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Simplification of Integers

A collection of numbers connected by one or more operations of addition, subtraction,
multiplication, division, and ‘of’ is called a numerical expression. A numerical expression may
also involve brackets.
For example: (a) 12 × 3 + 15 ÷ 3 (b) 16 – 8 ÷ 2 – 5 × (–7) (c) 25 + 12 ÷ (8 – 2) + 5 × (–8)

These are all numerical expressions.
Simplification of a Numerical Expression

Performing the operations involved in a numerical expression and getting its value is called
simplification.

Several operations are done with the help of the word ‘BODMAS’ taken in order.
B — Brackets. First, carry out the operations inside brackets.

O — Of. Change ‘of’ into ‘×’ (multiply) and carry it out.
D — Division. After ‘of’ work out division. The order of removing these

M — Multiplication. After division, carry out multiplication. Note: brackets never changes in
the absence of any bracket
A — Addition. After multiplication, carry out addition. in a numerical expression.

S — Subtraction. Finally, work out subtraction.
Example 11: Simplify the following:

(a) 25 – 18 ÷ 6 × 5 (b) 1 – 2 of {– 6 – (5 + 3)} ÷ 2
(c) 16 – {8 × 3 – (–6) × 12 ÷ (–3)} (d) 12 × 5 – [8 – {11 + 30 ÷ (7 − 7 – 5)}]
Solution: (a) 25 – 18 ÷ 6 × 5 = 25 – 3 × 5 [Division: 18 ÷ 6 = 3]

= 25 – 15 [Multiplication: 3 × 5 = 15]

= 10 [Subtraction: 25 – 15 = 10]
(b) 1 – 2 of {– 6 – (5 + 3)} ÷ 2 = 1 − 2 of { − 6 − 8} ÷ 2 [Solving ( )]
= 1 − 2 of {–14} ÷ 2 [Solving { }]

= 1 + 28 ÷ 2 [Solving ‘of’: –2 of {–14} = 28]
= 1 + 14 = 15 [Division: 28 ÷ 2 = 14]

(c) 16 – {8 × 3 – (–6) × 12 ÷ (–3)} = 16 – {8 × 3 – (–6) × (–4)} [Division: 12 ÷ (–3)]
= 16 – {24 – 24} [Multiplication: 8 × 3 and (–6) × (–4)]

= 16 – 0 = 16 [Solving { }]
(d) 12 × 5 – [8 – {11 + 30 ÷ (7 − 7 – 5)}] = 12 × 5 – [8 – {11 + 30 ÷ (7 – 2)}] [7 – 5 = 2]

= 12 × 5 – [8 – {11 + 30 ÷ 5}] Solving ( )
= 12 × 5 – [8 – {11 + 6}] [Division: 30 ÷ 5 = 6]

= 12 × 5 – [8 – 17] Solving { }
= 60 – [– 9] = 60 + 9 = 69

[Multiplying 12 × 5 = 60 and solving [8 – 17] = –9]

21 Integers