Mathematical Operations — SSC MTS Study Notes
Overview
Mathematical Operations is a reasoning topic that tests your ability to substitute symbols and balance equations according to given rules. Unlike regular arithmetic, these questions replace standard operators (+, −, ×, ÷) with unusual symbols or interchange them, then ask you to evaluate expressions or find which equation becomes true under the new system.
This topic appears regularly in SSC MTS Paper 1, typically 1–2 questions per exam. The questions are usually straightforward once you understand the substitution pattern, but errors happen when students apply the wrong order of operations or misread the symbol key. Mastering this topic means careful reading, systematic substitution, and strict adherence to BODMAS rules after substitution. Most questions are solvable in 30–45 seconds with practice, making them high-value scoring opportunities.
The key skill is mental flexibility — training your brain to treat a symbol like @ or ★ as a stand-in for a mathematical operation, then computing accurately. Success requires precision in transcription and calculation, not advanced mathematical concepts.
Key Concepts
- **Symbol Substitution**: The question provides a coding scheme where unusual symbols represent standard operations. For example, "If + means ×, − means ÷, × means +, and ÷ means −" then you must mentally or physically replace each symbol before computing.
- **Interchanged Operators**: Sometimes the question interchanges two standard operators (e.g., + and × are swapped, − and ÷ are swapped). Apply these swaps to the entire expression before evaluating.
- **BODMAS After Substitution**: Always apply BODMAS/BODMAS order (Brackets, Orders, Division/Multiplication left-to-right, Addition/Subtraction left-to-right) to the expression *after* you've made all symbol substitutions. The substitution changes what the symbols mean, not the order of operations.
- **Balancing Equations**: Some questions ask which arrangement of operators makes a given numerical equation true (e.g., 12 ? 3 ? 4 = 8, find the correct operators). Test each option by substituting and calculating.
- **Single vs. Multiple Substitutions**: Simpler questions substitute all occurrences of one or two symbols. Harder questions may involve four-way or circular swaps. Read the key carefully and apply every rule.
- **No Partial Credit**: These are multiple-choice questions. An error in one substitution or one calculation step yields a wrong answer. Accuracy is everything.
Key Facts
1. **Standard Operation Order**: Division and Multiplication are performed left to right before Addition and Subtraction (also left to right). This applies after substitution.
2. **Common Symbol Sets**: Expect symbols like @, #, $, %, ★, ÷, ×, +, − either as replacements or as the operators being replaced.
3. **Four-Operator Interchange**: The most common format gives you four statements: "A means B, C means D, E means F, G means H" covering +, −, ×, ÷. Decode all four before computing.
4. **Equation Format**: Questions may present expressions like "15 × 3 + 6 − 2 ÷ 1" and ask for the result after applying the given substitutions, or they may ask "Which equation is correct?" and list options.
5. **Integer Results**: Almost all SSC MTS questions yield integer or simple decimal results. If your answer is a complex fraction or irrational, recheck your substitutions.
6. **Time Management**: Allocate 30–45 seconds per question. If substitution takes longer, you may have misread the key or chosen a complicated path.
Worked Examples
**Example 1: Basic Symbol Substitution**
**Question**: If + means ×, − means +, × means ÷, and ÷ means −, then find the value of: 18 + 6 − 12 × 4 ÷ 3
**Solution**:
- Replace symbols as per the key:
- + becomes × → 18 × 6
- − becomes + → (result) + 12
- × becomes ÷ → 12 ÷ 4
- ÷ becomes − → (result) − 3
- Rewrite the expression: 18 × 6 + 12 ÷ 4 − 3
- Apply BODMAS (multiplication and division first, left to right):
- 18 × 6 = 108
- 12 ÷ 4 = 3
- Expression becomes: 108 + 3 − 3
- Addition and subtraction left to right:
- 108 + 3 = 111
- 111 − 3 = 108
**Answer**: 108
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**Example 2: Balancing Equations**
**Question**: If × means +, + means ÷, ÷ means −, and − means ×, which of the following equations is correct?
(A) 16 × 4 + 2 − 3 ÷ 4 = 16 (B) 16 × 4 + 2 − 3 ÷ 4 = 20 (C) 16 × 4 + 2 − 3 ÷ 4 = 18 (D) 16 × 4 + 2 − 3 ÷ 4 = 22
**Solution**:
- Substitute symbols in the expression 16 × 4 + 2 − 3 ÷ 4:
- × becomes + → 16 + 4
- + becomes ÷ → (result) ÷ 2
- − becomes × → (result) × 3
- ÷ becomes − → (result) − 4
- Rewrite: 16 + 4 ÷ 2 × 3 − 4
- Apply BODMAS (division and multiplication left to right):
- 4 ÷ 2 = 2
- 2 × 3 = 6
- Expression becomes: 16 + 6 − 4
- Addition and subtraction left to right:
- 16 + 6 = 22
- 22 − 4 = 18
**Answer**: (C) 18
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**Example 3: Interchanged Signs**
**Question**: If the signs + and ÷ are interchanged, and the numbers 2 and 4 are interchanged, what is the value of: 4 + 2 ÷ 8
**Solution**:
- Interchange signs: + becomes ÷ and ÷ becomes +
- Interchange numbers: 4 becomes 2 and 2 becomes 4
- Original: 4 + 2 ÷ 8
- After number swap: 2 + 4 ÷ 8
- After sign swap: 2 ÷ 4 + 8
- Apply BODMAS (division first):
- 2 ÷ 4 = 0.5
- Then addition:
- 0.5 + 8 = 8.5
**Answer**: 8.5
Common Mistakes
1. **Forgetting to Apply All Substitutions** → Students substitute two operators but miss the other two in a four-way swap. *Fix*: Write out all four substitution rules before touching the expression. Check each symbol in the expression against your list.
2. **Ignoring BODMAS After Substitution** → After substituting × for +, students add first instead of multiplying. *Fix*: Treat the substituted expression as a brand-new arithmetic problem. Multiplication and division always come before addition and subtraction, left to right.
3. **Left-to-Right Without Respecting Precedence** → Students compute "18 × 6 + 12 ÷ 4" as (18 × 6 + 12) ÷ 4 by going strictly left to right. *Fix*: Do all multiplications and divisions first (in any order or left-to-right among themselves), then all additions and subtractions.
4. **Mixing Up the Direction of Substitution** → If the question says "+ means ×", students sometimes reverse it and think "× means +". *Fix*: Write "original → new" clearly. The statement "+ means ×" translates every + in the expression to ×.
5. **Rushing Through Symbol Keys** → The symbol key is often presented in a long sentence. Students skim it and misread one substitution. *Fix*: Underline or rewrite the key in shorthand (e.g., + → ×, − → ÷, × → +, ÷ → −) before solving.
Quick Reference
- **Read the key carefully**: Write down or mentally note all substitutions before touching the expression.
- **Substitute first, compute second**: Replace every symbol, then treat it as a normal arithmetic problem.
- **BODMAS is non-negotiable**: Multiply/Divide before you Add/Subtract, always left-to-right within each precedence level.
- **Check integer results**: SSC MTS answers are usually whole numbers or simple decimals. Complex fractions signal an error.
- **Practice 10–15 questions**: Symbol substitution becomes automatic with repetition. Aim for under 40 seconds per question.
- **Balancing equations**: Test each option by substituting and calculating. Eliminate wrong answers quickly to save time.