Mathematical Operation

Table of Contents

1. Syllabus Overview
2. 1: Foundation & Basic Concepts
3. 2: Substitution / Replacement of Operators

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Syllabus Overview

1. Foundation & Basic Concepts: Understanding BODMAS rules, coded operators, and defined functional operations.
2. Substitution / Replacement: Techniques for evaluating expressions after symbol replacement and verifying equality in equations.
3. Interchanging Operators: Logic for swapping meanings between two specific operators (e.g., + ↔ –) and solving multi-step problems.

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1: Foundation & Basic Concepts

1.1 What Are Mathematical Operation Problems?

These problems present standard operations ($+$, $-$, $\times$, $\div$) in a coded form or define new functional rules. You must decode the logic or apply the given definitions to evaluate expressions or solve complex equations.

Two main categories:

• Coding of Operations: Standard symbols are replaced by other symbols (e.g., “$+$” means “$-$”, “$\times$” means “$\div$”).
• Defined Operations: A new operator (like $@, #, $$) is defined using a formula (e.g., $a \ @ \ b = a^2 + b - 5$). You must compute values based on this specific definition.

1.2 BODMAS / Order of Operations

When evaluating any mathematical expression, the standard order of operations remains paramount:

• Brackets (Parentheses)
• Orders (Powers, roots, indices)
• Division and Multiplication (Priority is left to right)
• Addition and Subtraction (Priority is left to right)

1.3 Common Notation in Coding

In coding problems, you are typically provided a mapping key:

• “If ‘$+$’ means ‘$\times$’, ‘$-$’ means ‘$\div$’, ‘$\times$’ means ‘$-$’, ‘$\div$’ means ‘$+$’, then find...”

1.4 Worked Examples – Foundation

Example 1 – Simple Substitution

Question: If ‘$+$’ means ‘$\times$’, ‘$-$’ means ‘$\div$’, ‘$\times$’ means ‘$-$’, ‘$\div$’ means ‘$+$’, find the value of: $8 + 4 - 2 \times 3 \div 5$.

Step 1 – Replace Operators:
$$8 \times 4 \div 2 - 3 + 5$$

Step 2 – Apply BODMAS:

1. Multiplication/Division (Left to Right): $8 \times 4 = 32$, then $32 \div 2 = 16$.
2. Expression now: $16 - 3 + 5$.
3. Addition/Subtraction (Left to Right): $16 - 3 = 13$, then $13 + 5 = 18$.

Answer: $18$

Example 2 – Defined Operation

Question: If $a * b = a^2 + b - 3$, find the value of $4 * 2$.

Step 1 – Identify variables:
$a = 4, \ b = 2$.

Step 2 – Apply formula:
$$4^2 + 2 - 3 = 16 + 2 - 3 = 15$$

Answer: $15$

1.5 Common Mistakes to Avoid

1.6 Quick Practice – Foundation

1. If ‘$\times$’ means ‘$+$’, ‘$\div$’ means ‘$-$’, ‘$+$’ means ‘$\times$’, ‘$-$’ means ‘$\div$’, find: $12 \times 3 \div 4 + 2 - 1$.
2. If $a \ # \ b = a + b - ab$, find: $5 \ # \ 3$.
3. If ‘$-$’ means ‘$\times$’, ‘$\div$’ means ‘$+$’, ‘$+$’ means ‘$\div$’, ‘$\times$’ means ‘$-$’, evaluate: $16 - 4 + 2 \times 3 \div 5$.
4. Define $p \ @ \ q = (p + q)^2$. Find: $2 \ @ \ 3$.

Answers:

1. 7 ($12 + 3 - [4 \times 2 \div 1] = 12 + 3 - 8 = 7$)
2. $-7$ ($5 + 3 - [5 \times 3] = 8 - 15 = -7$)
3. 34 ($16 \times 4 \div 2 - 3 + 5 \rightarrow 64 \div 2 - 3 + 5 = 32 - 3 + 5 = 34$)
4. 25 ($(2 + 3)^2 = 5^2 = 25$)

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2: Substitution / Replacement of Operators

2.1 Core Concepts

In section 2, we delve into more complex substitution mapping. The problem defines a specific mapping from standard symbols to different operations. Your task is to process these systematically:

• Step 1: Identify the replacement key.
• Step 2: Re-write the numeric expression.
• Step 3: Execute BODMAS.

2.2 Types of Substitution Problems

• Type 1 – Direct Evaluation: Numeric expression with coded symbols (evaluate to a number).
• Type 2 – Equation Validation: Determine which option is correct by evaluating both sides of the equal sign.
• Type 3 – Double Interchange: Swapping two symbols (e.g., $+$ becomes $-$ AND $-$ becomes $+$).
• Type 4 – Solving for Unknowns: Finding a missing variable $x$ in a coded equation.

2.3 Methodology for Success

1. Substitution Table: Draw a quick $2 \times 4$ grid for mappings.
2. Parentheses Priority: If original brackets exist, keep them; they override other operations.
3. Stepwise Calculation: Don't skip steps; compute Division, then Multiplication, then Subtraction/Addition.

2.4 Worked Examples

Example 1 – Multi-Operator Replacement

Question: If ‘$+$’ means ‘$-$’, ‘$-$’ means ‘$\times$’, ‘$\times$’ means ‘$\div$’, ‘$\div$’ means ‘$+$’, find: $15 + 3 - 2 \times 4 \div 2$.

Step 1 – Replace:
$$15 - 3 \times 2 \div 4 + 2$$

Step 2 – Apply BODMAS:

• $3 \times 2 = 6$
• $6 \div 4 = 1.5$
• $15 - 1.5 + 2 = 15.5$

Answer: $15.5$

Example 2 – Solving for Unknowns

Question: If ‘$\times$’ means ‘$+$’, ‘$\div$’ means ‘$-$’, ‘$+$’ means ‘$\times$’, ‘$-$’ means ‘$\div$’, find $x$ in: $(12 \times 4) - 3 + x \div 2 = 15$.

Step 1 – Replacement:
$$(12 + 4) \div 3 \times x - 2 = 15$$

Step 2 – Isolate $x$:

1. $(16) \div 3 \times x - 2 = 15$
2. $\frac{16}{3}x = 17$
3. $x = \frac{51}{16} = 3.1875$

Answer: $3.1875$

2.5 Pro Tips for Exam Speed

• The "Table Trick": Write the mapping key on your rough sheet immediately.
• Left-to-Right Rule: For operators of equal precedence ($\times$ and $\div$), always move left to right.
• Mental Bracketing: Group multiplication and division terms mentally before calculating sums.

2.6 Practice Set – Substitution

1. If ‘$+$’ means ‘$\times$’, ‘$-$’ means ‘$\div$’, ‘$\times$’ means ‘$-$’, ‘$\div$’ means ‘$+$’, find: $24 - 6 + 2 \times 4 \div 3$.
2. If ‘$+$’ and ‘$-$’ are interchanged, and ‘$\times$’ and ‘$\div$’ are interchanged, find: $16 \times 4 \div 2 + 3 - 5$.
3. Solve: $(15 - 3) + 4 \times 2 \div 1$ if ‘$-$’ means ‘$\times$’, ‘$\div$’ means ‘$+$’, ‘$\times$’ means ‘$-$’, ‘$+$’ means ‘$\div$’.

Answers:

1. 7 ($24 \div 6 \times 2 - 4 + 3 \rightarrow 4 \times 2 - 4 + 3 \rightarrow 8 - 4 + 3 = 7$)
2. 10 ($16 \div 4 \times 2 - 3 + 5 \rightarrow 4 \times 2 - 3 + 5 \rightarrow 8 - 3 + 5 = 10$)
3. 10.25 ($(15 \times 3) \div 4 - 2 + 1 \rightarrow 45 \div 4 - 2 + 1 \rightarrow 11.25 - 2 + 1 = 10.25$)

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3: Inequalities / Conditional Equations

3.1 Core Concepts

In these problems, you are given a set of statements with mathematical operators ($+$, $-$, $\times$, $\div$) and relational operators ($>$, $<$, $=$, $\ge$, $\le$) that may be coded or replaced. The task is to evaluate the truth of a given inequality or to determine the relationship between two expressions.

Common variations:

• Coded Relational Operators: For example, “$P \ @ \ Q$” means “$P > Q$”, etc.
• Coded Arithmetic Operators with Inequalities: “If ‘$+$’ means ‘$>$’, ‘$-$’ means ‘$<$’, etc.” Then an expression like “$A + B$” is interpreted as “$A > B$”.
• Conditional Equations: Equations given with conditions like “If $2x + 3 = 7$, then find $x$” but with coded operations.

3.2 Types of Problems

• Type 1 – Coded Relational Operators: Given symbols like $@, #, $$ represent $>$, $<$, $=$, $\ge$, $\le$.
• Type 2 – Arithmetic Operators Coded as Inequalities: “$+$” means “$>$”, “$-$” means “$<$”. Expressions like “$5 + 3 - 2$” become a chain of logical comparisons: $5 > 3$ AND $3 < 2$.
• Type 3 – Combined Numerical Evaluation: Given an inequality with coded symbols, determine if it is True/False.
• Type 4 – Solving Conditional Equations: Find the value of a variable $x$ after substituting coded operators.

3.3 Methodology

1. Decode all symbols according to the given legend.
2. Replace each coded symbol with its intended operation or relation.
3. Evaluate using standard arithmetic (BODMAS) and logic.
4. Transitivity: If $A > B$ and $B > C$, then $A > C$.
5. Chain Logic: A chain like $A > B < C$ is True only if both $A > B$ and $B < C$ are independently true.

3.4 Worked Examples

Example 1 – Coded Relational Operators

Question: If ‘$@$’ means ‘$>$’, ‘$#$’ means ‘$<$’, ‘$$$’ means ‘$=$’, then evaluate: $A \ @ \ B$, $B \ # \ C$, $C \ $ \ D$. Is $A > D$?

• Step 1 – Decode: $A > B$, $B < C$, $C = D$.
• Step 2 – Combine: From $B < C$ and $C = D$, we get $B < D$.
• Step 3 – Analysis: From $A > B$ and $B < D$, we cannot directly relate $A$ and $D$.
• Answer: Cannot be determined.

Example 2 – Solving Conditional Equations

Question: If ‘$+$’ means ‘$-$’, ‘$-$’ means ‘$+$’, ‘$\times$’ means ‘$\div$’, ‘$\div$’ means ‘$\times$’, then solve for $x$: $(15 + 3) \times 4 - 2 = x$.

• Step 1 – Replace: $(15 - 3) \div 4 + 2 = x$
• Step 2 – Compute: $12 \div 4 + 2 = 3 + 2 = 5$.
• Answer: $x = 5$

3.5 Common Mistakes

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4: Symbolic / Defined Operations

4.1 Core Concepts

A Defined Operation is a new mathematical rule created specifically for a problem. It follows a formula involving standard operators.
Examples:

• $a * b = a^2 + b - 3$
• $p \ \oplus \ q = (p + q) \times (p - q)$
• $x # y = 2x + y$

4.2 Types of Problems

• Type 1 – Direct Evaluation: Substitute numbers into the formula.
• Type 2 – Chaining: Computing from the innermost parentheses outward (e.g., $a * (b * c)$).
• Type 3 – Mixed Operations: Combine defined ops with standard arithmetic ($[2 # 3] + 5$).
• Type 4 – Solving for Unknowns: Finding $x$ when an equation like $x \oplus 5 = 20$ is given.

4.3 Worked Examples

Example 1 – Chaining

Question: If $a \ @ \ b = a + 2b$, find $(3 \ @ \ 4) \ @ \ 5$.

• Step 1: $3 \ @ \ 4 = 3 + 2(4) = 11$.
• Step 2: $11 \ @ \ 5 = 11 + 2(5) = 21$.
• Answer: $21$

Example 2 – Solving for Unknown

Question: If $x \ \oplus \ y = x \times y + y$, and $a \ \oplus \ 4 = 20$, find $a$.

• Step 1: Substitute: $a \times 4 + 4 = 4a + 4$.
• Step 2: $4a + 4 = 20 \rightarrow 4a = 16 \rightarrow a = 4$.
• Answer: $4$

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5: Data Sufficiency & Advanced Puzzles

5.1 Core Concepts

Data Sufficiency (DS) questions present a primary question followed by two numbered statements. Your objective is not necessarily to solve for a value, but to determine if the provided statements are sufficient to answer the question uniquely.

In the context of mathematical operations, this may involve:

• Evaluating Expressions: Deciding if an expression like $A \ @ \ B$ yields a specific, unique value.
• Determining Relationships: Verifying if $X > Y$ or $P = Q$ based on coded operators.
• Equation Validity: Checking if a coded equation balances correctly using the provided conditions.

Advanced Puzzles combine multiple concepts:

• Multiple defined operations (e.g., $a \ \oplus \ b$ and $a \ \odot \ b$).
• Coded operators mixed with complex inequalities.
• Conditional equations with multiple unknowns.

5.2 Types of DS in Mathematical Operations

• Type 1 – Evaluating an Expression: Determines if a coded rule yields a single unique value.
• Type 2 – Relationship between Variables: Questions like “Is $A > B$?” with statements giving coded inequalities.
• Type 3 – Equation Validity: Is a specific coded equation (e.g., $5 + 3 - 2 \times 4 = 10$) true?
• Type 4 – Solving for Unknowns: Determining if an unknown $x$ can be found uniquely when operators are coded.

5.3 Methodology for Data Sufficiency

1. Analyze Statement I alone: Can you answer the question uniquely?
2. Analyze Statement II alone: Can you answer the question uniquely?
3. Combine statements: Only if neither alone is sufficient.
4. A "No" is still a definite answer: If a statement proves something is definitively false, it is sufficient.
5. Beware of multiple solutions: If Statement I gives $x = \pm 3$, and the question asks for the value of $x$, Statement I is insufficient.

5.4 Worked Examples – Data Sufficiency

Example 1 – Defined Operation

Question: What is the value of $3 \ @ \ 4$?

• Statement I: $a \ @ \ b = a + b - 3$.
• Statement II: $2 \ @ \ 5 = 4$.

Analysis:

• I alone: Directly gives the definition: $3 + 4 - 3 = 4$. Unique value. Sufficient.
• II alone: Only one coordinate $(2, 5)$ is given. Multiple formulas could fit (e.g., $a \times b - 6$ or $a + b - 3$). We cannot find $3 \ @ \ 4$ uniquely. Insufficient.
• Answer: Statement I alone is sufficient.

Example 2 – Solving for Unknown

Question: What is the value of $x$?

• Statement I: $x \ @ \ 3 = 12$, where $a \ @ \ b = a^2 + b$.
• Statement II: $2 \ @ \ x = 8$, where the same definition applies.

Analysis:

• I alone: $x^2 + 3 = 12 \rightarrow x^2 = 9 \rightarrow x = +3$ or $-3$. Not unique. Insufficient.
• II alone: $2^2 + x = 8 \rightarrow 4 + x = 8 \rightarrow x = 4$. Unique value. Sufficient.
• Answer: Statement II alone is sufficient.

5.5 Advanced Puzzles

These tasks require you to deduce the logic from examples.

Example – Pattern Deduction

Puzzle: If $2 \ # \ 3 = 7$, $3 \ # \ 4 = 13$, and $4 \ # \ 5 = 21$, find $5 \ # \ 6$.

• Step 1 – Detect Pattern: $2 \times 3 + 1 = 7$, $3 \times 4 + 1 = 13$, $4 \times 5 + 1 = 21$.
• Step 2 – Apply Pattern: $5 \times 6 + 1 = 31$.
• Answer: $31$

5.6 Pro Tips for DS & Puzzles

• Test Independence: Never use Statement I’s rule to evaluate Statement II unless you are explicitly combining them.
• The "Rule of Three": In advanced puzzles, verify your formula on at least three examples before applying it to the question.
• Chain Logic: When evaluating complex inequality chains like $5 > 3 < 2 = 4$, every comparison ($5 > 3$, $3 < 2$) must be independently true for the chain to be valid.

5.7 Practice Set – DS & Advanced Puzzles

1. Question: Is $4 * 2 > 3 * 5$? ('*' is defined)
   • I: $a * b = a + b$.
   • II: $a * b = a \times b$.
   • Answer: D (I alone: $6 > 8$ False; II alone: $8 > 15$ False. Both provide a definite "No").
2. Puzzle: $2 \ \odot \ 3 = 11$, $3 \ \odot \ 4 = 19$, $4 \ \odot \ 5 = 29$. Find $5 \ \odot \ 6$.
   • Formula: $(a \times b) + (a + b)$.
   • Answer: $30 + 11 = 41$.

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Mock Test: Mathematical Operation Practice Lab

Welcome to the Mathematical Operation Practice Lab. This mock test features 30 interactive challenges designed to test your mastery in decoding operations and applying BODMAS accurately.