Mathematical Operations

Every question in this topic has two separate jobs, and marks are lost by mixing them: first rewrite the expression with the real signs, then calculate it with BODMAS. Never calculate while you are still replacing signs.

1. Substitution. "If + means ×, − means ÷ …" — rewrite the whole expression on the rough sheet, sign by sign, before touching a number. Letter codes such as P = + and Q = − work exactly the same way.
2. Interchange. "If + and × are interchanged" — swap both ways: every + becomes × and every × becomes +. When two numbers are interchanged as well, make both changes before calculating.
3. Which equation is correct? Convert the left side of each option and evaluate it; only one matches its right side. A tempting wrong option is often correct as printed, without the substitution.
4. Which two signs to interchange? Try the pairs one at a time on the given equation; the right pair makes both sides equal.
5. Defined operations. "a ★ b = 2a + 3b" — put the numbers in the places of a and b in that order, and work brackets from the inside out.

BODMAS: brackets first, then division and multiplication from left to right, then addition and subtraction from left to right.

The single most common mistake is working straight from left to right after substituting: 12 + 4 × 3 is 24, not 48, because the multiplication comes first — and the left-to-right number is always among the options.

How to check: before calculating, read the rewritten expression against the given meanings once more, sign by sign.

Rewrite first, calculate second · an interchange works both ways (+ ↔ ×) · BODMAS: brackets → ÷ and × left to right → + and − left to right · for "which equation is correct", convert every option · a ★ b rules: put the numbers in order, inner bracket first

If + means ÷, − means ×, × means − and ÷ means +, find the value of 20 × 4 + 2 − 3 ÷ 6. → Rewrite first: 20 − 4 ÷ 2 × 3 + 6. Division and multiplication: 4 ÷ 2 = 2, then 2 × 3 = 6. Then 20 − 6 + 6 = 20. Sanity check: working straight from left to right gives 30 instead — that is the trap option, because it ignores BODMAS.