If you read BODMAS as a strict left-to-right priority list — Brackets first, then Orders, then Division, then Multiplication, then Addition, then Subtraction — you will get 24 \div 4 \times 2 wrong. You will first do 4 \times 2 = 8 (because M is after D, so multiplication is lower priority? or higher? you just swapped the rule twice and now nothing is certain) and get 24 \div 8 = 3. The correct answer is 12. The acronym has misled you. This is one of the most common — and costly — misconceptions in school arithmetic.

The short answer

In BODMAS, the D and M are tied. They sit on the same priority level. When both appear in an expression, you evaluate them left-to-right in the order they are written, not in the order the acronym lists them.

So 24 \div 4 \times 2 is not 24 \div (4 \times 2) = 3. It is (24 \div 4) \times 2 = 6 \times 2 = 12.

The same tie holds for A and S: addition and subtraction share a single level, also evaluated left-to-right. 10 - 3 + 2 is (10 - 3) + 2 = 9, not 10 - (3 + 2) = 5.

Why the acronym puts D before M

It is alphabetical convenience, not priority. "BODMAS" is easier to say than "BOMDAS" would be, and in British schools in the 1800s the order got baked in. Had history gone the other way, you might be learning "BOMDAS" and seeing the misconception flipped in the opposite direction ("Oh, M comes first, so multiplication beats division"). The order of the letters within the tied pair is arbitrary.

American schools use PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) and meet the mirror-image misconception: "M is before D, so multiplication comes first, right?" Wrong for the same reason. The M and D are tied in PEMDAS too.

Both acronyms list tied operations in an arbitrary internal order. Both convey the same underlying rule:

  1. Brackets (Parentheses).
  2. Orders (Exponents).
  3. Multiplication and Division — tied, left-to-right.
  4. Addition and Subtraction — tied, left-to-right.

Four priority levels, not six.

The misconception, spelled out

The misconception: "In BODMAS, D is before M, so in 24 \div 4 \times 2 you do the \div before the \times. And because \div is above \times in priority, if they appear in different orders — like a \times b \div c — you should still do \div first."

Why it feels true: Students who see BODMAS as a vertical priority list naturally read its order as strict. The word "before" is how every other priority list works — "priority boarding before regular boarding" means priority strictly beats regular. The acronym is silent about the tie, and the tie is not intuitive.

The correct rule: BODMAS is actually four priority levels with two internal ties. The D and M share a level; the A and S share a level. Within a tied level, left-to-right wins.

The counter-example: 8 \div 2 \times 4. Left-to-right: (8 \div 2) \times 4 = 16. If you had read BODMAS as strict D-before-M, you would have done 8 \div (2 \times 4) = 1. These two numbers are a factor of 16 apart — miss this rule and half your arithmetic is silently wrong.

The picture: four levels, not six

BODMAS visualised as four priority levels with two internal tiesA vertical stack of four horizontal bars representing the four priority levels. From top to bottom: Brackets, Orders, Division and Multiplication tied on the same bar, Addition and Subtraction tied on the same bar. Each bar has an arrow reading left-to-right next to it when the bar contains two operations, indicating the tie-breaking rule. B — Brackets (Parentheses) top priority O — Orders (Exponents) D ——— tied ——— M evaluate left-to-right A ——— tied ——— S evaluate left-to-right bottom priority
BODMAS is four levels. The top two — Brackets and Orders — are single operations. The bottom two are each a tied pair, evaluated left-to-right when both operations appear. The acronym's ordering of "D then M" and "A then S" is alphabetical, not operational.

Concrete computations, both ways

See the tie in action on a handful of expressions.

expression correct (left-to-right at tied level) wrong (strict D-before-M)
24 \div 4 \times 2 (24 \div 4) \times 2 = 12 24 \div (4 \times 2) = 3
8 \div 2 \times 4 (8 \div 2) \times 4 = 16 8 \div (2 \times 4) = 1
20 \div 5 \times 2 (20 \div 5) \times 2 = 8 20 \div (5 \times 2) = 2
6 \times 4 \div 3 (6 \times 4) \div 3 = 8 you can't apply "D before M" here — there's no ambiguity, and both readings agree

The last row shows something subtle: the strict-D-before-M mistake only changes the answer when the expression starts with a \div followed by a \times. When it starts with a \times followed by a \div, the left-to-right rule and the wrong strict-D rule happen to land at the same answer by accident. So the misconception hides half the time — which is why students can carry it for years without noticing.

The deeper reason: division is multiplication in disguise

There is an even cleaner way to see why D and M tie. Division is secretly multiplication by the reciprocal:

24 \div 4 \;=\; 24 \times \tfrac{1}{4}.

Why this matters: rewritten with reciprocals, the expression 24 \div 4 \times 2 becomes 24 \times \tfrac{1}{4} \times 2 — three multiplications in a row. Multiplication is associative, so you can group them any way you like. (24 \times \tfrac{1}{4}) \times 2 = 6 \times 2 = 12. Or 24 \times (\tfrac{1}{4} \times 2) = 24 \times \tfrac{1}{2} = 12. Same answer either way.

The "tie" between \div and \times isn't an arbitrary convention — it is the fact that they are the same operation on different notation. Treating them as separate priority levels would break this, which is why no coherent arithmetic rule does.

The same thing is true of subtraction: a - b is a + (-b), so subtraction is secretly addition. A tie between + and - makes sense for exactly the same reason.

The exam cost

On every Board exam and every JEE paper, the D-before-M misconception shows up at least once, usually inside a slightly longer expression where it is easy to miss. A student who has internalised "D and M are tied" reads 48 \div 6 \times 4 + 2 correctly as

(48 \div 6) \times 4 + 2 \;=\; 8 \times 4 + 2 \;=\; 32 + 2 \;=\; 34.

A student with the misconception reads it as

48 \div (6 \times 4) + 2 \;=\; 48 \div 24 + 2 \;=\; 2 + 2 \;=\; 4.

Same expression, two answers that are a factor of nearly 10 apart. One of them is the intended answer on the mark scheme. The other is usually no partial credit.

The reflex, in one line

Treat BODMAS as four priority levels, not six. Brackets, Orders, and then two tied pairs — \div/\times and -/+ — each broken by left-to-right reading. If someone asks you "does D come before M?" the honest answer is "neither — they tie, and left-to-right wins."

Related: Operations and Properties · Expression Trees: Watching Precedence Collapse · BODMAS Drift: What Happens When You Break the Order · Is Subtraction Just "Adding a Negative"?