Presented with 10 + 20 ÷ 5 × 4, people may rush to an answer, assuming familiar symbols and small numbers make the calculation automatic. The expression is a compact test of whether the standard order of operations is applied consistently: multiplication and division share the same precedence, as do addition and subtraction. Within either pair, operations are evaluated from left to right.
A common wrong turn is to add 10 and 20 first, effectively inserting parentheses that are not in the expression. That changes the calculation: (10 + 20) ÷ 5 × 4 equals 24. Treating multiplication as higher priority than division can also lead to a different result. These are errors in applying the procedure, not evidence that someone lacks general mathematical ability.
Order Matters
For 10 + 20 ÷ 5 × 4, work from left to right through the division and multiplication before adding: 20 ÷ 5 = 4, then 4 × 4 = 16, and finally 10 + 16 = 26. Written in stages, the calculation is 10 + 20 ÷ 5 × 4 → 10 + 4 × 4 → 10 + 16 → 26. Khan Academy describes the rule as the same order of operations used elsewhere: “It is the same as the order of operations for anything else, so PEMDAS.”
Parentheses make clear why the notation matters. The expression (10 + 20) ÷ 5 × 4 evaluates to 24, while 10 + 20 ÷ (5 × 4) evaluates to 11. Neither set of parentheses appears in the original, so neither altered grouping should be assumed.
Why the Mistake Happens
The familiar format can encourage a quick, intuitive response instead of a deliberate check of each operation. Confidence and accuracy do not always track together: researchers Don A. Moore and Paul J. Healy distinguish overestimation of one’s performance from overconfidence based on comparisons with others or excessive precision. Their structure helps explain why confidence alone is not a reliable check on a calculation.
A World Economic Forum account of logic-test research involving hundreds of participants, conducted by psychologists affiliated with James Madison University and the University of Toronto, describes cases in which highly intelligent participants answered incorrectly after moving too quickly. That research offers broader context about reasoning under pressure; it was not an experiment on this particular arithmetic expression.
In this problem, the basic distinction is between a simple calculation slip and a mistaken rule. Adding 10 and 20 first changes the grouping, while multiplying before the earlier division applies an incorrect precedence rule. Slowing down makes it easier to spot which kind of error has occurred.
Building Better Habits
Short arithmetic exercises can check whether a rule is being recalled and applied correctly. Read the full expression, then write the intermediate steps while checking which operations share precedence, rather than relying on mental intuition. Here, checking that division and multiplication proceed from left to right leads directly to 26.
A pause before answering can also help counter the impulse to prioritize speed over sequence. A single puzzle cannot measure a person’s mathematical ability; visible steps make the reasoning easier to follow and check.
Even a basic-looking expression can expose how quickly assumptions affect a result. In this case, the reliable method is straightforward: follow the written order, give multiplication and division equal footing, and add only after those operations are complete.
