## Selina Concise Mathematics Class 6 ICSE Solutions – Simple (Linear) Equations (Including Word Problems)

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**IMPORTANT POINTS**

**1. Simple Equations :** A mathematical statement, which states that two expressions are equal, is called simple equation.

**2. Properties of Simple Equation :**

**(i)** If same quantity is added to both the sides of simple equation, the sums are equal.

**For Example :**

x = 6 ⇒ x + a = 6 + a [Adding a on both the sides]

**(ii)** If same quantity is subtracted from both the sides of simple equation, the remainders are equal.

**For Example :**

x = 6 ⇒ x-a = 6 – a [Subtracting a on both the sides]

**(iii)** If both the sides of an equation are multiplied by the same quantity, the products are equal.

For Example :

x = 6 ⇒ a x x = a x 6 i.e. ax = 6a [Multiplying both the sides by a]

**(iv)** If both the sides of simple equation are divided by the same quantity, the quotients are equal.

**For Example :**

x = 6 ⇒ = [Dividing both the sides by a]

**EXERCISE 22 (A)**

**Question 1.**

**Solve:**

**Solution:**

**Question 2.**

**Solve:**

**Solution:**

**Question 3.**

**Solve:**

**Solution:**

**Question 4.**

**Solve:**

**Solution:**

**Question 5.**

**Solve:**

**Solution:**

**EXERCISE 22 (B)**

**Question 1.**

**Solve:**

**Solution:**

**Question 2.**

**Solve:**

**Solution:**

**Question 3.**

**Solve:**

**Solution:**

**EXERCISE 22 (C)**

**Question 1.
**

**5 – x = 3**

**Solution:**

**Question 2.
**

**2 – y = 8**

**Solution:**

**Question 3.
**

**8.4 – x = -2**

**Solution:**

**Question 4.**

**Solution:**

**Question 5.**

**Solution:**

**Question 6.**

**Solution:**

**Question 7.
**

**1.6z = 8**

**Solution:**

**Question 8.
**

**3a = – 2.1**

**Solution:**

**Question 9.**

**Solution:**

**Question 10.**

**Solution:**

**Question 11.
**

**– 5x = 10**

**Solution:**

**Question 12.
**

**2.4z = -4.8**

**Solution:**

**Question 13.
**

**2y – 5 = -11**

**Solution:**

**Question 14.
**

**2x + 4.6 = 8**

**Solution:**

**Question 15.
**

**5y – 3.5 = 10**

**Solution:**

**Question 16.
**

**3x + 2 = -2.2**

**Solution:**

**Question 17.**

**Solution:**

**Question 18.**

**Solution:**

**Question 19.**

**Solution:**

**Question 20.
**

**-3y – 2 = 10**

**Solution:**

**Question 21.
**

**4z – 5 = 3 – z**

**Solution:**

**Question 22.
**

**7x -3x +2 =22**

**Solution:**

**Question 23.**

**6y + 3 = 2y + 11**

**Solution:**

**Question 24.
**

**3 (x+5) = 18**

**Solution:**

**3 (x+5) = 18**

**Question 25.
**

**5 (x-2) -2 (x+2) = 3**

**Solution:**

**Question 26.
**

**(5x-3) 4=3**

**Solution:**

**Question 27.
**

**3(2x+1) -2(x-5) -5 (5-2x) = 16**

**Solution:**

**EXERCISE 22 (D)**

**Question 1.**

**A number increased by 17 becomes 54. Find the number.**

**Solution:**

Let the required number = x

∴ According to the sum :

x+17 = 54

⇒ x = 54-17

⇒ x = 37

Required number = 37

**Question 2.**

**A number decreased by 8 equals 26, find the number.**

**Solution:**

Let required number = A

∴According to the sum :

x – 8 = 26

⇒A = 26 + 8

⇒A = 34

∴Required number = 34

**Question 3.**

**One-fourth of a number added to two- seventh of it gives 135; find the number.**

**Solution:**

**Question 4.**

**Two-fifths of a number subtracted from three-fourths of it gives 56, find the number.**

**Solution:**

**Question 5.**

**A number is increased by 12 and the new number obtained is multiplied by 5. If the resulting number is 95, find the original number.**

**Solution:**

**Question 6.**

**A number is increased by 26 and the new number obtained is divided by 3. If the resulting number is 18; find the original number.**

**Solution:**

**Question 7.**

**The age of a man is 27 years more than the age of his son. If the sum of their ages is 47 years, find the age of the son and his father.**

**Solution:**

**Question 8.**

**The difference between the ages of Gopal and his father is 26 years. If the sum of their ages is 56 years, find the ages of Gopal and his father.**

**Solution:**

**Question 9.**

**When two consecutive natural numbers are added, the sum is 31; find the numbers.**

**Solution:**

**Question 10.**

**When three consecutive natural numbers are added, the sum is 66, find the numbers.**

**Solution:**

**Question 11.**

**A natural number decreased by 7 is 12. Find the number.**

**Solution:**

**Question 12.**

**One fourth of a number added to one- sixth of itself is 15. Find the number.**

**Solution:**

**Question 13.**

**A whole number is increased by 7 and the new number so obtained is multiplied by 5; the result is 45. Find the number.**

**Solution:**

**Question 14.**

**The age of a man and the age of his daughter differ by 23 years and the sum of their ages is 41 years. Find the age of the man.**

**Solution:**

**Question 15.**

**The difference between the ages of a woman and her son is 19 years and the sum of their ages is 37 years; find the age of the son.**

**Solution:**

**Question 16.**

**Two natrual numbers differ by 6 and sum of them is 36. Find the larger number.**

**Solution:**

**Question 17.**

**The difference between two numbers is 15. Taking the smaller number as x; find:**

**(i) the expression for larger number.**

**(if) the larger number, if the sum of these numbers is 71.**

**Solution:**

**Question 18.**

**The difference between two numbers is 23. Taking the larger number as x, find:**

**(i) the expression for smaller number.**

**(ii) the smaller number, if the sum of these two numbers is 91.**

**Solution:**

**Question 19.**

**Find three consecutive integers such that their sum is 78.**

**Solution:**

Sum of three consecutive numbers = 78

Let first number = x

Then second number = x + 1

and third number = x + 2

Then x + x+1+x + 2 = 78

⇒ 3x + 3 = 78

⇒ 3x = 78 – 3 = 75

⇒ x = =25

∴First number=25

Second number = 25 + 1 = 26

and third number = 26 + 1 = 27

Then the three required numbers are 25, 26,27

**Question 20.**

**The sum of three consecutive numbers is 54. Taking the middle number as x, find:**

**(i) expression for the smallest number and the largest number.**

**(ii) the three numbers.**

**Solution:**

Sum of three consecutive numbers = 54

Middle number = x

(i) The first number = x – 1

and third number = x + 1

(ii) ∴x + x-1+x+1 = 54

⇒ 3x = 54

⇒ x= = 18

∴First number =18-1 = 17

and third number =18 + 1 = 19

∴Three required numbers are 17, 18,19

**REVISION EXERCISE**

**Question 1.**

**Solve each of the following equations :**

**Question i.
**

**2x + 3 = 7**

**Solution:**

**Question ii.
**

**2x – 3 = 7**

**Solution:**

**Question iii.
**

**2x ÷ 3 = 7**

**Solution:**

**Question iv.
**

**3x – 8 = 13**

**Solution:**

**Question v.
**

**3y + 8 = 13**

**Solution:**

**Question vi.
**

**3y ÷ 8 = 13**

**Solution:**

**Question vii.**

**Solution:**

**Question viii.**

**Solution:**

**Question ix.**

**Solution:**

**Question x.
**

**5x – 2.4 = 4.9**

**Solution:**

**Question xi.
**

**5y + 4.9 = 2.4**

**Solution:**

**Question xii.
**

**48 z + 3.6 = 1.2**

**Solution:**

**Question xiii.**

**Solution:**

**Question xiv.**

**Solution:**

**Question xv.**

**Solution:**

**Question xvi.
**

**-3x + 4 = 10**

**Solution:**

**Question xvii.
**

**5 = x – 3**

**Solution:**

**Question xviii.
**

**8y = 3- 3y**

**Solution:**

**Question xix.
**

**4x = 4.9 = 6.5**

**Solution:**

**Question xx.
**

**3z + 2 = -4**

**Solution:**

**Question xxi.
**

**7y -18 = 17**

**Solution:**

**Question xxii.**

**Solution:**

**Question xxiii.**

**Solution:**

**Question xxiv.**

**Solution:**

**Question xxv.
**

**7x -2 = 4x +7**

**Solution:**

**Question xxvi.
**

**3y -(y -+2) =4**

**Solution:**

**Question xxvii.
**

**3z – 18 = z – (12 -4z)**

**Solution:**

**Question xxiii.**

**Solution:**

**Question xxix.**

**Solution:**

**Question xxx.**

**Solution:**

**Question xxxi.
**

**5x – 2x +15 = 27**

**Solution:**

**Question xxxii.
**

**5y – 15 = 27 -2y**

**Solution:**

**Question xxxiii.
**

**7z + 15 = 3z – 13**

**Solution:**

**Question xxxiv.
**

**2 (x -3) – 3 (x-4) =12**

**Solution:**

**Question xxxv.
**

**(7y +8) +7= 8**

**Solution:**

**Question xxxvi.
**

**2(z-5) +3 (z+2) -(3-5z) =10**

**Solution:**

**Question 2.**

**A natural number decreased by 7 is 12. Find the number.**

**Solution:**

**Question 3.**

**One-fourth of a number added to one-sixth of It is 15. Find the number.**

**Solution:**

**Question 4.**

**A whole number is increased by 7 and the number so obtained is multiplied by 5; the result is 45. Find the whole number.**

**Solution:**

**Question 5.**

**The age of a man and the age of his daughter differ by 23 years and the sum of their ages is 41 years. Find the age of the man**.

**Solution:**

**Question 6.**

**The difference between the ages of a woman and her son is 19 years and the sum of their ages is 37 years; find the age of the son.**

**Solution:**

**Question 7.**

**Two natural numbers differ by 6 and their sum is 36. Find the larger number.**

**Solution:**

**Question 8.**

**The difference between two numbers is 15. Taking the smaller number as x; find :**

**(i) the expression for the larger number.**

**(ii) the larger number, if the sum of these numbers is 71.**

**Solution:**

**Question 9.**

**The difference between two numbers is 23. Taking the larger number as x, find :**

**(i) the expression for smaller number.**

**(ii) the smaller number, if the sum of these two numbers is 91.**

**Solution:**

**Question 10.**

**Find the three consecutive integers whose sum is 78.**

**Solution:**

**Question 11.**

**The sum of three consecutive numbers is 54. Taking the middle number as x, find :**

**(i) the expressions for the smallest number and the largest number.**

**(ii) the three numbers.**

**Solution:**

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