Mathematical Operations

A question where "plus means multiply" is not a trick — it is a gift: there is nothing to calculate you do not already know, and the only skill tested is whether you can swap symbols calmly and then follow the rules you have used since Class 5.

Definition: A mathematical-operations (or operator-substitution) question temporarily redefines one or more arithmetic symbols, then asks you to evaluate an expression under the new definitions while still using normal BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) precedence.

The two-step method — non-negotiable

Every operator-substitution question, however it is phrased, yields to the same two steps:

1. Translate: Rewrite the entire expression on your rough paper, replacing each symbol with the operator it now stands for.
2. Evaluate: Apply BODMAS to the new expression exactly as you would any normal arithmetic.

Never attempt Step 2 while doing Step 1 in your head. The brain's default is to read "+" as plus even when you "know" it means multiply — the cognitive load of simultaneous translation and evaluation causes systematic errors. Write it out; it takes five seconds and earns the full mark.

Worked examples — all standard types

Type 1 — All four operators redefined

Question: If +→×, −→+, ×→÷, ÷→−, evaluate: 12 ÷ 6 × 4 − 2 + 3.
Solution:
Step 1 (translate): 12 − 6 ÷ 4 + 2 × 3
Step 2 (BODMAS): Division first → 6÷4 = 1.5; Multiplication → 2×3 = 6; Now left-to-right: 12 − 1.5 + 6 = 16.5.

Type 2 — Letter codes for operators

Question: A=+, B=−, C=×, D=÷. Evaluate: 60 D 4 C 5 B 8 A 2.
Solution:
Step 1: 60 ÷ 4 × 5 − 8 + 2
Step 2: 60÷4 = 15; 15×5 = 75; 75 − 8 = 67; 67 + 2 = 69.

Type 3 — Two symbols interchanged (bilateral swap)

Question: If + and − are interchanged, evaluate: 14 − 6 + 8 − 3 + 12.
Solution:
Step 1: Replace every − with + and every + with −: 14 + 6 − 8 + 3 − 12
Step 2: 14 + 6 = 20; 20 − 8 = 12; 12 + 3 = 15; 15 − 12 = 3.

Why this type trips students: they swap only the first sign they see, not every occurrence. "Interchange" means all instances of both symbols change — re-read the full expression after swapping.

Type 4 — Digit swap combined with operator swap

Question: If 3 is replaced by 7 and 7 by 3, AND + is replaced by −, evaluate: 27 + 73.
Solution:
Step 1 (digit swap): 27 → 23 (7 became 3); 73 → 37 (3 became 7).
Step 2 (operator swap): + → −.
Rewritten: 23 − 37 = −14.

Note: swap all digits and operators fully before touching any arithmetic.

Type 5 — Defined (new) operator

Question: If a ★ b = a² + b, evaluate: (3 ★ 4) ★ 2.
Solution:
Step 1: Inner bracket first: 3 ★ 4 = 3² + 4 = 9 + 4 = 13.
Step 2: Now 13 ★ 2 = 13² + 2 = 169 + 2 = 171.

Key insight: A defined operator must be re-evaluated at every application, including nested ones. The student who calculates 3 ★ 4 = 13 but then computes 13 ★ 2 as 13 + 2 (forgetting to square) gets 15 — a very common error.

Type 6 — Non-standard symbols

Question: <→+, >→−, =→×. Evaluate: 6 = 4 < 8 > 2.
Solution:
Step 1: 6 × 4 + 8 − 2
Step 2 (BODMAS): 6×4 = 24; 24 + 8 = 32; 32 − 2 = 30.

BODMAS — the rules that never change

Even after symbol substitution, BODMAS applies to the new expression:

Priority	Operation	Example
1	Brackets	Evaluate innermost first
2	Orders (powers, roots)	3² = 9
3	Division and Multiplication	Left to right, equal priority
4	Addition and Subtraction	Left to right, equal priority

"Division and Multiplication" are treated at equal priority — work left to right when both appear.

Traps and how to dodge them

Trap 1 — Applying old precedence after new symbol: you write + now means ×, but when you see 4 + 2 + 3, your eyes read it left-to-right and your hand adds rather than multiplying. Fix: rewrite first; evaluate the new line with no memory of the original.

Trap 2 — Incomplete swap in bilateral exchanges: "interchange × and ÷" means every × becomes ÷ AND every ÷ becomes ×. Swapping only one direction gives the wrong answer. Read the swapped expression back to yourself to verify.

Trap 3 — Not re-applying a defined operator in nested questions: for a ★ b = a² + b, once you find 3 ★ 4 = 13, the next application (13 ★ 2) requires squaring 13, not 3. The operator rule stays the same; the inputs change.

Trap 4 — Confusing "interchange" with "replace": "replace + with ×" means all + become ×, but × stays as ×. "Interchange + and ×" means +→× AND ×→+. Check the exact wording.

Speed and exam strategy

These questions are among the fastest to score in any competitive paper because:

• No formulas to memorise — just a substitution rule.
• Computation is simple arithmetic once correctly rewritten.
• Error rate is high in a hurry, so careful students stand out.

Target: 45–60 seconds per question. If a question takes longer, the most likely cause is in-head substitution rather than writing it out. The 5 seconds spent writing the new expression saves 30 seconds of recomputing.

Real-world example: Think of a currency conversion: a traveller's cheque in yen shows ¥12,000, but you need to pay in rupees. You do not multiply while staring at the yen number — you first convert (translate), write ₹7,200, then pay (evaluate). Mixing translation and payment causes the wrong amount to leave your wallet. Operator substitution is the same cognitive procedure: translate once, evaluate cleanly.

Common misconception: A student reasons: "+ now means ×, so I should multiply whenever I see it — and since × comes before + in BODMAS, I should multiply before adding." This is wrong on both counts. After substitution, the symbol you wrote down follows normal BODMAS for that symbol — the new × (which was originally +) is still evaluated at multiplication priority. Changing what + means does not change when you evaluate it; writing it as ×, not +, already handles the priority correctly.

:::keypoints Key points

• Translate the full expression on rough paper before evaluating — never compute while swapping mentally.
• After translation, apply standard BODMAS to the new expression without exception.
• "Interchange" is bilateral: both symbols swap with each other, across every occurrence.
• Defined operators (a★b = …) must be re-applied at each step, including nested brackets.
• Division and Multiplication are equal priority — work left to right when both appear together.
• These are high-accuracy, low-difficulty marks — methodical students consistently outscore intuitive ones here.
  :::

:::memory
T-E-B: Translate → Evaluate → BODMAS. Three words, every question.
:::

:::recap

• The entire skill is: write the substituted expression, then calculate normally.
• BODMAS does not change because symbols do — write the correct operator, then the rules apply automatically.
• Bilateral swaps affect every instance of both symbols; reread the entire expression to confirm.
• Defined operators must be computed fresh at every application; nested questions demand two separate applications.
  :::

Reasoning sections in SSC, RRB and Banking exams routinely disguise basic arithmetic behind unfamiliar symbols — the moment you decode the substitution, the question collapses into a simple BODMAS exercise. This lesson gives you the step-by-step method, covers every variant you will encounter, and drills the BODMAS pitfalls that cause most wrong answers.

Definition: Mathematical operations / Symbol substitution — a question type where standard arithmetic operators (+, −, ×, ÷) are replaced by arbitrary symbols (@, #, $, &, P, Q, letters, or even words). The task is to rewrite the expression with actual operators and then evaluate it using the correct order of operations.

Definition: BODMAS — the order of operations in arithmetic: Brackets → Of (powers and roots) → Division → Multiplication → Addition → Subtraction. Division and Multiplication are equal in priority and resolved left-to-right; so are Addition and Subtraction.

The four variants you will see

Variant 1 — Symbol-for-operator substitution
A key gives each symbol its meaning. You substitute and compute.

Variant 2 — "If + and × are interchanged"
Swap the two named operators everywhere in the expression, then evaluate.

Variant 3 — Letter-as-operator
Letters (P, Q, R, S) or words ("star", "hash") stand in for operators.

Variant 4 — "Which equation is correct?"
Given several equations, find which one holds true after applying the given substitution rule.

Step-by-step method (works for all variants)

Step 1: Read the substitution key carefully — mistakes here invalidate every subsequent step.
Step 2: Rewrite the expression, replacing each symbol with its actual operator.
Step 3: Apply BODMAS strictly: resolve brackets first, then powers/roots, then × and ÷ from left to right, then + and − from left to right.
Step 4: Double-check the sign of the answer — negative results are common and frequently set as traps.

Worked examples: Variant 1

Question: @ means +, # means −, $ means ×, & means ÷. Evaluate: 6 $ 4 @ 8 & 2 # 1.

Solution:
Step 1: Substitute: 6 × 4 + 8 ÷ 2 − 1.
Step 2: Division and multiplication first (left to right): 6 × 4 = 24, 8 ÷ 2 = 4.
Step 3: Expression becomes 24 + 4 − 1.
Step 4: Left to right: 24 + 4 = 28, 28 − 1 = 27.
Conclusion: Answer = 27.

Question: * means −, ↑ means ×, → means +, ← means ÷. Evaluate: 15 ← 3 ↑ 2 → 4 * 6.

Solution:
Step 1: Substitute: 15 ÷ 3 × 2 + 4 − 6.
Step 2: Division and multiplication left to right: 15 ÷ 3 = 5, then 5 × 2 = 10.
Step 3: Addition and subtraction left to right: 10 + 4 = 14, 14 − 6 = 8.
Conclusion: Answer = 8.

Worked examples: Variant 2 — Operator interchange

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

Solution:
Step 1: Swap: wherever you see +, write ×; wherever you see ×, write +.
Step 2: Expression becomes 6 × 4 + 2.
Step 3: Multiplication first: 6 × 4 = 24.
Step 4: 24 + 2 = 26.
Conclusion: Answer = 26.

Question: If − and ÷ are interchanged, find: 48 ÷ 6 − 4 + 2.

Solution:
Step 1: Swap ÷ and −: 48 − 6 ÷ 4 + 2.
Step 2: Division first: 6 ÷ 4 = 1.5.
Step 3: Left to right: 48 − 1.5 + 2 = 48.5.
Conclusion: Answer = 48.5.

Why it matters: In Variant 2 questions, the most common error is swapping only one of the two operators and forgetting to swap the other. Work systematically: underline every occurrence of both operators before rewriting.

Worked examples: Variant 3 — Letter operators

Question: P means ÷, Q means ×, R means +, S means −. Evaluate: 36 P 12 Q 2 R 4 S 1.

Solution:
Step 1: Substitute: 36 ÷ 12 × 2 + 4 − 1.
Step 2: Division and multiplication left to right: 36 ÷ 12 = 3, then 3 × 2 = 6.
Step 3: 6 + 4 − 1 = 9.
Conclusion: Answer = 9.

Question: A means +, B means −, C means ×, D means ÷. If 25 C 4 D 5 A 3 B 2 = x, find x.

Solution:
Step 1: Substitute: 25 × 4 ÷ 5 + 3 − 2.
Step 2: Multiplication and division left to right: 25 × 4 = 100, 100 ÷ 5 = 20.
Step 3: 20 + 3 − 2 = 21.
Conclusion: x = 21.

Worked example: Variant 4 — "Find the correct equation"

Question: If @ means +, # means ×, and $ means −, which of the following is correct?
(A) 4 # 3 @ 2 = 16 (B) 4 # 3 $ 2 = 10 (C) 4 @ 3 # 2 = 14 (D) 4 $ 3 @ 2 = 5

Solution:
Step 1: Substitute each option:
(A) 4 × 3 + 2 = 12 + 2 = 14 ≠ 16 ✗
(B) 4 × 3 − 2 = 12 − 2 = 10 = 10 ✓
Step 2: Option (B) checks out immediately; no need to test further.
Conclusion: Answer = (B).

Speed tip: In Variant 4, start with the option that has the fewest operations or the cleanest numbers. One match is all you need.

BODMAS — the most common error source

The single biggest reason students get symbol-substitution questions wrong is not the substitution step — it is applying BODMAS incorrectly after substituting.

The three traps:

Trap 1 — Left-to-right without priority
10 + 2 × 3 computed as (10 + 2) × 3 = 36. Wrong. Correct: 10 + (2 × 3) = 10 + 6 = 16.

Trap 2 — Division and multiplication treated as different priorities
6 × 8 ÷ 2 computed as 6 × (8 ÷ 2) = 6 × 4 = 24. This is actually correct here because ÷ comes after × left-to-right. But 6 ÷ 2 × 8: the mistake is doing 2 × 8 first. Correct: (6 ÷ 2) × 8 = 3 × 8 = 24.

Trap 3 — Brackets created by operator interchange
When you interchange + and ×, then see an expression like 2 + 3 × (4 + 5), be careful: after interchange the brackets still take priority. Do not strip them.

Real-world example: A shop owner codes his pricing as: "P means 10% off, Q means add ₹50, R means double the price." A customer reads: "coat price R Q P." If R = ×2, Q = +50, P = −10%:

• Original price ₹400.
• After R: 400 × 2 = 800.
• After Q: 800 + 50 = 850.
• After P: 850 − 85 = ₹765.
  The order (BODMAS doesn't apply here since each step is sequential) is crucial; reverse the steps and you get a completely different price. In exam questions the operators interact via BODMAS, not sequential application — but the analogy makes clear why order matters.

Quick-reference summary

Question type	What to do first	Biggest trap
Symbol substitution	Rewrite with real operators	Mis-reading the key
Operator interchange	Swap ALL occurrences of both operators	Swapping only one
Letter operators	Treat exactly like symbol substitution	Forgetting left-to-right rule
Find correct equation	Test each option; stop at first match	Testing all options (waste of time)

Common misconception: Students believe BODMAS means "do + before −." It does not. Addition and subtraction are equal in priority; execute them left to right. Similarly, multiplication and division are equal in priority; execute them left to right. The acronym BODMAS only separates the four levels, not the operations within a level.

:::keypoints Key points

• Symbol substitution: rewrite with real operators, then apply BODMAS — two separate, sequential steps.
• BODMAS levels: Brackets > Of/Powers > (÷ and × equal, left-to-right) > (+ and − equal, left-to-right).
• In "operator interchange" questions, swap every occurrence of both named operators before evaluating.
• In "find the correct equation" questions, test options one at a time and stop at the first match.
• Addition and subtraction are equal priority — neither comes before the other; use left-to-right.
• The most common error: computing left-to-right without applying multiplication/division first.
• Negative answers are valid and frequently appear as answer options — do not discard them.
  :::

:::memory
"BODMAS: Big Orange Dinosaurs Mostly Avoid Sunlight" — Brackets, Of, Division, Multiplication, Addition, Subtraction. Division and Multiplication are on the same level; Addition and Subtraction are on the same level.
:::

:::recap

• Read the substitution key carefully before touching any numbers.
• Rewrite the full expression with real operators, then evaluate with strict BODMAS.
• Operator interchange: swap every instance of both stated operators; missing one is the classic mistake.
• To find the correct equation from four options: compute each in turn and stop at the first that works.
• Within the same priority level (× and ÷; or + and −), always work left to right.
  :::