You read PEMDAS as P-E-MD-AS and noticed + and - sit on the same rung. Same level, so the order should not matter, right? Compute them in whichever order, same answer? That is the trap. "Same precedence" and "order doesn't matter" are two different statements, and confusing them is the single most common PEMDAS mistake in class tests.

State the misconception

The misconception: because addition and subtraction share a level in PEMDAS/BODMAS, you can evaluate them in any order and get the same answer.

Why it's tempting: your teacher told you + and - are "at the same rank," which sounds symmetric. Addition is commutative — 3 + 7 = 7 + 3. It feels like subtraction, being "just addition of a negative," should share that freedom.

The counter-example

Take the expression

10 - 3 + 2

Left-to-right evaluation — what PEMDAS actually prescribes for tied operations — gives

10 - 3 + 2 \;=\; (10 - 3) + 2 \;=\; 7 + 2 \;=\; 9

If "same level" really meant "any order," you could do the addition first:

10 - 3 + 2 \;\stackrel{?}{=}\; 10 - (3 + 2) \;=\; 10 - 5 \;=\; 5

The two answers differ by 4. The misconception would cost you the whole mark on this problem. The rule for tied operations is not "any order" — it is "strict left-to-right."

Two readings of ten minus three plus two diverge by fourThe expression ten minus three plus two is shown at the top. Below it, two paths fork. The left path labelled correct evaluates left to right: ten minus three equals seven, then seven plus two equals nine. The right path labelled misconception groups the addition first: three plus two equals five, then ten minus five equals five. A red cross marks the misconception path. The bottom note reads: tied precedence means strict left to right, not any order. 10 − 3 + 2 Correct: left-to-right (10 − 3) + 2 = 9 Wrong: group + first 10 − (3 + 2) = 5 ✗ tied precedence = strict left-to-right, not any order
Same expression, two readings, two different answers. The correct reading treats $+$ and $-$ as tied in precedence *but* evaluated strictly left-to-right. The wrong reading treats "tied" as "order doesn't matter," groups the $+$ first, and lands four units away from the correct answer.

Why left-to-right, specifically?

Subtraction is not commutative and not associative on its own. 10 - 3 \neq 3 - 10, and (10 - 3) - 2 \neq 10 - (3 - 2). Since you cannot freely rearrange subtraction, written-arithmetic notation needs a tie-breaker. Mathematicians picked "left-to-right" because that is how we read text — no extra rule to memorise.

There is a cleaner way to think about it, too. Every subtraction a - b can be rewritten as an addition of a negative: a + (-b). Rewrite the example that way:

10 - 3 + 2 \;=\; 10 + (-3) + 2

Now every operation is +, and addition is associative and commutative — so the order genuinely does not matter any more. You can add in any order you like: 10 + (-3) + 2 = 9, (-3) + 2 + 10 = 9, 2 + 10 + (-3) = 9. Same answer every time.

Why: the moment you convert subtractions into additions of negatives, the whole expression becomes pure addition, which is commutative and associative. The "left-to-right" rule is really a warning sign: until you make the conversion, do not rearrange. After the conversion, you can.

Try it: slide the second operand

Interactive: watch left-to-right and wrong readings drift apartA draggable red point on a horizontal line from zero to ten sets a value b. Above the line, two readouts show ten minus b plus two (the correct left-to-right reading) and ten minus the quantity b plus two (the wrong grouping). As b changes, the two readouts stay exactly four apart — the gap is constant and non-zero for every b, proving the two readings are never equivalent. 0 5 10 ↔ drag b
Drag $b$ and watch the two readings. No matter what you pick, the correct answer exceeds the wrong one by exactly $4$, because the wrong reading absorbs an extra $-2 \cdot 2 = 4$ flip of sign on the last term. The gap is never zero — there is no value of $b$ for which the two readings agree.

Same trap in division-and-multiplication

Everything said above applies identically to \times and \div. They share a PEMDAS level, and the rule is strict left-to-right, not any order. Try

24 \div 4 \times 2

Left-to-right: (24 \div 4) \times 2 = 6 \times 2 = 12. "Any order": 24 \div (4 \times 2) = 24 \div 8 = 3. Same shape of mistake, factor-of-four error this time. The rule is identical because division, like subtraction, is not commutative.

The correct mental model

Reframe "same level" in PEMDAS as: same priority, not same freedom. Tied operations still have to be evaluated in written order, because they are non-commutative (for - and \div) or because their intended grouping is what the writer wrote, and reading rules must not second-guess it.

If you want the actual freedom to rearrange, convert every subtraction into an addition-of-a-negative and every division into a multiplication-by-a-reciprocal. Once every operation in the expression is + or \times, commutativity and associativity kick in for real, and you can rearrange to your heart's content.

Quick self-check

Before you read the answer, evaluate these in your head:

Answers: 15, 12, 10. If any one of those didn't match, re-read the "left-to-right" part of the rule above. The expressions are designed to trip exactly this misconception — if you grouped the plus or the times first, you'd get 9, 3, 10 for the three. The third one happens to agree by coincidence; the first two do not.

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 · Is Subtraction Just 'Adding a Negative' — or Is It Its Own Operation?