Symbol & Relationship Test — UP Police Constable Study Notes

Overview

Symbol & Relationship Test is a unique reasoning topic that blends mathematical operations with logical thinking. In UP Police Constable exams, expect 2–4 questions where symbols replace standard arithmetic operators (+, −, ×, ÷) or where you must establish relationships between numbers using given symbolic rules.

This topic tests your ability to decode patterns, substitute operations correctly, and apply BODMAS principles under non-standard notation. Unlike pure arithmetic, these questions require careful reading and systematic substitution — rushing leads to mistakes. Mastering this topic builds confidence in problem-solving under altered conditions, a skill valuable across the reasoning section.

Students often find these questions tricky initially, but with practice in recognizing substitution patterns and maintaining operational order, scoring becomes consistent. Focus on accuracy over speed — each question typically takes 30–45 seconds once you've internalized the approach.

Key Concepts

• **Symbol substitution**: Common operators (+, −, ×, ÷) are replaced by symbols (@, #, *, $, etc.) or interchanged with each other. You must decode which symbol represents which operation.

• **BODMAS preservation**: Even when symbols change, the order of operations (Brackets, Orders, Division/Multiplication, Addition/Subtraction) remains unchanged. Always apply BODMAS after substitution.

• **Statement-based operations**: Questions provide a key (e.g., "If + means ×, − means +, × means ÷, ÷ means −") followed by an expression to evaluate using the new meanings.

• **Relationship validation**: Some questions ask you to determine if a given mathematical relationship becomes true after symbol substitution. You test each option systematically.

• **Equation balancing**: Questions may present an equation with symbolic operators and ask which number or operation makes it valid.

• **Combination puzzles**: Advanced variants combine symbol substitution with series, inequalities, or logical conditions requiring multi-step reasoning.

• **Time management**: These questions reward methodical work. Write down the substitution key, replace symbols carefully, then calculate. Avoid mental shortcuts that skip steps.

Key Facts

• **Standard operator symbols**: + (addition), − (subtraction), × (multiplication), ÷ (division), = (equals), > (greater than), < (less than).

• **Common symbol replacements**: @, #, *, $, %, &, !, ~ are frequently used as substitute operators. No universal standard exists — each question defines its own key.

• **BODMAS rule**: Brackets first, then Orders (powers/roots), then Division and Multiplication (left to right), then Addition and Subtraction (left to right).

• **Interchange vs. new symbols**: Questions either replace operators with new symbols or interchange existing operators (e.g., + now means × and × now means +). Read carefully.

• **Verification approach**: After substitution, perform the calculation step-by-step. Compare your result with the answer choices only after completing the math.

• **Double-check substitution**: The most common error is incorrect symbol substitution. Always verify you've applied the given key correctly before calculating.

• **Practice variety**: UP Police papers feature both simple substitution (3–4 symbols) and complex puzzles (5+ symbols with nested operations). Practice both types.

Worked Examples

**Example 1: Basic Symbol Substitution**

*If + means ÷, − means +, × means −, and ÷ means ×, then find the value of: 16 + 4 − 10 × 5 ÷ 2*

**Solution:** Step 1 — Write the substitution key clearly:

• + → ÷
• − → +
• × → −
• ÷ → ×

Step 2 — Replace symbols in the expression: 16 ÷ 4 + 10 − 5 × 2

Step 3 — Apply BODMAS (division and multiplication first, left to right): = 4 + 10 − 5 × 2 = 4 + 10 − 10

Step 4 — Now addition and subtraction (left to right): = 14 − 10 = 4

**Answer: 4**

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**Example 2: Equation Validation**

*If × means +, + means ÷, − means ×, and ÷ means −, which of the following equations is correct?*

(a) 18 × 6 + 3 − 2 ÷ 1 = 29 (b) 18 − 6 + 3 × 2 ÷ 1 = 33 (c) 18 + 6 × 3 − 2 ÷ 1 = 8 (d) 18 ÷ 6 − 3 × 2 + 1 = 40

**Solution:** Substitution key:

• × → +
• + → ÷
• − → ×
• ÷ → −

Test option (b): 18 − 6 + 3 × 2 ÷ 1 After substitution: 18 × 6 ÷ 3 + 2 − 1

Calculate using BODMAS: = 18 × 2 + 2 − 1 (since 6 ÷ 3 = 2) = 36 + 2 − 1 = 37

This doesn't equal 33, so (b) is incorrect.

Test option (c): 18 + 6 × 3 − 2 ÷ 1 After substitution: 18 ÷ 6 + 3 × 2 − 1

Calculate: = 3 + 3 × 2 − 1 = 3 + 6 − 1 = 8 ✓

**Answer: (c)**

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**Example 3: Finding the Missing Number**

*If @ means ×, # means +, $ means ÷, and * means −, find the value of: 20 @ 3 # 15 $ 5 * 7 = ?*

**Solution:** Substitution:

• @ → ×
• # → +
• $ → ÷
• * → −

Expression becomes: 20 × 3 + 15 ÷ 5 − 7

Apply BODMAS: = 20 × 3 + 3 − 7 = 60 + 3 − 7 = 56

**Answer: 56**

Common Mistakes

**Mistake 1: Ignoring BODMAS after substitution** *Wrong thinking:* "After substitution, just calculate left to right." *Correct fix:* Always apply BODMAS rules to the substituted expression. Multiplication and division come before addition and subtraction, regardless of symbols used in the original question.

**Mistake 2: Reversing the substitution** *Wrong thinking:* "If + means ×, then × means +," assuming symmetry. *Correct fix:* Read the key exactly as stated. If only "+ means ×" is given, you cannot assume what × means unless explicitly stated. Each symbol's new meaning is independent.

**Mistake 3: Mental calculation errors under time pressure** *Wrong thinking:* "I can do simple math in my head to save time." *Correct fix:* Write down each step, especially for expressions with 4+ operations. Mental shortcuts in symbol-substitution questions lead to frequent errors. Spend 10 extra seconds writing to ensure accuracy.

**Mistake 4: Not verifying all options in "which is correct" questions** *Wrong thinking:* "Option (a) looks right, I'll mark it." *Correct fix:* In validation questions, test each option systematically. Sometimes multiple options seem plausible, but only one satisfies the equation after proper substitution and calculation.

**Mistake 5: Misreading symbol definitions** *Wrong thinking:* Confusing @ with #, or misremembering which symbol means what midway through calculation. *Correct fix:* Write the substitution key on scratch paper before starting. Refer back to it for every operation rather than relying on memory. Cross out each symbol as you substitute to track progress.

Quick Reference

• **Core skill**: Replace symbols → Apply BODMAS → Calculate carefully → Verify answer.
• **Time per question**: 30–45 seconds with practice. Don't rush the substitution step.
• **BODMAS reminder**: Brackets → Orders → Division/Multiplication (L to R) → Addition/Subtraction (L to R).
• **Symbol substitution format**: "If X means Y" defines new meanings; apply literally without assumptions.
• **Validation strategy**: For "which equation is correct" questions, test options one by one using full calculation.
• **Practice priority**: Solve 20–30 varied questions to internalize substitution patterns and build calculation speed under altered notation.