Mathematical Operations

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:

1. Read the substitution rule carefully. Write the translated expression in one shot.
2. 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.

:::keypoints Key points

• 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.
  :::

:::memory
"SUBSTITUTE then OPERATE" — two steps, always in that order.
"Swap is two-way, not one-way" — if + and − swap, every + becomes − and every − becomes +.
:::

:::recap

• 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.
  :::

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:

1. Accurately translate one code to another without error.
2. Apply BODMAS correctly after substitution.
3. 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:

1. Substitute: Replace each symbol with its operator: 6 × 4 + 8 ÷ 2 − 1
2. 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
3. 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:

1. Swap the two operators everywhere in the expression: 6 × 4 + 2
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.

1. Swap + ↔ × and − ↔ ÷: 12 × 6 + 3 ÷ 9 − 3 (every + becomes ×, every × becomes +, every − becomes ÷, every ÷ becomes −)
2. BODMAS: First × and ÷: 12 × 6 = 72; 3 ÷ 9 = 0.333...
3. 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:

1. Substitute: 36 ÷ 12 × 2 + 4 − 1
2. 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?

1. Replace *: 15 ? 3 + 2 = 4 → 15 ? 3 = 2
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

1. Write the substituted expression first before calculating. Do not try to do substitution and BODMAS simultaneously.
2. Circle division and multiplication in the substituted expression; resolve them first from left to right.
3. For sign-interchange questions: underline each operator in the original expression and replace it explicitly. Don't trust your memory.
4. Time budget: these questions should take 30–40 seconds max. If you exceed 60 seconds, mark and move on.

:::keypoints Key points

• 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.
  :::

:::memory
"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.
:::

:::recap

• 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.
  :::