Mathematical Operations

In mathematical operations problems, normal symbols (+, −, ×, ÷) are replaced with other symbols (@, #, $, &). You evaluate expressions using the redefined operations under BODMAS.

Standard substitution format: given a key like

• '@' means '+'
• '#' means '−'
• '$' means '×'
• '&' means '÷'

Evaluate: 6 $ 4 @ 8 & 2 # 1 = 6 × 4 + 8 ÷ 2 − 1 = 24 + 4 − 1 = 27 (apply BODMAS to the substituted operators).

Step-by-step approach:

1. Write the expression with original operators substituted in.
2. Apply BODMAS: brackets, division, multiplication, addition, subtraction.
3. Be careful with left-to-right rule for same-priority operations.

Variant — "If signs are interchanged": e.g. "if + and × are interchanged, what is the value of 6 + 4 × 2?"
Method: swap symbols mentally → 6 × 4 + 2 = 24 + 2 = 26.

Letter-as-operator: same idea but with letters or words. "If P stands for ÷, Q for ×, R for +, S for −, then 36 P 12 Q 2 R 4 S 1 = ?"
= 36 ÷ 12 × 2 + 4 − 1 = 3 × 2 + 4 − 1 = 6 + 4 − 1 = 9.

Number-to-symbol substitution (less common): replace digits using a code.

BODMAS reminder:
B → Brackets
O → Of (or powers/roots)
DM → Division/Multiplication (left to right)
AS → Addition/Subtraction (left to right)

Common error: applying operators in left-to-right order instead of BODMAS. Always solve × and ÷ before + and − unless brackets force a different order.

Example 1:
If '+' means '×', '−' means '+', '×' means '÷', '÷' means '−', then 12 ÷ 6 × 4 − 2 + 3 = ?
Method: Substitute: 12 − 6 ÷ 4 + 2 × 3 = 12 − 1.5 + 6 = 16.5.

Example 2:
If A means +, B means −, C means ×, D means ÷, then 60 D 4 C 5 B 8 A 2 = ?
Method: 60 ÷ 4 × 5 − 8 + 2 = 15 × 5 − 8 + 2 = 75 − 8 + 2 = 69.

Example 3 — Sign swap:
If signs '+' and '−' are interchanged, find 14 − 6 + 8 − 3 + 12.
Method: swap: 14 + 6 − 8 + 3 − 12 = 20 − 8 + 3 − 12 = 12 + 3 − 12 = 3.

Example 4 — Digit-and-sign swap:
If 3 is replaced with 7 and 7 with 3, and + with −, find 27 + 73.
Method: 27 → 23. 73 → 37. + → −. So 23 − 37 = −14.

Example 5 — Numerical operator pairs:
If a * b = a² + b, find (3 * 4) * 2.
Method: 3 * 4 = 9 + 4 = 13. Then 13 * 2 = 169 + 2 = 171.

Example 6 — Multi-step swap:
If '<' means '+', '>' means '−', '=' means '×', then 6 = 4 < 8 > 2 = ?
Method: 6 × 4 + 8 − 2 = 24 + 8 − 2 = 30.

Speed tip: write the substitution clearly on rough sheet — 12 ÷ 6 × 4 − 2 + 3 becomes 12 − 6 ÷ 4 + 2 × 3 — then apply BODMAS. Most errors come from applying operators in the original order.

Trap: questions that say "interchange" or "swap" are bilateral — both symbols change places. Re-reread the swap before computing.