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Evaluate Expressions with Parentheses

Order of Operations

We follow the PEMDAS rule for solving arithmetic expressions. The hierarchy for solving such expressions is determined as follows:

Step I: First of all, we need to perform operations inside parentheses or brackets. There are different types of brackets viz big bracket or square bracket denoted as [ ], curly bracket denoted as { }, round or small bracket denoted as ( ), line bracket denoted as ¯z, where z is any number. The expression is evaluated from inside out, starting from line bracket to small bracket to curly bracket and finally to big bracket respectively.

Step II: Solving any exponent in the expression.

Step III: Multiplication and division to be performed from left to right.

Step IV: Addition and subtraction to be performed from left to right.

Example: Evaluate {20 + (15 + 5)} – 5

First, we solve the brackets from inside out.

{20 + (15 + 5)} – 5 = {20 + 20} – 5 = 40 – 5 = 35

Example: Simplify: 96 ÷ 2 × 24 – 2

Here, we will perform multiplication and division from left to right.

96 ÷ 2 × 24 – 2 = 48 × 24 – 2 = 1152 – 2 = 1150

Example: Simplify: 4[127 – 5{113 – 2(15 × 9 ÷ 3)}]

4[127 – 5{113 – 2(15 × 9 ÷ 3)}] = 4[127 – 5{113 – 2(135 ÷ 3)}] = 4[127 – 5{113 – 2(45)}]

= 4[127 – 5{113 – 90}] = 4[127 – 5{23}] = 4[127 – 115] = 4[12] = 48

Check Point

Simplify:

1. (10 × 8) ÷ (20 ÷ 5)
2. 12 × 3 ÷ 3 × 4 – 2 + 6
3. {(18 + 17) ÷ 5} ÷ 7
4. 121 ÷ 11 + 29 – 2 × 10
5. 10[2 × {45 + (29 – 18)}]

1. (10 × 8) ÷ (20 ÷ 5) = 80 ÷ 4 = 20
2. 12 × 3 ÷ 3 × 4 – 2 + 6 = 36 ÷ 3 × 4 – 2 + 6 = 12 × 4 – 2 + 6 = 48 – 2 + 6 = 46 + 6 = 52
3. {(18 + 17) ÷ 5} ÷ 7 = {35 ÷ 5} ÷ 7 = 7 ÷ 7 = 1
4. 121 ÷ 11 + 29 – 2 × 10 = 11 + 29 – 2 × 10 = 11 + 29 – 20 = 40 – 20 = 20
5. 10[2 × {45 + (29 – 18)}] = 10[2 × {45 + 11}] = 10[2 × 56] = 10[112] = 1120

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