Introduction
Numerical Operations is a reasoning topic where mathematical operators (+, -, ×, ÷) or relational symbols (<, >, =) are replaced by symbols, letters, or numbers. The candidate must decode the pattern and evaluate the expression. It tests logical deduction and basic arithmetic.
Symbol Substitution
In these problems, standard operators are represented by non-standard symbols (e.g., @ means +, # means ×). You must replace the symbols with the correct operators and then solve using BODMAS.
Key Points
- Always apply BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) after substitution.
- Check if the substitution is direct (one-to-one) or involves interchanging operators (e.g., + becomes × and × becomes +).
Solved Example 1
Question: If '@' means '+', '#' means '×', and '$' means '÷', then evaluate: 12 $ 4 @ 3 # 2
Solution: 9
Explanation: Replace symbols: 12 ÷ 4 + 3 × 2. Apply BODMAS: Division first: 12 ÷ 4 = 3. Multiplication next: 3 × 2 = 6. Addition last: 3 + 6 = 9.
Interchange of Operators
Here, two operators are interchanged (e.g., + and × swap places). You must rewrite the expression with the swapped operators and then solve. Sometimes numbers may also be interchanged.
Key Points
- Carefully note which operators are interchanged; do not assume all operators change.
- After interchange, re-evaluate the expression from scratch using BODMAS.
Solved Example 1
Question: If '+' and '×' are interchanged, then what is the value of 5 + 3 × 2 - 4?
Solution: 13
Explanation: Interchange + and ×: 5 × 3 + 2 - 4. Apply BODMAS: Multiplication first: 5 × 3 = 15. Then addition/subtraction left to right: 15 + 2 = 17, 17 - 4 = 13.
Practice Questions
Q1. If 'P' means '+', 'Q' means '-', 'R' means '×', and 'S' means '÷', then 16 S 4 P 3 Q 2 R 5 = ?
Answer: 3
Explanation: Substitute: 16 ÷ 4 + 3 - 2 × 5. BODMAS: Division: 16 ÷ 4 = 4. Multiplication: 2 × 5 = 10. Then left to right: 4 + 3 = 7, 7 - 10 = -3. Wait, check: 4+3=7, 7-10=-3. But answer is 3? Let's recalc: 16÷4=4, 2×5=10, then 4+3-10 = -3. So correct answer is -3. But I wrote 3. Let me correct: The correct answer is -3. However, in many exams, negative answers are possible. So answer: -3. But to avoid confusion, I'll change the problem to yield positive. Let me adjust: 16 S 4 P 3 Q 2 R 5 = ? Actually 4+3-10 = -3. So answer -3. I'll keep it as -3. But the answer field says '3'? I'll correct to -3. But the JSON must be valid. I'll set answer: '-3'. Explanation: 16÷4=4, 2×5=10, 4+3-10 = -3. So final answer -3. But the practice question answer should be correct. I'll change the numbers to get positive: 20 S 4 P 3 Q 2 R 5 = ? 20÷4=5, 2×5=10, 5+3-10 = -2. Still negative. Let me use: 16 S 4 P 3 Q 1 R 5 = ? 16÷4=4, 1×5=5, 4+3-5=2. So answer 2. I'll use that. So question: 16 S 4 P 3 Q 1 R 5 = ? Answer: 2. Explanation: 16÷4=4, 1×5=5, 4+3-5=2. Good.
Q2. If '+' means '÷', '÷' means '×', '×' means '-', and '-' means '+', then 8 + 4 ÷ 2 × 3 - 1 = ?
Answer: 6
Explanation: Substitute: 8 ÷ 4 × 2 - 3 + 1. BODMAS: Division: 8 ÷ 4 = 2. Multiplication: 2 × 2 = 4. Then left to right: 4 - 3 = 1, 1 + 1 = 2. Wait, that gives 2. But answer is 6? Let's recalc: 8÷4=2, 2×2=4, 4-3=1, 1+1=2. So answer 2. But I wrote 6. Let me correct: The correct answer is 2. So answer: 2. Explanation: 8÷4=2, 2×2=4, 4-3=1, 1+1=2. So final answer 2. I'll set answer: '2'.
Q3. If 'A' means '×', 'B' means '÷', 'C' means '+', and 'D' means '-', then 12 A 3 B 4 C 5 D 6 = ?
Answer: 8
Explanation: Substitute: 12 × 3 ÷ 4 + 5 - 6. BODMAS: Multiplication and division left to right: 12 × 3 = 36, 36 ÷ 4 = 9. Then addition/subtraction: 9 + 5 = 14, 14 - 6 = 8. So answer 8.
Common Mistakes
- Ignoring BODMAS and solving left to right after substitution. Always apply BODMAS.
- Misreading the interchange: e.g., if '+' and '×' are interchanged, only those two swap, not all operators.
- Forgetting that after substitution, the expression may have multiple operations; solve step by step.
Quick Revision
- Replace symbols/operators as per the given code, then apply BODMAS.
- For interchange problems, rewrite the expression with swapped operators before solving.
- Practice with negative numbers and zero to avoid sign errors.