A student reads "BODMAS" and internalises six strict priority tiers: brackets first, then orders, then division, then multiplication, then addition, then subtraction. Six tiers, six letters, one answer per expression. This reading is wrong. It produces wrong answers on simple problems like 24 \div 4 \times 2, and the error compounds on anything longer. The fix is a single habit: for operations of the same rank, work left-to-right, in the order they appear on the page.

The rule

There are really only four precedence tiers, not six:

  1. Brackets — highest.
  2. Orders (exponents and roots) — next.
  3. Multiplication and division — tied. Go left-to-right.
  4. Addition and subtraction — tied. Go left-to-right.

Multiplication and division share tier 3. Neither outranks the other. The same is true of addition and subtraction on tier 4. When you meet a row of operations from the same tier, you process them in the order you read them.

The canonical left-to-right trap

24 \div 4 \times 2 = ?

Acronym-strict (wrong) reading: BODMAS says D then M, so do the division first and the multiplication second. 24 \div 4 = 6, then 6 \times 2 = 12. Answer: 12. Lucky — the correct answer is 12, but only because left-to-right happened to agree with the bad rule this time.

Same-rank (correct) reading: division and multiplication are tied. Process left-to-right. The division sits on the left of the multiplication, so do it first. 24 \div 4 = 6, then 6 \times 2 = 12. Answer: 12.

Now try the version that exposes the trap:

8 \times 4 \div 2 = ?

Acronym-strict (wrong) reading: "Do all multiplications, then all divisions." 8 \times 4 = 32, then 32 \div 2 = 16. Answer: 16. Correct again — by coincidence.

Now flip the operand order:

8 \div 4 \times 2 = ?

Acronym-strict (wrong) reading: "D before M." Interpret as 8 \div (4 \times 2) = 8 \div 8 = 1. Wrong answer.

Same-rank (correct) reading: left-to-right. 8 \div 4 = 2, then 2 \times 2 = 4. Correct answer: 4.

The first two problems gave the same answer either way, so a student who misread BODMAS might have gone years without noticing. This third problem is where the habit breaks down — and problems of this exact shape are exam staples.

Two readings of eight divided by four times twoA table with two columns showing the two possible readings of the expression eight divided by four times two. The left column shows the wrong reading that treats M as a strict priority after D: four times two equals eight, so eight divided by eight equals one. The right column shows the correct left-to-right reading: eight divided by four equals two, then two times two equals four. The correct answer four is highlighted and labelled the answer, and the wrong answer one is labelled a trap. M "outranks" D (wrong) left-to-right (correct) 8 ÷ 4 × 2 8 ÷ 4 × 2 do × first: 4 × 2 = 8 do ÷ first: 8 ÷ 4 = 2 = 8 ÷ 8 = 2 × 2 now the division now the multiplication = 1 = 4 (the trap) (the answer) tied-rank operations go left-to-right, always
Two readings of $8 \div 4 \times 2$ side by side. The left column is the acronym-strict reading that does multiplication before division — wrong answer $1$. The right column is the left-to-right reading that respects the tied rank — correct answer $4$. Every exam board, every textbook, every calculator agrees with the right column. The acronym is just a memory aid for the *tiers*, not the order within a tier.

The addition and subtraction version

The same trap lives in tier 4.

10 - 3 + 2 = ?

Acronym-strict (wrong) reading: "A before S." 3 + 2 = 5 first, then 10 - 5 = 5. Wrong.

Left-to-right (correct) reading: 10 - 3 = 7, then 7 + 2 = 9. Correct answer: 9.

This one catches even more students because subtraction is already a little stressful, and the acronym's suggestion to group 3 + 2 "feels helpful." It isn't — it silently swaps the sign of the 2, costing 4 units.

Why the rule is left-to-right and not right-to-left

Because that is the convention mathematicians agreed on, and nothing in the structure of arithmetic forces it. Subtraction and division are not associative (see Operations and Properties) — the grouping matters — and in the absence of brackets, some convention has to break the tie. The convention chosen, worldwide, is to read expressions the way a sentence is read: left to right, top to bottom.

This is the same reason English is read left to right and Arabic is read right to left: a choice with no cosmic basis, made consistent so everyone agrees. Mathematical expressions happen to follow English reading order, regardless of the language of the surrounding text.

A quick mental check

When you meet a row of operations from the same tier, run this three-step check before committing to an answer.

  1. Spot the tier. Is the row \times and \div (tier 3), or is it + and - (tier 4)? If it mixes tiers, tier 3 goes first — always.

  2. Go left-to-right within the tier. Pick the leftmost operation in the row and evaluate it. Replace that sub-expression with its value.

  3. Repeat. Continue left-to-right until the row is reduced to a single number.

Try this on 18 - 5 \times 2 + 6 \div 3.

The trap version of this is to do the - and + in the wrong order on tier 4 and come out with 6. Left-to-right prevents that.

Try it — drag the operands and see the two readings disagree

Interactive comparison of two reading orders for a divided by b times cA horizontal number line from one to twenty with three draggable red points labelled a, b, c. Two readouts above the line show the left-to-right answer (a divided by b, then times c) and the wrong reading (a divided by the quantity b times c). As the reader drags the three points, the two readouts almost always disagree, showing how the left-to-right rule gives a different answer from the wrong "M before D" reading. they match only when c = 1 — a special case that fools the eye 1 10 20 ↔ drag any point
Drag the three points. The left readout is the correct answer to $a \div b \times c$ — the left-to-right reading. The right readout is the wrong "M before D" reading, $a \div (bc)$. The two match only when $c = 1$ (the multiplication does nothing) or when $b$ exactly divides $a$ in a special way. Every other configuration — the majority — shows a visible gap. The wrong reading is not a rounding error; it is a different expression with a different answer.

Why: the two readings a \div b \times c and a \div (bc) differ by a factor of c^2. Specifically, (a/b) \cdot c = ac/b, but a/(bc) is a/(bc). The ratio of the two is c^2. So whenever c \neq \pm 1, the two answers are meaningfully different — often dramatically so.

A side note on fraction bars

One place where the same-rank rule looks like it gets suspended is the fraction bar. When you write

\frac{6 + 4}{2 + 3}

the long horizontal bar behaves like a giant set of brackets around both the numerator and the denominator. You compute the numerator (10), the denominator (5), and then divide (2). This is a separate rule from the in-line division sign \div. If you rewrote the same expression in-line as 6 + 4 \div 2 + 3, BODMAS would read it as 6 + (4 \div 2) + 3 = 11 — a completely different answer.

This is one reason mathematicians prefer fraction bars over \div for anything non-trivial. The bar has built-in brackets; the \div sign doesn't. See Brackets Are 'Do Me First' Signs for the free-bracketing habit that prevents this kind of slippage.

The single reliable rule

Forget the mnemonic drama. Carry these four lines in your head.

Three BODMAS letters became a single rule ("tied — left-to-right"), and two common exam traps evaporate.

Related: Operations and Properties · In BODMAS, Doesn't Division Come Before Multiplication Because D Comes First? · Addition and Subtraction Share a Level in PEMDAS — Does That Mean Order Doesn't Matter? · How Is 'Left-to-Right When Same Rank' a Rule — Nobody Told Me That Part · Brackets Are 'Do Me First' Signs — When in Doubt, Add More