202_html_m64d8549d.gif  to R.H.S, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2945/Chapter%202_html_14fc5d6.gif

On multiplying both sides byhttps://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2945/Chapter%202_html_m6f03c315.gif , we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2945/Chapter%202_html_78983eb.gif

Hence, the rational number ishttps://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2945/Chapter%202_html_5400acf9.gif .

Question 14:

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

Answer:

Let the common ratio between the numbers of notes of different denominations be x. Therefore, numbers of Rs 100 notes, Rs 50 notes, and Rs 10 notes will be2x, 3x, and 5x respectively.

Amount of Rs 100 notes = Rs (100 × 2x) = Rs 200x

Amount of Rs 50 notes = Rs (50 × 3x)= Rs 150x

Amount of Rs 10 notes = Rs (10 × 5x) = Rs 50x

It is given that total amount is Rs 400000.

∴ 200x + 150x + 50x = 400000

⇒ 400x = 400000

On dividing both sides by 400, we obtain

x = 1000

Number of Rs 100 notes = 2x = 2 × 1000 = 2000

Number of Rs 50 notes = 3x = 3 × 1000 = 3000

Number of Rs 10 notes = 5x = 5 × 1000 = 5000

Question 15:

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

Answer:

Let the number of Rs 5 coins be x.

Number of Rs 2 coins = 3 × Number of Rs 5 coins = 3x

Number of Re 1 coins = 160 − (Number of coins of Rs 5 and of Rs 2)

= 160 − (3x + x) = 160 − 4x

Amount of Re 1 coins = Rs [1 × (160 − 4x)] = Rs (160 − 4x)

Amount of Rs 2 coins = Rs (2 × 3x)= Rs 6x

Amount of Rs 5 coins = Rs (5 × x) = Rs 5x

It is given that the total amount is Rs 300.

∴ 160 − 4x + 6x + 5x = 300

160 + 7x = 300

On transposing 160 to R.H.S, we obtain

7x = 300 − 160

7x = 140

On dividing both sides by 7, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2952/Chapter%202_html_3e6c5a05.gif

x = 20

Number of Re 1 coins = 160 − 4x = 160 − 4 × 20 = 160 − 80 = 80

Number of Rs 2 coins = 3x = 3 × 20 = 60

Number of Rs 5 coins = x = 20

Question 16:

The organizers of an essay competition decide that a winner in the competition gets a prize of Rs 100 and a participant who does not win gets a prize of Rs 25. The total prize money distributed is Rs 3000. Find the number of winners, if the total number of participants is 63.

Answer:

Let the number of winners be x. Therefore, the number of participants who did not win will be 63 − x.

Amount given to the winners = Rs (100 × x) = Rs 100x

Amount given to the participants who did not win = Rs [25(63 − x)]

= Rs (1575 − 25x)

According to the given Question,

100x + 1575 − 25= 3000

On transposing 1575 to R.H.S, we obtain

75= 3000 − 1575

75= 1425

On dividing both sides by 75, we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/8/5/65/2956/Chapter%202_html_m394ac5cf.gif

x = 19

Hence, number of winners = 19

Also Read : Exercise-2.3-Chapter-2-Linear-Equations-In-One-Variable-class-8-ncert-solutions-Maths

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