You learnt BODMAS as a rhyme. Brackets, Orders, Division, Multiplication, Addition, Subtraction — six ranks, top to bottom, done. Then a problem like 24 \div 4 \times 2 shows up and the class splits into two camps with two answers. You ask your teacher, and suddenly there is this extra rule — "left-to-right when same rank" — that was never in the rhyme. Where did it come from? Why is it real? And why didn't anyone explain it the first time?

The rhyme is incomplete

The acronym BODMAS lists six categories, but two of those "categories" are actually pairs of operations sharing one rank:

So what looks like six ranks is really four:

  1. Brackets
  2. Orders (exponents and roots)
  3. \times and \div (tied)
  4. + and - (tied)

Within a tied rank, you need a rule for which operation to do first when both appear in the same expression. That rule is: read left-to-right, in the order written.

BODMAS is really four ranks, two of them tiedA vertical ladder with four rungs. The top rung is labelled Brackets, one operation. The second is Orders, one operation. The third rung is labelled times and divide, tied — evaluate left to right. The fourth rung is labelled plus and minus, tied — evaluate left to right. The two tied rungs are highlighted. B — Brackets O — Orders (powers, roots) × and ÷ — tied evaluate left-to-right + and − — tied evaluate left-to-right
Four ranks, not six. The middle two are shared between pairs of operations. On a tied rank, the operation written first is the operation done first — that is the tie-break nobody mentioned.

Why the rule has to exist

Consider 10 - 3 + 2. With + and - genuinely tied at the same level, there are two candidate readings:

Those are different answers. One expression cannot have two correct values. So somebody has to pick, and mathematicians picked the one that matches how we read English: left-to-right, in the order the symbols appear on the page. That choice makes 10 - 3 + 2 = 9 the canonical answer.

The same thing happens with 24 \div 4 \times 2:

Different answers. Left-to-right wins, so the canonical answer is 12.

Why: subtraction is not commutative (you cannot swap 10 and 3), and division is not commutative (you cannot swap 24 and 4). When you mix non-commutative operations at the same level, a tie-break rule is required — otherwise the same expression means two things. "Left-to-right" is the tie-break chosen by universal convention.

Try it: slide the middle number

Interactive: see left-to-right and right-to-left readings differA horizontal line from one to ten with a draggable red point labelled b. Readouts above show the expression twenty-four divided by b times two evaluated left-to-right, and the same expression evaluated right-to-left. The two readings agree only at specific values of b — everywhere else they differ, showing why a tie-break rule is necessary. 1 5.5 10 ↔ drag b
The two readings of $24 \div b \times 2$ differ by a factor of four for almost every $b$. If notation had no tie-break, the expression would have two legal meanings. Left-to-right is the convention — the right-to-left reading is *not wrong arithmetic*, it is just reading the wrong rule.

Why nobody told you

School-level teaching often compresses the rule into "BODMAS, in that order" for simplicity, especially for class 6-7 students who mostly see expressions where the order doesn't matter anyway. Problems like 3 + 2 \times 4 can be read strictly top-to-bottom (multiplication before addition), and left-to-right never comes up — the multiplication was never tied with anything.

The rule only bites when two operations at the same rank appear in one expression. 24 \div 4 \times 2, 10 - 3 + 2, 16 \div 8 \div 2 — these are the tests. Many class tests don't include them, so you can coast through the year without ever needing the tie-break. Then JEE-level questions or viral internet tweets start using such expressions on purpose, and the gap in your training opens up.

A clean restatement of BODMAS

If you want to carry one version of the rule for the rest of school and into JEE, here is the one that is actually complete:

  1. Brackets — innermost first, if they are nested.
  2. Orders — exponents and roots.
  3. Multiplication and division — evaluate in the order written, reading left to right.
  4. Addition and subtraction — evaluate in the order written, reading left to right.

That fourth bullet (and the left-to-right clause inside bullets three and four) are the parts the rhyme forgot. Once you internalise them, ambiguous-looking expressions stop being confusing — you read them like a sentence, one operation at a time, from left to right, within each rank.

Quick sanity test

Evaluate each expression two ways — correctly (left-to-right on ties) and incorrectly (ignoring the tie-break). If the answers differ, you have caught the rule in action.

Expression Correct (L-to-R) Wrong (ignore tie-break)
16 \div 8 \div 2 1 4
20 - 6 - 4 10 18
3 \times 6 \div 2 9 1
12 - 5 + 3 - 1 9 3 or 13

Without the left-to-right rule, the last expression has multiple possible readings; with the rule, it has exactly one. That uniqueness is the whole point of having the rule. Good notation cannot afford ambiguity, and the tie-break is how BODMAS closes the last loophole.

Related: Operations and Properties · In BODMAS, Doesn't Division Come Before Multiplication Because D Comes First? · BODMAS Drift: What Happens When You Break the Order · Addition and Subtraction Share a Level in PEMDAS — Does That Mean Order Doesn't Matter?