Math-ops — quick-fire examples
Symbol-substitution questions have a perfect strategy: they cannot be solved in your head in the order the symbols appear — you must substitute every operator first, write the new expression clearly, then apply the order of operations. One careless step costs the mark; the two-step discipline guarantees it.
Definition: A symbol-substitution problem redefines one or more arithmetic operators (e.g. "+" actually means "×"), requiring you to replace each symbol with its true meaning before computing.
Definition: BODMAS (Brackets, Orders/Exponents, Division, Multiplication, Addition, Subtraction) is the universal precedence rule applied to any arithmetic expression — it governs the order of calculation after substitution.
Why the two-step method is mandatory
Consider "12 + 4" where "+" means "÷": the answer is 3, not 16. If you compute using the symbols as written and only change the meaning in your head mid-calculation, you will likely revert to habit. Writing the fully substituted expression on rough paper before touching numbers eliminates this risk.
The two-step method:
- Read the substitution rule carefully. Write the translated expression in one shot.
- Apply BODMAS to the translated expression exactly as you would any standard maths problem.
Worked examples — operator substitution
Question: If '+' means '×', '−' means '+', '×' means '÷', and '÷' means '−', evaluate 12 ÷ 6 × 4 − 2 + 3.
Solution:
Step 1: Substitute every symbol: ÷→−, ×→÷, −→+, +→×. The expression becomes: 12 − 6 ÷ 4 + 2 × 3.
Step 2: Apply BODMAS — division and multiplication before addition and subtraction: 6÷4 = 1.5; 2×3 = 6.
Step 3: Left to right: 12 − 1.5 + 6 = 16.5.
Conclusion: 16.5.
Question: If A means '+', B means '−', C means '×', D means '÷', evaluate 60 D 4 C 5 B 8 A 2.
Solution:
Step 1: Replace letters: 60 ÷ 4 × 5 − 8 + 2.
Step 2: BODMAS — ÷ and × first (left to right): 60÷4 = 15; 15×5 = 75.
Step 3: 75 − 8 + 2 = 69.
Conclusion: 69.
Question: If '<' means '+', '>' means '−', and '=' means '×', evaluate 6 = 4 < 8 > 2.
Solution:
Step 1: Substitute: 6 × 4 + 8 − 2.
Step 2: Multiplication first: 6×4 = 24.
Step 3: 24 + 8 − 2 = 30.
Conclusion: 30.
Sign-swap questions — the bilateral rule
When two operators are interchanged or swapped, the exchange is two-way: every occurrence of A becomes B and every occurrence of B becomes A. A common error is swapping only one direction.
Question: If '+' and '−' are interchanged, find the value of 14 − 6 + 8 − 3 + 12.
Solution:
Step 1: Every '−' becomes '+' and every '+' becomes '−': 14 + 6 − 8 + 3 − 12.
Step 2: Left to right: 14+6 = 20; 20−8 = 12; 12+3 = 15; 15−12 = 3.
Conclusion: 3.
Digit-swap questions
Some questions swap specific digits (e.g. "3 and 7 swap") and may combine this with an operator swap.
Question: If digits 3 and 7 swap, and '+' swaps with '−', find 27 + 73.
Solution:
Step 1: Digit swap: 27 → 23 (the 7 becomes 3); 73 → 37 (the 3 becomes 7).
Step 2: Operator swap: '+' → '−'. The expression becomes 23 − 37.
Conclusion: −14.
Custom-operator questions — the formula approach
Some questions define a brand-new operator with an explicit algebraic formula. Apply the definition literally — do not attempt to guess based on the symbol used.
Question: If a * b = a² + b, find (3 * 4) * 2.
Solution:
Step 1: Resolve the inner bracket first: 3 * 4 = 3² + 4 = 9 + 4 = 13.
Step 2: Now the outer: 13 * 2 = 13² + 2 = 169 + 2.
Conclusion: 171.
Question: If a # b = (a + b)/(a − b), find (6 # 2) # (3 # 1).
Solution:
Step 1: 6 # 2 = (6+2)/(6−2) = 8/4 = 2.
Step 2: 3 # 1 = (3+1)/(3−1) = 4/2 = 2.
Step 3: 2 # 2 = (2+2)/(2−2) = 4/0 → undefined (a valid exam answer; check for this trap).
Conclusion: Undefined (or "cannot be determined"). This tests whether you apply the formula rather than assume symmetry.
Why it matters: Symbol-substitution questions appear in the Reasoning section of every SSC (CGL, CHSL, MTS), RRB (NTPC, Group D), and banking exam. They are fast marks — the whole question takes 30–60 seconds if you use the two-step method. They are lost marks if you try to hold the substitution in your head.
Real-world example: Spreadsheet logic follows identical rules. Type =12-6/4+2*3 into any spreadsheet cell and it returns 16.5, because the application applies BODMAS automatically — division (6/4 = 1.5) and multiplication (2×3 = 6) before subtraction and addition. The exam is testing the same algorithmic discipline that software follows.
Common misconception: After doing the substitution, many students still solve the expression left-to-right, ignoring BODMAS. They get 12 − 6 = 6; 6 ÷ 4 = 1.5; 1.5 + 2 = 3.5; 3.5 × 3 = 10.5 — wrong. Division and multiplication must be resolved before addition and subtraction regardless of their position in the expression.
- ✓- Step 1: substitute every symbol/digit. Step 2: apply BODMAS. Never skip or merge these steps.
- ✓- Write the substituted expression on rough paper before computing — errors almost always come from not doing this.
- ✓- "Interchange" or "swap" is bilateral — both operators change places throughout the expression.
- ✓- Custom operators: apply the formula definition literally, resolving inner brackets first.
- ✓- BODMAS precedence: Brackets → Orders → Division/Multiplication → Addition/Subtraction.
- ✓- Division and multiplication are equal priority; resolve left to right among them.
"SUBSTITUTE then OPERATE" — two steps, always in that order.
"Swap is two-way, not one-way" — if + and − swap, every + becomes − and every − becomes +.
- ✓- Substitute first (completely, on paper), then solve using BODMAS.
- ✓- Swap questions require bilateral changes to all instances of both symbols.
- ✓- Custom-operator rules are followed exactly as given; inner brackets first.
- ✓- The most common error is applying operators in the original (pre-substitution) sequence — discipline prevents it.
Mathematical operations & symbol substitution
One of the cleverest tricks examiners play is to swap the symbols you've been calculating with all your life — suddenly '+' becomes '×' and '÷' becomes '-' — and the only students who breeze through are those who have made the symbol-swap procedure completely automatic.
Definition: Mathematical Operations / Symbol Substitution — a reasoning question type where standard arithmetic operators (+, −, ×, ÷) are replaced by other symbols (letters, @, #, $, etc.), and the student must evaluate the expression using the redefined operations.
Definition: BODMAS/BODMAS Rule — the order in which operations must be performed: Brackets → Of (powers/roots) → Division → Multiplication → Addition → Subtraction. Division and multiplication have equal priority (left to right); same for addition and subtraction.
Why This Topic Exists in Exams
Symbol substitution tests whether a candidate can:
- Accurately translate one code to another without error.
- Apply BODMAS correctly after substitution.
- Work quickly without making careless "muscle-memory" errors (the brain wants to treat + as addition, not multiplication).
It appears in SSC CGL/CHSL/MTS, RRB NTPC/Group D, banking prelims, and all similar aptitude sections.
Type 1 — Symbol-to-Operator Substitution
You are given a key mapping symbols to operators, then asked to evaluate an expression.
Standard format:
If @ means +, # means −, $ means ×, & means ÷, evaluate: 6 $ 4 @ 8 & 2 # 1
Step-by-step method:
- Substitute: Replace each symbol with its operator:
6 × 4 + 8 ÷ 2 − 1 - Apply BODMAS:
- Division and multiplication first (left to right): 6 × 4 = 24; 8 ÷ 2 = 4
- Expression becomes: 24 + 4 − 1
- Addition then subtraction (left to right): 24 + 4 = 28; 28 − 1 = 27
- Answer: 27
Common error: Doing 24 + 4 = 28, then 28 − 1 = 27 is correct, but some students do 4 − 1 = 3 first, then 24 + 3 = 27 (same answer here, but this is WRONG method — can give wrong answer in other cases). Always proceed strictly left to right once you've resolved all × and ÷.
Type 2 — Interchanged Signs
The question says two operators are interchanged in the equation. You must swap them mentally.
Example: If + and × are interchanged, what is the value of 6 + 4 × 2?
Method:
- Swap the two operators everywhere in the expression:
6 × 4 + 2 - Apply BODMAS: 6 × 4 = 24; 24 + 2 = 26
More complex example: If − and ÷ are interchanged and + and × are interchanged, evaluate 12 + 6 × 3 − 9 ÷ 3.
- Swap + ↔ × and − ↔ ÷:
12 × 6 + 3 ÷ 9 − 3(every + becomes ×, every × becomes +, every − becomes ÷, every ÷ becomes −) - BODMAS: First × and ÷: 12 × 6 = 72; 3 ÷ 9 = 0.333...
- Then + and −: 72 + 0.333 − 3 = 69.33
Critical tip: When two pairs are interchanged simultaneously, map every single operator. Many exam papers have only one pair swapped — read the question carefully every time.
Type 3 — Letter-as-Operator Substitution
Example: If P stands for ÷, Q for ×, R for +, S for −, evaluate: 36 P 12 Q 2 R 4 S 1
Method:
- Substitute:
36 ÷ 12 × 2 + 4 − 1 - BODMAS: 36 ÷ 12 = 3; 3 × 2 = 6; 6 + 4 = 10; 10 − 1 = 9
This is identical in principle to Type 1 — just letters instead of symbols.
Type 4 — Balancing with Symbol Substitution
Some questions ask "which expression is equal to X?" or "which operator makes the equation true?"
Example: If * means +, what should replace ? in 15 ? 3 * 2 = 4?
- Replace *:
15 ? 3 + 2 = 4→15 ? 3 = 2 - What gives 15 ? 3 = 2? If ? = ÷, then 15 ÷ 3 = 5 ≠ 2. If ? = −, then 15 − 3 = 12 ≠ 2. No standard operator works here cleanly — but in real exam format, choices are given and one works.
Type 5 — Finding the Wrong Equation
Some questions give four equations (a, b, c, d) and ask which one is correct after substituting the given symbols.
Method: Evaluate each option with the given key; the one that gives a true equation is the answer.
BODMAS Deep Dive — The Non-Negotiable Rules
B — Brackets: Innermost first. (2 + 3) × 4 = 5 × 4 = 20, NOT 2 + 12 = 14.
O — Of (powers, roots): 2 + 3² = 2 + 9 = 11, NOT (2 + 3)² = 25.
DM — Division and Multiplication (left to right, equal priority):
12 ÷ 4 × 3: Start left → 12 ÷ 4 = 3 → 3 × 3 = 9 (NOT 12 ÷ 12 = 1).
AS — Addition and Subtraction (left to right, equal priority):
10 − 3 + 2: Start left → 10 − 3 = 7 → 7 + 2 = 9 (NOT 10 − 5 = 5).
Common misconception: Many students think division always comes before multiplication (because D appears before M in BODMAS). In reality, they are equal priority — process left to right. Same for A and S.
Worked Examples
Question 1: If @ means ×, # means ÷, % means +, $ means −, find the value of:16 # 4 @ 2 % 8 $ 3
Solution:
Step 1: Substitute: 16 ÷ 4 × 2 + 8 − 3
Step 2: BODMAS — Division first (left): 16 ÷ 4 = 4
Step 3: Multiplication: 4 × 2 = 8
Step 4: Expression: 8 + 8 − 3
Step 5: Left to right: 8 + 8 = 16; 16 − 3 = 13
Question 2: The signs + and − are interchanged and the signs × and ÷ are interchanged. What is the value of 48 × 12 + 16 ÷ 4 − 8?
Solution:
Step 1: Swap: × ↔ ÷ and + ↔ −: 48 ÷ 12 − 16 × 4 + 8
Step 2: BODMAS: 48 ÷ 12 = 4; 16 × 4 = 64
Step 3: Expression: 4 − 64 + 8
Step 4: Left to right: 4 − 64 = −60; −60 + 8 = −52
Conclusion: −52. Notice that negative results are fully valid — don't be alarmed.
Question 3 (Finding correct equation): Given: A means +, B means ×, C means −, D means ÷. Which equation is correct?
(a) 3 B 4 A 2 D 2 C 1 = 13
(b) 3 B 4 A 2 D 2 C 1 = 12
Evaluate: 3 × 4 + 2 ÷ 2 − 1 = 12 + 1 − 1 = 12. Option (b) is correct.
Speed Strategy for Exam Hall
- Write the substituted expression first before calculating. Do not try to do substitution and BODMAS simultaneously.
- Circle division and multiplication in the substituted expression; resolve them first from left to right.
- For sign-interchange questions: underline each operator in the original expression and replace it explicitly. Don't trust your memory.
- Time budget: these questions should take 30–40 seconds max. If you exceed 60 seconds, mark and move on.
- ✓- Always write the fully substituted expression before applying BODMAS — never compute on-the-fly.
- ✓- Division and multiplication have equal priority; so do addition and subtraction — process each pair left to right.
- ✓- In sign-interchange questions, replace EVERY occurrence of the swapped operators — missing even one gives the wrong answer.
- ✓- Negative results are valid — don't reject an option just because it's negative.
- ✓- The most common error is treating BODMAS as strictly ordered (D before M, A before S) rather than equal-priority left-to-right pairs.
- ✓- For multi-swap problems (two pairs interchanged), map out all replacements explicitly before computing.
- ✓- This question type rewards speed — practice until substitution is a reflex (target: <30 seconds per question).
- ✓- Letter-as-operator and symbol-as-operator are identical in method — just different notation.
"Substitute Then BODMAS" — STB. That's the entire method in three words. Never skip step 1 (full substitution) even if it seems like one step; that's where most errors enter.
- ✓- Symbol substitution: replace each symbol/letter with its defined operator, then evaluate with BODMAS.
- ✓- BODMAS order: Brackets → Of → (Division = Multiplication, left to right) → (Addition = Subtraction, left to right).
- ✓- Sign interchange: swap all instances of the stated operators before calculating.
- ✓- Common exam variants: symbol key given, letters as operators, signs interchanged, find the correct equation.
- ✓- Negative answers are valid; don't second-guess if calculation is correct.
- ✓- Practice making substitution automatic — the exam rewards those who do this without thinking.
Mathematical Operations — RRB Technician Reasoning
Mathematical Operations is one of the most predictable scoring areas in RRB Technician CBT-1. Reasoning carries 25 of 100 marks, and 1–2 of those are almost always symbol-substitution or BODMAS questions. They need zero GK — only clean arithmetic — so a steady method makes them near-guaranteed marks in ~30 seconds each.
Why it matters
Every candidate can attempt these, so accuracy (not knowledge) decides who scores. A single BODMAS slip costs the mark, so the skill is disciplined order of operations, not tricks.
The method — BODMAS
Solve strictly in this order:
| Order | Operation | Symbol |
|---|---|---|
| B | Brackets | ( ) { } [ ] |
| O | Of / Orders (powers, roots) | x², √ |
| D | Division | ÷ |
| M | Multiplication | × |
| A | Addition | + |
| S | Subtraction | − |
Division and Multiplication rank equally — do them left to right. Same for Addition and Subtraction.
The three question types
- Symbol substitution: "If + means ×, − means ÷, × means +, ÷ means −, find 8 + 4 − 2 × 6 ÷ 3." Replace each symbol first, then apply BODMAS.
- Interchange of signs/numbers: pick the option that makes the equation balance after the given swap.
- Correct-the-equation: choose which pair of signs to interchange so LHS = RHS.
Exam Tricks & Tips
- 🎯 Rewrite before you compute. Physically translate every coded symbol to its real operator on your rough sheet, then apply BODMAS — never do both steps in your head.
- 🎯 Division before multiplication only if it comes first left-to-right — they are co-equal; direction decides.
- 🎯 In interchange questions, verify by plugging into the equation, don't eyeball it.
- 🎯 A minus sign before a bracket flips every sign inside when you open it.
- 🎯 "Of" means multiply and is done with O/powers, before ÷ — e.g. 1/2 of 8 ÷ 2 = 4 ÷ 2 = 2.
- ❌ Common mistake: doing addition before division because it appears first left-to-right — BODMAS overrides reading order; ÷ and × always beat + and −.
Expected exam pattern
Expect 1–2 questions: usually one symbol-substitution and sometimes one balance-the-equation. Options are close, so a BODMAS error lands you on a trap choice. Low difficulty, high certainty.
Quick recap
Translate coded symbols first, then apply B-O-DM-AS (÷ and × left-to-right, + and − left-to-right). Verify interchanges by substitution. Among the safest marks in the paper — bank them.
Mathematical Operations — Rapid Recall
Cover the answer, recall, then check. 12 cards on the rules and traps for RRB Technician math-operations questions.
Q1. State the full BODMAS order.
A1. Brackets → Of/Orders (powers, roots) → Division → Multiplication → Addition → Subtraction.
Q2. Do you always divide before you multiply?
A2. No. ÷ and × are co-equal; do whichever appears first left-to-right. Same rule for + and −.
Q3. First step in a symbol-substitution question?
A3. Rewrite the expression replacing each coded symbol with its real operator, THEN apply BODMAS.
Q4. If + means ×, − means ÷, × means +, ÷ means −: solve 8 + 4 − 2.
A4. 8 × 4 ÷ 2 = 32 ÷ 2 = 16.
Q5. What does "of" mean and where does it sit in BODMAS?
A5. "Of" means multiply, handled at the O (Orders) stage — before ÷ and ×.
Q6. Opening a bracket that has a minus sign in front — what happens?
A6. Every sign inside the bracket flips: −(a − b) = −a + b.
Q7. Solve 12 ÷ 4 × 3 + 2.
A7. 12 ÷ 4 = 3; 3 × 3 = 9; 9 + 2 = 11.
Q8. How do you verify an interchange-of-signs answer?
A8. Substitute the swap into the equation and check LHS = RHS numerically — never eyeball.
Q9. If ÷ and × are interchanged in 16 ÷ 4 × 2, what's the value?
A9. Becomes 16 × 4 ÷ 2 = 64 ÷ 2 = 32.
Q10. Solve (6 + 2) × 3 − 4.
A10. Bracket first: 8 × 3 = 24; 24 − 4 = 20.
Q11. Biggest scoring trap in these questions?
A11. Doing addition before division just because it appears first — order of operations overrides reading order.
Q12. 1/2 of 18 ÷ 3 = ?
A12. "Of" first: 1/2 of 18 = 9; then 9 ÷ 3 = 3.