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Chapter 15: Mathematical Operations

What You Will Learn in This Chapter

• Core principles of Mathematical Operations and Symbol Substitution
• Equation Balancing, Sign Interchange, and accurate application of the BODMAS rule
• Detecting the logical flow in tricky expressions to double your problem-solving speed
• Spotting the examiner's favorite psychological traps to answer with total confidence

1. Introduction: The Exam Hall Hook

Imagine sitting in the exam hall as the timer ticks down. On your screen appears a question:
*If '+' means '-', '-' means 'x', 'x' means '/', and '/' means '+', then what is the value of: 24 / 8 x 2 - 5 + 3 = ?*

Your mind spins for a brief second. While the symbols appear simple, speed and precision decide who wins the race. The secret is simple: never perform math on disguise symbols. First strip away the disguise, then let standard BODMAS take over.

2. The Core Logic: Change the Symbols, Not the Rules of Mathematics

In Verbal Reasoning, the real challenge is that the symbols represent custom operations, but the fundamental axioms of mathematics remain unchanged. Your three-step blueprint:

1. Read the given substitution rules methodically.
2. Replace each coded sign with its real arithmetic operator on scratch paper.
3. Apply standard BODMAS strictly to the rewritten expression.

Active Recall Check 1

Without looking ahead: If '+' means '-' and '-' means '+', what is 12 + 7 - 4?
*(Hint: Rewrite as 12 - 7 + 4 = 5 + 4 = 9. Excellent!)*

3. The BODMAS Hierarchy: Your Unshakable Foundation

Unless a question explicitly states "do not apply BODMAS", always follow the order:

• B: Brackets first ( ), { }, [ ]
• O: Orders / Powers / Roots
• D: Division (left to right)
• M: Multiplication (left to right)
• A: Addition
• S: Subtraction

Active Recall Check 2

In the expression `20 - 8 / 2 + 3`, if `/` means `+` and `+` means `/`, what is the very first step?
*(Answer: First rewrite as `20 - 8 + 2 / 3`, then apply division before addition or subtraction.)*

4. Sign Interchange Problems: The Balancing Act

In sign interchange problems, you are given an incorrect equation and four option pairs of signs (e.g., "+" and "x", "-" and "/"). Your mission is to find which swap balances the equation:

• Strategy: Reverse Engineering via Options.
  Instead of guessing blindly, test the options systematically. Look at the division sign first: if a division results in a fraction that cannot be simplified to match the RHS integer, eliminate that option immediately!
• Mnemonic: "Swap the Sign, Balance the Line!"

Active Recall Check 3

If `18 x 2 + 4 - 5 = 21` needs to be corrected, which two signs should be tested first?
*(Testing options where multiplication is reduced or division is introduced brings the large value closer to 21.)*

5. Double-Relation & Inequality Coding

In some questions, letters represent relational operators:

• 'P' denotes 'greater than' (>)
• 'Q' denotes 'equal to' (=)
• 'R' denotes 'less than' (<)
  Convert each option into a clear mathematical inequality chain, then evaluate its mathematical truth.

Chapter Summary Table

Examiner's Traps & Pitfalls

1. The Incomplete Replacement Trap: Replacing some symbols while forgetting others mid-equation. Always rewrite the full expression.
2. The Fractional Division Trap: Forgetting that division takes precedence over multiplication, leading to erroneous integer rounding.
3. The Left-to-Right Blindness: Treating all operations in left-to-right reading order instead of following BODMAS precedence.