BIDMAS, BODMAS and PEMDAS Are the Same Rule

The three acronyms, letter by letter

BIDMAS is standard in English schools, BODMAS is the older British form still used across much of the Commonwealth including Pakistan and India, and PEMDAS is American. Canadian and New Zealand schools often teach BEDMAS. Line them up and the mapping is exact:

A student who learned BODMAS in Lahore and moves to a school teaching BIDMAS has nothing to relearn. The only genuinely different letter is the O, and that is dealt with below.

• B or P — Brackets, Parentheses. Same thing, different side of the Atlantic.
• I, O, E — Indices, Orders, Exponents. All three mean powers and roots.
• D and M, or M and D — Division and Multiplication, ranked equally, taken left to right.
• A and S — Addition and Subtraction, ranked equally, taken left to right.

The mistake every version of the acronym invites

Every one of these mnemonics is a list, and a list looks like a ranking of six or seven steps. It is not. It is four tiers, two of which contain two operations of equal rank. Reading DM as do all the divisions, then all the multiplications produces wrong answers, and so does reading MD the other way.

Take 24 ÷ 4 × 3. Reading the letters literally as multiply-before-divide gives 4 × 3 = 12 and then 24 ÷ 12 = 2. Worked correctly, left to right within the tier, it is 24 ÷ 4 = 6 and then 6 × 3 = 18. The same trap sits in the last tier: 10 − 4 + 3 is 6 + 3 = 9, not 10 − 7 = 3.

This is where nearly every lost mark on the topic comes from. The student who gets it wrong usually knows the acronym perfectly — that is the problem.

Why those letters share a rank

The pairing is not an arbitrary tidying-up. Division is multiplication in disguise: 24 ÷ 4 is 24 × ¼. Subtraction is addition in disguise: 10 − 4 is 10 + (−4). Rewrite the two problems above and the difficulty evaporates, because multiplication and addition can be done in any order.

24 ÷ 4 × 3 becomes 24 × ¼ × 3, and 24 × 3 × ¼ gives 18 just as readily. 10 − 4 + 3 becomes 10 + (−4) + 3, which is 9 whichever way you group it. There are not four operations at the bottom of the hierarchy. There are two, each with an inverse written as though it were separate.

Teaching this rewrite is more useful than drilling the acronym, because it removes the need to remember a left-to-right rule at all.

What the O in BODMAS actually stands for

In British usage the O is Orders, an older word for powers and roots — it means exactly what the I in BIDMAS means. In some Commonwealth teaching the O is taught as Of, as in a half of twelve, and that causes real confusion because students then treat of as a separate tier ranked above division.

It is not a separate tier. Of is simply multiplication in words: half of twelve is ½ × 12. Give it its own letter ranked above D and M, though, and expressions acquire two answers. Take 12 ÷ ½ of 6. Treating of as a priority operation gives ½ of 6 = 3 and then 12 ÷ 3 = 4. Treating it as ordinary multiplication and working left to right gives 12 ÷ ½ = 24 and then 24 × 6 = 144. Four, or a hundred and forty-four, from four symbols.

Examiners deal with this by never setting an expression of that shape. Of appears in questions attached to a single quantity — find ¾ of 40 — where nothing can be misread. Anything genuinely ambiguous that you meet outside an exam is badly written rather than difficult, and the repair is a bracket rather than a rule.

The brackets nobody prints

Several pieces of standard notation act as brackets without looking like them, and they trip up students who are typing an expression into a calculator rather than reading it off a page. A fraction bar groups everything above it and everything below it, so (7 + 5) ÷ (2 × 3) is what a printed fraction with 7 + 5 over 2 × 3 means. Typed into a calculator as 7 + 5 ÷ 2 × 3 it gives something else entirely.

The square root sign does the same job: √(16 + 9) is 5, while √16 + 9 is 13. And a minus sign in front of a power binds more loosely than the power itself, so −3² is −9 while (−3)² is 9. That last one appears in substitution questions every series and costs marks quietly.

The habit worth building is to insert brackets whenever an expression moves from paper to a keypad, even where they are not strictly needed. It is faster than checking.