Mathematical Operations

"If + means ×, and − means ÷..." — mathematical-operations questions deliberately scramble the symbols to test one skill above all: the discipline to decode first and compute second, never the other way around.

Definition: A mathematical-operations (symbol-substitution) problem redefines arithmetic symbols, swaps them in pairs, or introduces a custom operator, then asks you to evaluate an expression using the new meanings while still respecting BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction).

Why This Topic Exists in Exams

Symbol-substitution questions appear in SSC CGL, CHSL, RRB NTPC, and almost every state PSC reasoning section. They need zero advanced mathematics — only disciplined substitution and BODMAS awareness. That makes them quick, reliable marks for students with method, and an easy trap for those who rush.

The Core Method (Four Steps)

Step 1 — Read the substitution rule carefully. Write it down as a mapping: old symbol → new symbol.

Step 2 — Rewrite the entire expression on rough paper with every symbol replaced. Do this before doing any arithmetic.

Step 3 — Apply BODMAS to the rewritten expression. Resist the urge to compute left to right.

Step 4 — Check for "interchange" vs. "replace". "Interchange + and −" means both swap simultaneously (+ becomes − and − becomes +). "Replace + with −" means only + changes; − stays as −. These are different.

Type 1: Full Redefinition (four symbols each given a new meaning)

Question: If '+' means ×, '−' means +, '×' means ÷, '÷' means −, evaluate 12 ÷ 6 × 4 − 2 + 3.
Solution:
Step 1: Map each symbol: ÷ → −, × → ÷, − → +, + → ×.
Step 2: Rewrite: 12 − 6 ÷ 4 + 2 × 3.
Step 3: BODMAS first — division before addition/subtraction: 6 ÷ 4 = 1.5; multiplication before addition: 2 × 3 = 6. Now: 12 − 1.5 + 6.
Conclusion: 12 − 1.5 + 6 = 16.5.

Common misconception: A common error is computing 12 − 6 = 6 first (left to right), then 6 ÷ 4, etc. BODMAS must govern the rewritten expression, not the original order of writing.

Type 2: Letter Codes for Operators

Question: If A means +, B means −, C means ×, D means ÷, evaluate 60 D 4 C 5 B 8 A 2.
Solution:
Step 1: Decode: 60 ÷ 4 × 5 − 8 + 2.
Step 2: BODMAS — left to right for × and ÷: 60 ÷ 4 = 15; 15 × 5 = 75. Then: 75 − 8 + 2.
Conclusion: 75 − 8 + 2 = 69.

Why it matters: Letter-code questions appear in SSC CGL Tier I and RRB reasoning papers. Once you recognise the type, the method is identical — decode, then BODMAS.

Type 3: Interchange (Bilateral Swap)

Question: If '+' and '−' are interchanged, find the value of 14 − 6 + 8 − 3 + 12.
Solution:
Step 1: Both symbols swap: every + becomes − and every − becomes +.
Step 2: Rewrite: 14 + 6 − 8 + 3 − 12.
Step 3: Left to right (only + and − remain, so order doesn't matter here): 14 + 6 = 20; 20 − 8 = 12; 12 + 3 = 15; 15 − 12 = 3.
Conclusion: 3.

Common misconception: Students sometimes swap only the first occurrence of each symbol, or treat "interchange" as a one-way replacement. Interchange means every instance of both symbols swaps simultaneously.

Type 4: Digit Swap Combined with Operator Swap

Question: If 3 and 7 are interchanged and '+' is replaced by '−', find 27 + 73.
Solution:
Step 1: In 27, the digit 7 → 3, giving 23. In 73, the digit 3 → 7, giving 37.
Step 2: '+' → '−': expression becomes 23 − 37.
Conclusion: 23 − 37 = −14.

Real-world example: Think of encoding and decoding a message. A railway booking system internally uses its own codes (PNR, status codes) that must be decoded before you can read the information. Symbol-substitution is the same — you must decode the legend first.

Type 5: Custom-Defined Operators

Question: If a ★ b = a² + b, find (3 ★ 4) ★ 2.
Solution:
Step 1: Compute the inner bracket first. 3 ★ 4 = 3² + 4 = 9 + 4 = 13.
Step 2: Use the result as the new 'a'. 13 ★ 2 = 13² + 2 = 169 + 2 = 171.
Conclusion: (3 ★ 4) ★ 2 = 171.

Note: custom operators are almost never commutative (a ★ b ≠ b ★ a unless the definition is symmetric), so never swap the operands.

Type 6: Multi-Symbol Expression with Mixed Codes

Question: If '<' means '+', '>' means '−', '=' means '×', find 6 = 4 < 8 > 2.
Solution:
Step 1: Decode: 6 × 4 + 8 − 2.
Step 2: BODMAS — multiplication first: 6 × 4 = 24. Then: 24 + 8 − 2.
Conclusion: 24 + 8 − 2 = 30.

Speed Strategies for Competitive Exams

1. Write the full substitution row on your rough sheet before touching the numbers. One careless substitution throws off all four operations.
2. Circle BODMAS-priority operations in the rewritten expression (× and ÷ before + and −) before computing.
3. Test with known quantities: if the answer options are small integers and you get a fraction, re-check your substitutions.
4. "Interchange" ≠ "replace": exams exploit this — always re-read the instruction.
5. For questions with 4–5 options that are all different, any arithmetic error takes you to a wrong option that still looks plausible. Double-check once.

:::keypoints Key points

• Always substitute first, compute second — never both simultaneously.
• After substitution, BODMAS governs the expression; never compute left to right blindly.
• "Interchange" is bilateral — both symbols swap everywhere.
• "Replace X with Y" is unilateral — only X changes.
• Digit-swap questions modify the numbers themselves, not just the operators.
• Custom operators (a ★ b = …) must be applied as defined each time, with correct operand order.
• Check whether the operator is commutative before assuming a ★ b = b ★ a.
• Write the decoded expression clearly on rough paper to avoid sequence errors.
  :::

:::memory
"Decode before you compute" — D.C. Like a DC current that flows in one direction only, your method must flow in one direction: Decode → BODMAS → Answer. Never shortcut this sequence.
:::

:::recap

• Four-step method: read rule → rewrite expression → apply BODMAS → compute.
• Bilateral swap means every instance of both symbols changes; verify the instruction word carefully.
• Custom operators: apply the formula exactly as given, innermost bracket first.
• Most errors in this topic come from left-to-right computation instead of BODMAS — that single habit change is worth multiple marks per paper.
  :::

When a question paper replaces the familiar "+" with "@" or "#", the only thing that changes is the costume — the underlying mathematics is identical, and BODMAS is still the rulebook.

Definition: Mathematical operations / symbol substitution is a category of reasoning questions in which standard arithmetic operators (+, −, ×, ÷) are replaced by arbitrary symbols or letters, and you must decode the substitution and evaluate the expression correctly.

Definition: BODMAS (also written PEMDAS in some curricula) is the hierarchy of operations: Brackets → Orders (powers, roots) → Division → Multiplication → Addition → Subtraction. Division and Multiplication have equal priority (left to right); so do Addition and Subtraction (left to right).

---

Types of Symbol Substitution Questions

There are four common variants you will encounter in SSC, RRB, and banking exams:

Type 1 — Symbols given with a key

A legend assigns each symbol to an operation. You substitute and apply BODMAS.

Example key:

• @ → +
• # → −
• $ → ×
• & → ÷

Question: Evaluate 6 $ 4 @ 8 & 2 # 1

Solution:
Step 1: Substitute — 6 × 4 + 8 ÷ 2 − 1
Step 2: Apply BODMAS — first handle × and ÷ (left to right):

• 6 × 4 = 24
• 8 ÷ 2 = 4
• Expression becomes 24 + 4 − 1
  Step 3: Handle + and − (left to right): 24 + 4 = 28, then 28 − 1 = 27
  Conclusion: The answer is 27.

---

Type 2 — "If signs are interchanged"

Two or more operators are swapped with each other throughout the expression. You mentally replace every instance of sign A with sign B and vice versa, then evaluate.

Question: "If + and × are interchanged, what is the value of 6 + 4 × 2?"

Solution:
Step 1: Swap every + with × and every × with +: 6 × 4 + 2
Step 2: BODMAS — multiplication first: 6 × 4 = 24
Step 3: Addition: 24 + 2 = 26
Conclusion: The answer is 26.

Common trap: Students apply BODMAS to the original expression first, then swap — wrong order. Always swap first, then apply BODMAS.

---

Type 3 — Letters as operators

Letters are defined to represent operations, then a multi-term expression is given.

Question: If P = ÷, Q = ×, R = +, S = −, find the value of 36 P 12 Q 2 R 4 S 1.

Solution:
Step 1: Substitute letters: 36 ÷ 12 × 2 + 4 − 1
Step 2: BODMAS — left to right for ÷ and ×:

• 36 ÷ 12 = 3
• 3 × 2 = 6
  Step 3: Addition and subtraction: 6 + 4 − 1 = 9
  Conclusion: The answer is 9.

---

Type 4 — Which expression is correct? (Balancing type)

A number (say 24) is given and you must choose which symbolic expression, when decoded, equals 24. This is reverse substitution — try each option.

Question: Using the key @ = +, # = −, $ = ×, & = ÷, which equals 24?
(a) 4 $ 5 @ 4 (b) 4 $ 3 @ 12 (c) 50 & 5 @ 14 (d) 30 # 10 @ 4

Solution:

• (a) = 4 × 5 + 4 = 20 + 4 = 24 ✓ — stop here in the exam!

---

BODMAS Deep Dive: The Rule That Never Changes

Priority	Operation	Example	Note
1	Brackets ( ) then [ ] then { }	(3 + 2) × 4 = 20	Innermost bracket first
2	Orders: powers, roots	2³ = 8; √9 = 3
3	Division & Multiplication	12 ÷ 3 × 2 = 8	Left to right
4	Addition & Subtraction	10 − 3 + 2 = 9	Left to right

Common misconception: Many students believe Division always comes before Multiplication because the D appears before M in BODMAS. That is incorrect. D and M have the same priority — work left to right. Similarly, A and S have the same priority. Only B and O, then DM, then AS are separate tiers.

Example to illustrate: 12 ÷ 4 × 3

• Wrong: 12 ÷ (4 × 3) = 12 ÷ 12 = 1
• Right: (12 ÷ 4) × 3 = 3 × 3 = 9 ✓

---

Left-to-Right Rule for Equal-Priority Operations

When two operations at the same tier appear, process them strictly left to right.

Example: 20 − 5 + 3

• Left to right: (20 − 5) + 3 = 15 + 3 = 18
• Wrong (right to left): 20 − (5 + 3) = 20 − 8 = 12 ✗

---

Worked Example: Multi-symbol with Brackets

Question: Using the key @ = +, # = −, $ = ×, & = ÷:
Evaluate (8 @ 4) $ 3 # 10 & 2

Solution:
Step 1: Substitute: (8 + 4) × 3 − 10 ÷ 2
Step 2: Brackets first: 12 × 3 − 10 ÷ 2
Step 3: × and ÷ left to right: 36 − 5
Step 4: Subtraction: 31
Conclusion: The answer is 31.

---

Speed Tactics for the Exam

1. Write out the substitution before doing any arithmetic. Spend 5 seconds writing 6 × 4 + 8 ÷ 2 − 1 rather than trying to track two layers in your head.
2. Circle the × and ÷ symbols in the substituted expression before starting — you will process them first.
3. For "balanced" (choose the correct expression) questions, start with the option that looks simplest; if it matches, choose it.
4. Beware "if + and − are interchanged AND × and ÷ are interchanged" — two simultaneous swaps. Replace all four symbols before evaluating.

---

Practice Problems

Question 1: Key: @ = ×, # = +, $ = −, & = ÷. Evaluate 5 @ 3 # 20 & 4 $ 2.
Solution:
Step 1: 5 × 3 + 20 ÷ 4 − 2
Step 2: 15 + 5 − 2
Step 3: 18
Answer: 18

Question 2: If − and ÷ are interchanged, evaluate 18 − 3 + 4 ÷ 2.
Solution:
Step 1: Swap: 18 ÷ 3 + 4 − 2
Step 2: Division first: 6 + 4 − 2
Step 3: Left to right: 8
Answer: 8

Question 3: If P = −, Q = +, R = ×, evaluate 12 R 3 Q 5 P 2.
Solution:
Step 1: 12 × 3 + 5 − 2
Step 2: 36 + 5 − 2 = 39
Answer: 39

---

:::keypoints Key points

• Symbol substitution questions change only the notation — BODMAS governs the arithmetic as always.
• In "signs interchanged" questions, swap first, then evaluate — never evaluate then swap.
• D and M are at the same BODMAS tier; so are A and S — use left-to-right order within each tier.
• Write out the substituted expression in full before calculating to avoid tracking errors.
• Brackets, even when formed by substitution, are processed first regardless of what operators are inside.
• In "which expression equals X" questions, evaluate each option in order and stop at the first match.
• Double swaps (two pairs interchanged simultaneously) require extra care — mark each symbol before replacing.
• These questions test BODMAS discipline far more than arithmetic speed.
  :::

:::memory
"SWAP before you SOLVE" — in any "interchanged" question, the number-one rule is to complete all symbol swaps on paper before touching the calculator in your mind.
:::

:::recap

• Four types: symbol key, signs interchanged, letters as operators, and choose-the-correct-expression.
• Always substitute fully in writing, then apply BODMAS in strict order.
• Division and Multiplication are equal priority (left to right); same for Addition and Subtraction.
• Common errors: wrong order of operations for D vs M; evaluating before swapping in "interchanged" type.
• Brackets in the substituted expression are processed first — even if they were formed by the substitution.
  :::