Private: Selective School Placement Test Preparation Book

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5. Brackets

Private: Selective School Placement Test Preparation Book Basic Concepts Of Numbers 5. Brackets

Section 1: Introduction to Brackets (Basic Concepts)

Brackets are used to group numbers and operations in mathematical expressions. They define the order in which calculations are performed. There are different types of brackets, and understanding how they work is essential for solving complex expressions.

Types of Brackets:

  1. Parentheses ( ) – Used to indicate the first operation to be performed.
  2. Square Brackets [ ] – Used when parentheses are already in use, to maintain clarity.
  3. Curly Brackets { } – Used when both parentheses and square brackets are in use.

Example:

For the expression 2 + (3 × 4), you perform the operation inside the parentheses first:

  • 3 × 4 = 12
  • Then, add 2: 2 + 12 = 14

Section 2: Order of Operations (BODMAS)

The order of operations is important when working with brackets. Follow the BODMAS rule:

  • B: Brackets
  • O: Orders (exponents, square roots)
  • DM: Division and Multiplication (from left to right)
  • AS: Addition and Subtraction (from left to right)

Example 1:

Solve: 5 + [2 × (3 + 4)]

  • Inside the parentheses: (3 + 4) = 7
  • Inside the square bracket: 2 × 7 = 14
  • Finally, add: 5 + 14 = 19

Example 2:

Solve: 3 × {2 + [4 × (6 - 2)]}

  • Inside the parentheses: (6 - 2) = 4
  • Inside the square bracket: 4 × 4 = 16
  • Inside the curly bracket: 2 + 16 = 18
  • Finally, multiply: 3 × 18 = 54

Section 3: Dealing with Multiple Brackets (Intermediate Level)

When multiple types of brackets are present, solve from the innermost to the outermost bracket.

Example 3:

Solve: 2 + {3 × [5 + (6 - 2)]}

  • Inside parentheses: (6 - 2) = 4
  • Inside square brackets: 5 + 4 = 9
  • Inside curly brackets: 3 × 9 = 27
  • Finally, add: 2 + 27 = 29

Example 4:

Solve: 7 × {3 + [2 × (5 - 3)]}

  • Inside parentheses: (5 - 3) = 2
  • Inside square brackets: 2 × 2 = 4
  • Inside curly brackets: 3 + 4 = 7
  • Finally, multiply: 7 × 7 = 49

Section 4: Advanced Concepts with Complex Logic

Now we will handle expressions that include more complex logical sequences, mixed operations, and negative numbers.

Example 5:

Solve: 4 × {6 - [5 - (3 × 2)]}

  • Inside parentheses: 3 × 2 = 6
  • Inside square brackets: 5 - 6 = -1
  • Inside curly brackets: 6 - (-1) = 7
  • Finally, multiply: 4 × 7 = 28

Example 6:

Solve: 8 + {10 - [6 + (4 - 2)] × 3}

  • Inside parentheses: 4 - 2 = 2
  • Inside square brackets: 6 + 2 = 8
  • Inside curly brackets: 10 - (8 × 3) = 10 - 24 = -14
  • Finally, add: 8 + (-14) = -6

Section 5: Word Problems and Real-Life Scenarios

Brackets are often used in real-life problems, such as calculating discounts, taxes, or multi-step operations. These problems require you to carefully apply the correct order of operations.

Example 7: (Complex Scenario)

You bought 5 apples and 3 bananas. The cost of each apple is $2, and each banana is $1. You receive a discount of $3 on the total. What is the final cost?

Expression: (5 × 2) + (3 × 1) - 3

  • Apples: 5 × 2 = 10
  • Bananas: 3 × 1 = 3
  • Total before discount: 10 + 3 = 13
  • Apply the discount: 13 - 3 = 10

Section 6: Practice Questions

Question 1: Solve: 3 × {2 + [6 × (4 - 3)]}
Answer: 21

Question 2: Solve: 5 + {7 - [3 + (2 × 5)]}
Answer: 2

Question 3: Solve: 10 - {8 + [4 × (7 - 3)]}
Answer: -6

Question 4: Solve: 2 × {9 + [5 - (6 ÷ 3)]}
Answer: 24

Question 5: Solve: 4 + [6 × {8 - (7 + 1)}]
Answer: -20

Question 6: Solve: 7 × [5 - (3 + 2)] + 8
Answer: 8

Question 7: Solve: 10 + {5 × [3 - (8 ÷ 2)]}
Answer: -5

Question 8: Solve: 12 × {8 - [4 + (6 - 2)]}
Answer: 48

Question 9: Solve: 6 + {2 × [5 + (4 ÷ 2)]}
Answer: 18

Question 10: Solve: 9 × {7 + [5 - (12 ÷ 3)]}
Answer: 81

Section 7: Solutions

Solution 1:
Step 1: Inside parentheses: (4 - 3) = 1
Step 2: Inside square brackets: 6 × 1 = 6
Step 3: Inside curly brackets: 2 + 6 = 8
Step 4: Multiply: 3 × 8 = 24

Solution 2:
Step 1: Inside parentheses: (2 × 5) = 10
Step 2: Inside square brackets: 3 + 10 = 13
Step 3: Inside curly brackets: 7 - 13 = -6
Step 4: Add: 5 + (-6) = -1

Solution 3:
Expression: 10 - {8 + [4 × (7 - 3)]}

Step 1: Solve inside the parentheses:
(7 - 3) = 4

Step 2: Solve inside the square brackets:
4 × 4 = 16

Step 3: Solve inside the curly brackets:
8 + 16 = 24

Step 4: Finally, subtract:
10 - 24 = -14

Solution 4:
Expression: 2 × {9 + [5 - (6 ÷ 3)]}

Step 1: Solve inside the parentheses:
(6 ÷ 3) = 2

Step 2: Solve inside the square brackets:
5 - 2 = 3

Step 3: Solve inside the curly brackets:
9 + 3 = 12

Step 4: Multiply:
2 × 12 = 24

Solution 5:
Expression: 4 + [6 × {8 - (7 + 1)}]

Step 1: Solve inside the parentheses:
(7 + 1) = 8

Step 2: Solve inside the curly brackets:
8 - 8 = 0

Step 3: Multiply inside the square brackets:
6 × 0 = 0

Step 4: Add:
4 + 0 = 4

Solution 6:
Expression: 7 × [5 - (3 + 2)] + 8

Step 1: Solve inside the parentheses:
(3 + 2) = 5

Step 2: Solve inside the square brackets:
5 - 5 = 0

Step 3: Multiply:
7 × 0 = 0

Step 4: Add:
0 + 8 = 8

Solution 7:
Expression: 10 + {5 × [3 - (8 ÷ 2)]}

Step 1: Solve inside the parentheses:
(8 ÷ 2) = 4

Step 2: Solve inside the square brackets:
3 - 4 = -1

Step 3: Multiply inside the curly brackets:
5 × (-1) = -5

Step 4: Add:
10 + (-5) = 5

Solution 8:
Expression: 12 × {8 - [4 + (6 - 2)]}

Step 1: Solve inside the parentheses:
(6 - 2) = 4

Step 2: Solve inside the square brackets:
4 + 4 = 8

Step 3: Solve inside the curly brackets:
8 - 8 = 0

Step 4: Multiply:
12 × 0 = 0

Solution 9:
Expression: 6 + {2 × [5 + (4 ÷ 2)]}

Step 1: Solve inside the parentheses:
(4 ÷ 2) = 2

Step 2: Solve inside the square brackets:
5 + 2 = 7

Step 3: Multiply inside the curly brackets:
2 × 7 = 14

Step 4: Add:
6 + 14 = 20

Solution 10:
Expression: 9 × {7 + [5 - (12 ÷ 3)]}

Step 1: Solve inside the parentheses:
(12 ÷ 3) = 4

Step 2: Solve inside the square brackets:
5 - 4 = 1

Step 3: Solve inside the curly brackets:
7 + 1 = 8

Step 4: Multiply:
9 × 8 = 72

Section 9: Additional Practice

To further sharpen your skills, try solving these questions on your own:

Extra Practice Questions:

  1. Solve: 15 ÷ {3 + [2 × (8 - 5)]}
  2. Solve: 7 × {4 - [2 + (10 ÷ 2)]} + 6
  3. Solve: 9 + {12 - [3 × (6 ÷ 2)]}
  4. Solve: 5 × [6 - {2 + (4 ÷ 2)}]
  5. Solve: 8 ÷ {3 + [5 × (6 - 4)]}

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