Simplification

Simplification is not about being clever; it is about doing the steps in the right order and not losing a sign.

BODMAS fixes the order: Brackets, Of, Division, Multiplication, Addition, Subtraction. "Of" means multiply — "3/5 of 250" is 3/5 × 250.

Four patterns cover the paper:

1. Plain BODMAS chain — multiplication and division before addition and subtraction.
2. Nested brackets — open the innermost first: ( ) then { } then [ ].
3. Identities that save time — a² − b² = (a+b)(a−b), (a+b)² = a² + 2ab + b², (a−b)² = a² − 2ab + b².
4. Continued fractions — start at the bottom and work upwards, taking the reciprocal each time.

The single most common mistake is a minus sign in front of a bracket. It flips every sign inside: 50 − (20 − 8) = 50 − 12 = 38, not 50 − 20 − 8 = 22. Whenever you remove a bracket that has a minus in front of it, change every sign inside before you go on.

BODMAS · "of" = × · minus before a bracket flips every sign inside · a² − b² = (a+b)(a−b)

Simplify 87² − 13². → Use the identity instead of squaring: (87 + 13)(87 − 13) = 100 × 74 = 7400. Sanity check the long way: 87² = 7569 and 13² = 169, and 7569 − 169 = 7400. Same answer, four times the work.