Mathematical Operations

 Mathematical Operations is a reasoning topic where mathematical symbols (+, −, ×, ÷, =) are interchanged or used in coded forms, and you must identify the correct equation or perform logical symbol substitutions to make a statement correct.

🧠 Types of Questions

Type	Description	Example
1. Symbol Interchange	Replace symbols and solve	If ‘+’ means ‘×’, etc.
2. Verify Equation	Check if given equation is true after replacing operators
3. Coded Equations	Symbols represent operations	A @ B = A + B
4. Balance Equations	Find correct symbol to balance	3 ? 2 = 6
5. Sign-Number Interchange	Digits and signs are interchanged	2 + 3 = 6 becomes 2 × 3 = 6

🧮 Key Concepts and Symbols

Symbol	Meaning (can be changed in question)
+	Addition
−	Subtraction
×	Multiplication
÷	Division
=	Equals

🧰 Important Methods

✅ Step-by-Step Method to Solve

1. Read symbol definitions carefully

2. Rewrite the equation using the new meanings

3. Apply BODMAS rule for correct order

4. Check if both sides match

5. Answer based on validation

🧩 Examples with Diagrams & Explanation

🔷 Example 1: Basic Symbol Substitution

Q: If + means ×, − means ÷, × means −, and ÷ means +, then solve:
6 + 3 − 2 × 1 ÷ 5 = ?

✅ Solution:

Substitute:

• + → ×

• − → ÷

• × → −

• ÷ → +

So:
6 × 3 ÷ 2 − 1 + 5

Now apply BODMAS:
→ 6 × 3 = 18
→ 18 ÷ 2 = 9
→ 9 − 1 = 8
→ 8 + 5 = 13

✅ Answer: 13

🔷 Example 2: Equation Validation

Q: If ‘×’ means ‘+’, ‘+’ means ‘−’, ‘−’ means ‘÷’, and ‘÷’ means ‘×’, then is
8 × 3 + 2 = 13 correct?

✅ Solution:

Substitute:

• × → + → 8 + 3 = 11

• + → − → 11 − 2 = 9 ≠ 13

✅ Answer: No, equation is incorrect.

🔷 Example 3: Match the Operation

Q: Which symbol should replace ? to make the statement correct?
9 ? 3 = 3

Options: (A) + (B) − (C) ÷ (D) ×

Try:

• 9 + 3 = 12 ❌

• 9 − 3 = 6 ❌

• 9 ÷ 3 = 3 ✅

• 9 × 3 = 27 ❌

✅ Answer: (C) ÷

🔷 Example 4: Coded Operation

Q: If A @ B means A² + B, and A $ B means A + B², then what is value of:
3 @ 2 $ 1

✅ Solution:

Step 1: Evaluate inside out
→ First: 2 $ 1 = 2 + 1² = 2 + 1 = 3
→ Now: 3 @ 3 = 3² + 3 = 9 + 3 = 12

✅ Answer: 12

🧭 Diagrams to Visualize Substitution

🔄 Symbol Table:

Given Symbol	Actual Operation
+	× (Multiply)
−	÷ (Divide)
×	− (Subtract)
÷	+ (Add)

Visual Conversion:

Original: 6 + 3 − 2
Becomes:  6 × 3 ÷ 2
Apply BODMAS → Solve

📌 Key Tips for Exams

• Never solve directly — always rewrite with new symbols

• Use BODMAS after substitution

• Practice common operator swap patterns

• Watch for trick substitutions (e.g., + means ÷)

📝 Mini Practice Challenge

Q: If ‘+’ means ‘−’, ‘−’ means ‘×’, and ‘×’ means ‘+’, then what is the value of:
8 − 2 + 1 × 3?

✅ Step-by-step:

Original → 8 − 2 + 1 × 3
Convert:

• − → × → 8 × 2 = 16

• + → − → 16 − 1 = 15

• × → + → 15 + 3 = 18

✅ Answer: 18

✅ Practice Questions - Mathematical Operations 

🔹 Q1.

If ‘+’ means ‘×’, ‘×’ means ‘−’, ‘−’ means ‘÷’, and ‘÷’ means ‘+’, then what is the value of:
6 + 3 × 2 − 1 = ?

✅ Substitute:

• + → ×

• × → −

• − → ÷

So expression becomes:
6 × 3 − 2 ÷ 1
→ 6 × 3 = 18
→ 18 − 2 = 16
→ 16 ÷ 1 = 16

✅ Answer: 16

🔹 Q2.

If ‘×’ means ‘+’, ‘+’ means ‘−’, ‘−’ means ‘÷’, and ‘÷’ means ‘×’, then:
9 + 3 × 2 − 1 = ?

✅ Substitution:

• • → −

• × → +

• − → ÷

Now: 9 − 3 + 2 ÷ 1
→ 9 − 3 = 6
→ 6 + 2 = 8
→ 8 ÷ 1 = 8

✅ Answer: 8

🔹 Q3.

Which symbol should replace ? to make the equation correct:
15 ? 5 = 3

Try each:

• 15 + 5 = 20 ❌

• 15 − 5 = 10 ❌

• 15 ÷ 5 = 3 ✅

✅ Answer: ÷

🔹 Q4.

If ‘×’ means ‘−’, ‘−’ means ‘+’, ‘+’ means ‘÷’, and ‘÷’ means ‘×’, then:
12 + 6 − 3 = ?

✅ Substitution:

• → ÷, − → +
  Now: 12 ÷ 6 + 3 = ?
  → 2 + 3 = 5

✅ Answer: 5

🔹 Q5.

If ‘A @ B’ means A + B, and ‘A $ B’ means A − B, then solve:
5 $ 3 @ 2 = ?

→ First: 5 − 3 = 2
→ Then: 2 + 2 = 4

✅ Answer: 4

🔹 Q6.

If ‘×’ means ‘+’, ‘+’ means ‘×’, then:
6 + 3 × 2 = ?

→ Change: 6 × 3 + 2
→ 6 + 3 = 9
→ 9 × 2 = 18

✅ Answer: 18

🔹 Q7.

What will be the output of:
If ‘+’ means ‘÷’, ‘−’ means ‘×’, ‘×’ means ‘−’, and ‘÷’ means ‘+’, then
12 − 2 × 3 + 1 = ?

→ Substitution:
− → ×, × → −, + → ÷
So: 12 × 2 − 3 ÷ 1
→ 24 − 3 = 21
→ 21 ÷ 1 = 21

✅ Answer: 21

🔹 Q8.

If A # B = A² – B, find the value of:
6 # 3 = ?

→ 6² – 3 = 36 – 3 = 33

✅ Answer: 33

🔹 Q9.

If 5 + 3 = 24, 6 + 2 = 36, what is 7 + 1 = ?

Pattern: (First × Second) + (First + Second)

→ 5 × 3 = 15, 5 + 3 = 8 → 15 + 8 = 23 ❌
But in original: 5 + 3 = 24 → Try: 5 + 3 = 8 → 8 × 3 = 24
New Pattern: (A + B) × B

→ 7 + 1 = (7 + 1) × 1 = 8 × 1 = 8

✅ Answer: 8

🔹 Q10.

If ‘%’ means ‘+’, ‘$’ means ‘−’, then solve:
20 % 10 $ 5 = ?

→ 20 + 10 − 5 = 25

✅ Answer: 25