ICSE Class 8 Maths Chapter 02 Simplifications PDF Download

Chapter 2: Simplifications

2.1 Four Fundamental Operations

1. Addition

For any two numbers, if:

(i) both the numbers are positive, their sum is also positive. e.g. (+8) + (+5) = +13, i.e. 8 + 5 = 13.

(ii) both the numbers are negative, their sum is also negative. e.g. (-8) + (-5) = -13, i.e. -8 - 5 = -13

(iii) one number is positive and the other is negative; without considering their signs, subtract the smaller from the bigger and then take the sign of the bigger number. e.g. (+8) + (-5) = +3 i.e. 8 - 5 = 3 and (-8) + (+5) = -3 i.e. -8 + 5 = -3

2. Subtraction

Change the sign of the number to be subtracted and then add.

e.g. subtraction of:

(i) 8 from 12 = 12 - 8 = 4,

(ii) -8 from 12 = 12 + 8 = 20,

(iii) -8 from -12 = -12 + 8 = -4

(iv) 8 from -12 = -12 - 8 = -20 and so on.

3. Multiplication

(i) The multiplication of two positive numbers is positive. e.g. (+5) \times (+3) = +15; 7 \times 4 = 28 and so on.

(ii) The multiplication of two negative numbers is also positive. e.g. (-5) \times (-3) = +15; (-7) \times (-4) = 28 and so on.

(iii) The multiplication of a positive number and a negative number is negative. e.g. (-5) \times (+3) = -15; (7) \times (-4) = -28 and so on.

The multiplication of two or more numbers is

(i) positive; if each number is positive i.e. 2 \times 3 = 6, 4 \times 2 \times 5 = 40, 7 \times 3 \times 5 \times 6 = 630 and so on.

(ii) positive; if the number of negative numbers is even,

(iii) negative; if the number of negative numbers is odd.

e.g. (-3) \times (-4) \times (-5) \times (-6) = +360; as the no. of negative numbers is 4 (even).

(-3) \times (-4) \times (+5) \times (-6) = -360; as the no. of negative numbers is 3 (odd)

(+3) \times (+4) \times (-5) \times (-6) = +360; as the number of negative numbers is 2 (even)

(+3) \times (-4) \times 5 \times 6 = -360; as the number of negative numbers is 1 (odd)

4. Division

For dividing one number by the other, the same rules are applied as for multiplication. That is, the division between the two given numbers is:

(i) positive; if both the numbers are either positive or both are negative. e.g. \(\frac{+8}{+4} = 2\), i.e. \(\frac{8}{4} = 2\), \(\frac{-75}{-15} = 5\) and so on.

(ii) negative; if one number is negative and the other is positive. e.g. \(\frac{-8}{+4} = -2\) i.e. \(\frac{-8}{4} = -2\), \(\frac{75}{-15} = -5\) and so on.

Example 1:

Evaluate: (i) 8 - 3 + 15 - 7 - 14 (ii) -25 - 47 + 64 - 12 + 30

Solution:

Steps: 1. Add all the positive numbers together and all the negative numbers separately together.

2. Add or subtract the resulting numbers as the case may be.

(i) 8 - 3 + 15 - 7 - 14

= 23 - 24 [- 8 + 15 = 23 and -3 - 7 - 14 = -24]

= -1 (Ans.)

(ii) -25 - 47 + 64 - 12 + 30

= -84 + 94 = 10 (Ans.)

Example 2:

Evaluate: (i) \(\frac{-4 \times 20}{5 \times -8}\) (ii) \(\frac{6 \times (-30)}{4 \times 15}\) (iii) \(\frac{6 \times (-10) \times (-4)}{(-8) \times (-2) \times (-3)}\)

Solution:

(i) \(\frac{-4 \times 20}{5 \times -8} = \frac{-80}{-40} = 2\) (Ans.) \[\frac{-ve}{-ve} = +ve\]

(ii) \(\frac{6 \times (-30)}{4 \times 15} = \frac{-180}{60} = -3\) (Ans.) \[\frac{-ve}{+ve} = -ve\]

(iii) \(\frac{6 \times (-10) \times (-4)}{(-8) \times (-2) \times (-3)} = \frac{6 \times 10 \times 4}{-8 \times 2 \times 3} = \frac{240}{-48} = -5\) (Ans.)

2.2 Brackets

The signs for different types of brackets are:

(i) ________; Vinculum or bar brackets,

(ii) ( ); Parenthesis or small brackets,

(iii) { }; Curly brackets or middle brackets,

(iv) [ ]; Square brackets or big brackets.

In a combined operation, the brackets must be removed in the same order as given above.

Remember, if:

1. there is a + sign or there is no sign before a bracket, the bracket is removed without changing the signs of its terms. e.g. +(17 - 8) = 17 - 8; (-8 + 4) = -8 + 4 and so on.

2. there is a minus (-) sign before a bracket, the bracket is removed with changing the signs of its terms. e.g. -(17 - 8) = -17 + 8; -(-8 + 4) = 8 - 4 and so on.

Example 3:

Simplify: 18 - [5 - {6 + 2(7 - 8 - 5)}]

Solution:

18 - [5 - {6 + 2(7 - 8 - 5)}]

= 18 - [5 - {6 + 2(7 - 3)}] [Removing bar bracket first; 8 - 5 = 3]

= 18 - [5 - {6 + 2(4)}] [7 - 3 = 4]

= 18 - [5 - {6 + 8}] [2(4) = 2 \times 4 = 8]

= 18 - [5 - 14]

= 18 - [-9] [5 - 14 = -9]

= 18 + 9 = 27 (Ans.)

2.3 Principle Of 'BODMAS'

For the purpose of simplification (including several operations), we use 'BODMAS' rule. 'BODMAS', actually, is the abbreviation formed by taking the initial letters of six operations and it helps in remembering the order in which the combined operations should be done.

The rule of BODMAS (i.e. the order of operations) is:

1. B stands for bracket i.e. [ ( ( - ) ) ]

2. O stands for of (means multiply)

3. D stands for division (i.e. \(\div\))

4. M stands for multiplication (i.e. \(\times\))

5. A stands for addition (i.e. +)

6. S stands for subtraction (i.e. -)

Example 4:

Simplify: 7 - 5 \times 2 of 3 + (19 - 7 - 4) \(\div\) 4

Solution:

7 - 5 \times 2 of 3 + (19 - 7 - 4) \(\div\) 4

= 7 - 5 \times 2 of 3 + (19 - 3) \(\div\) 4 [Remove the brackets (B) first]

= 7 - 5 \times 2 of 3 + 16 \(\div\) 4

= 7 - 5 \times 6 + 16 \(\div\) 4 [Simplifying 'of' (O) i.e. 2 of 3 = 2 \times 3 = 6]

= 7 - 5 \times 6 + 4 [Division (D) is done]

= 7 - 30 + 4 [Multiplication (M) is done]

= 11 - 30 [Addition (A)]

= -19 [Subtraction (S)] (Ans.)

Teacher's Note

Understanding the order of operations (BODMAS) helps us solve real-world problems correctly, such as calculating discounts in shopping or splitting bills among friends.

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