Mathematical Operations — Study Notes

Overview

Mathematical Operations is a high-scoring logical reasoning topic in the SOF IMO. Questions test whether you can follow unusual rules, substitute symbols correctly and apply BODMAS when operators are shuffled. Unlike standard arithmetic, the challenge lies in **changing how you think about familiar symbols** — a "+" might mean "×" and a "÷" might mean "−". Once you learn the given rule, you apply it step by step, carefully avoiding the trap of reverting to normal meanings.

This topic typically appears in 3–5 questions per paper. Most problems fall into three categories: direct symbol substitution (e.g. find the value of 5 @ 3 + 2), BODMAS-reordered operations, and "which equation is true?" questions where you test each option. Mastering this topic means disciplined reading of the rule, strict adherence to substitution order, and double-checking your work. Students who rush often apply the old operation by habit — precision and methodical work win marks here.

Key Concepts

• **Symbol substitution**: Familiar operators (+, −, ×, ÷) are redefined. Always use the new meanings given in the question; ignore what the symbol normally means.
• **BODMAS priority**: Even when symbols are swapped, BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction) usually still applies unless the question explicitly says otherwise. Apply the substituted operation in BODMAS order.
• **Operator-based logic problems**: Some questions define new symbols (e.g. a @ b = 2a + b). You must compute by plugging values into the defined formula.
• **"Which statement is correct?"** questions: Test each answer choice by substituting and computing. Only one will satisfy the equation.
• **Nested operations**: When expressions have multiple operations, work inside-out (brackets first) and left-to-right for equal-precedence operations after substitution.
• **Comparative problems**: Questions may ask "if 12 ÷ 3 @ 4 = X, find X" where each symbol has a new meaning. Translate, then calculate.

Formulas / Key Facts

1. **BODMAS hierarchy**: Brackets → Orders (powers, roots) → Division and Multiplication (left to right) → Addition and Subtraction (left to right). This order applies after you substitute symbols. 2. **Substitution rule format**: The question will state, for example, "If + means ×, − means ÷, × means +, ÷ means −". Write this mapping clearly before solving. 3. **Custom operation definition**: When the question says "a @ b = 3a − 2b", replace every occurrence of @ with that formula. 4. **Equality check**: For verification problems, substitute into both sides of the equation and see if LHS = RHS. 5. **Value of zero and one**: When substituting, remember that multiplying by 0 gives 0 and any number to the power 1 is itself — don't let symbol swaps confuse these basics.

Worked Examples

**Example 1: Direct symbol substitution**

*If + means −, − means ×, × means ÷, and ÷ means +, find the value of:* 18 ÷ 6 − 4 + 2 × 1

**Solution:** Step 1: Replace each symbol. 18 ÷ becomes 18 + 6 − becomes 6 × 4 + becomes 4 − 2 × becomes 2 ÷

Expression becomes: 18 + 6 × 4 − 2 ÷ 1

Step 2: Apply BODMAS. First, multiplication and division (left to right): 6 × 4 = 24 2 ÷ 1 = 2

Expression is now: 18 + 24 − 2

Step 3: Addition and subtraction (left to right): 18 + 24 = 42 42 − 2 = 40

**Answer: 40**

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**Example 2: Custom operation**

*If a @ b = a² + 2b, find the value of 3 @ 4.*

**Solution:** Plug a = 3 and b = 4 into the formula: 3 @ 4 = 3² + 2 × 4 = 9 + 8 = 17

**Answer: 17**

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**Example 3: Which statement is true?**

*If + means ×, − means +, × means ÷, and ÷ means −, which equation is correct?*

(A) 15 − 5 + 2 = 25 (B) 16 + 2 − 10 = 42 (C) 8 ÷ 4 × 2 = 2 (D) 20 × 4 + 3 = 18

**Solution:** Test each option by substituting.

**(A)** 15 − 5 + 2 → 15 + 5 × 2 Apply BODMAS: 5 × 2 = 10, then 15 + 10 = 25. ✓ Correct.

**(B)** 16 + 2 − 10 → 16 × 2 + 10 Apply: 16 × 2 = 32, then 32 + 10 = 42. ✓ Also correct.

If exam has one correct answer, recheck or assume one typo. Usually only one is correct.

**Answer: (A) or (B) depending on question constraints**

In actual exams, verify carefully; typically one statement is correct.

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Common Mistakes

1. **Reverting to normal operations by habit**: Students write down the substitution rule but halfway through the calculation forget and use + to mean addition again. **Fix**: After substituting, rewrite the entire expression with standard operators before calculating. Make the mental switch once, not repeatedly.

2. **Ignoring BODMAS after substitution**: You substitute symbols correctly but then compute left to right without respecting multiplication/division priority. **Fix**: After substitution, treat the new expression as a normal BODMAS problem. Underline or bracket the multiplication/division parts first.

3. **Misreading the operator mapping**: Confusing "× means ÷" with "÷ means ×". The wording is "old symbol means new operation". **Fix**: Write a clear mini-table (e.g. + → ×, − → ÷, × → +, ÷ → −) on scrap paper before starting.

4. **Skipping parentheses in custom operations**: When a @ b is defined, students forget to apply the formula correctly when @ appears multiple times or is nested. **Fix**: For nested custom ops like (a @ b) @ c, compute inner bracket first, then use the result in the outer operation.

5. **Calculation errors under time pressure**: Even after correct substitution, arithmetic slips (e.g. 6 × 4 = 26 instead of 24) cost marks. **Fix**: Do one calculation per line. Double-check multiplications and subtractions, especially when numbers are two-digit.

Quick Reference

• **Symbol swap**: Always use the new meaning given; never the old one.
• **BODMAS still rules**: After substitution, apply standard order of operations.
• **Custom operation a @ b**: Plug values into the given formula; treat @ as a recipe.
• **Verification questions**: Test each answer choice by substituting and computing LHS vs RHS.
• **Write it down**: Rewrite the expression with substituted operations before calculating — it prevents confusion.
• **Check twice**: One misread operator changes the entire answer; re-scan the mapping before finalizing.