Simplification and BODMAS: The Order of Operations and Its Traps

Simplification questions give a long string of numbers and signs and ask for a single value. The arithmetic is easy; the trap is doing the steps in the wrong order. The order taught for exams is usually remembered as BODMAS, or VBODMAS when the bar is included:

1. Vinculum (the bar drawn over an expression)
2. Brackets, from the inside out: ( ), then { }, then [ ]
3. Of
4. Division and Multiplication, from left to right
5. Addition and Subtraction, from left to right

The key detail is in steps 4 and 5. Division and multiplication share the same rank, so you work them in the order they appear, left to right. Addition and subtraction also share a rank. Reading BODMAS as "add before subtract" is the single most common error in this chapter.

"Of" means multiplication, but in exam simplification it is worked before ordinary division and multiplication. That makes "of" behave as if it had its own bracket.

Compare 48 ÷ 4 × 3, where there is no "of": left to right gives 12 × 3 = 36. One word changes the answer ninefold.

A bar, or vinculum, drawn over part of an expression groups it even more tightly than a bracket, so it is worked first. Then clear brackets from the innermost outwards. Write each step on a new line instead of keeping all of it in your head.

A minus sign in front of a bracket changes the sign of every term inside it when the bracket is removed. Most bracket mistakes come from forgetting this when expanding rather than evaluating.

A safe habit is to think of each minus sign as belonging to the number after it: 15 + (−7) + (+3). Then the terms can be added in any order and the answer is always 11.

Simplification questions often hide an identity. When you see a difference of squares divided by a sum or difference, use a² − b² = (a + b)(a − b) instead of squaring.

To check a long multiplication without redoing it, compare digit sums. The digit sum of the answer must match the product of the digit sums, both reduced to a single digit.

The check catches most slips, though not a swap of two digits, which leaves the digit sum unchanged.

• ✘ Adding before subtracting. Work + and − left to right.
• ✘ Dividing before multiplying when multiplication comes first. Work ÷ and × left to right.
• ✘ Treating "of" like ordinary ×. Work it before ÷ and ×.
• ✘ Forgetting to change signs when removing a bracket after a minus.

Fast arithmetic also needs square and cube roots and the divisibility rules. The formula list is in the Quant cheat code.

Does BODMAS mean division always comes before multiplication?

No. Division and multiplication have equal priority and are worked from left to right, and the same is true for addition and subtraction. The letters only give a memory order; reading the D before the M literally can give wrong answers in other contexts.

What is the 'of' rule?

'Of' means multiplication, but in exam simplification it is worked before ordinary division and multiplication. So 48 ÷ 4 of 3 means 48 ÷ 12 = 4, while 48 ÷ 4 × 3 means 12 × 3 = 36.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

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