Working Through Order Of Operations Practice Problems
Most people mess these up because they memorize PEMDAS and then proceed to do exactly what the acronym tells them — left to right, no deviations. That works for clean textbook problems. Real practice problems are designed to catch you when you stop thinking. The actual method is straightforward but easy to botch under pressure. Handle parentheses and brackets first, working from the innermost outward. Then exponents. Then multiplication and division as they appear left to right — these two share the same tier and you do not prioritize one over the other. Finally, addition and subtraction, again left to right. The trap most students fall into is treating multiplication as always coming before division. They don't. Same level. Go left to right.
Common Order Of Operations Practice Problems
Take this problem: 3 + 4 × 2² ÷ (6 - 3). A lot of people will blast through it like this. Parentheses first: 6 minus 3 equals 3. Then they see the exponent and knock out 2 squared, getting 4. Now they have 3 + 4 × 4 ÷ 3. From here is where it falls apart. Some multiply 4 times 4 to get 16, divide by 3 to get roughly 5.33, then add 3 and end up at 8.33. Others grab the addition first and do 3 plus 4, which is completely wrong because addition is lowest priority here. The correct path: 4 times 4 is 16. 16 divided by 3 is 5.333 repeating. 3 plus 5.333 is 8.333. But if this appeared on a test that expects fractional answers, you'd write it as 25/3. Students lose points not because they don't know the rules, but because they rush through the final steps or write down intermediate decimals instead of keeping exact forms. I remember grading a set of practice problems once where about a third of the class treated implied multiplication differently. The expression 2x(3 + 4) had half the students multiplying 2 by x first, then distributing, and the other half doing the parenthesis first. When there's no explicit operator between a number and a variable or parenthesis, some curricula treat that as higher priority than explicit multiplication or division. This isn't universal and it's one of those things that varies by textbook and teacher. The workaround I used was simply to ask my students to write out every step with explicit operators — 2 × x × (3 + 4) — so there was no ambiguity in how they read it.
Here's another edge case that comes up constantly and barely gets explained anywhere: negative numbers inside parentheses. -3² is not the same as (-3)². The first one is negative nine. The second is positive nine. Students consistently write -9 when the answer is 9 because their brain reads the negative sign as part of the base rather than as a unary operator applied after the exponent. I started having them rewrite every expression with explicit grouping before solving, and it cut that error rate down dramatically. For practice, start with problems that mix all four operations plus exponents and parentheses. Work through at least ten before moving to harder territory. Then tackle problems with nested parentheses and fractions. The real test problems hide complexity in plain sight — things like 5 - [3 + (2 - 4)²] ÷ 2 where the nesting makes it easy to lose track of which bracket closes first. One thing nobody warns you about: calculators lie to you if you're not careful. A basic calculator will compute left to right and give you the wrong answer for anything beyond the simplest expression. You need a scientific calculator or a tool that respects order of operations. If you're doing this by hand for a test, write each step on its own line. It takes longer but it eliminates the most common error source — holding too many values in your head and mixing up the sequence.
Get the Full Details
The only reliable way to get better at these is repetition with error tracking. Keep a list of every problem you got wrong and note exactly which rule you violated. After about twenty mistakes logged this way, you'll start recognizing the patterns before you even finish reading the problem. That's when you actually know it.
