ACT® · Math: Integrating Essential Skills
Computation and Problem Solving with Real Numbers
Order of operations tells you which part of an expression to compute first so everyone gets the same single answer. Work parentheses, then exponents, then multiplication and division left to right, then addition and subtraction left to right.
These problems look small but punish any skipped step. The rule, often called PEMDAS, is really a priority list: parentheses first, then exponents, then multiplication and division together left to right, then addition and subtraction together left to right. A nested expression — parentheses inside parentheses, or a fraction bar stretched under a whole expression — gets peeled from the inside out. Whatever sits deepest inside the grouping symbols gets resolved before anything touches it. A fraction bar counts as its own grouping symbol: everything above it is one parenthesis, everything below it is another, each simplified completely before you divide.
Fractions follow the same hierarchy but add a second skill: dividing by a fraction means multiplying by its reciprocal. Flip the second fraction, multiply straight across, simplify, and only then handle any addition or subtraction waiting outside. Mixing up order here — subtracting before dividing, say — gives a confidently wrong answer that still looks reasonable, which is exactly why these items are unforgiving.
This topic sits right next to unit conversions, rates, and percent problems in the same cluster, because all of them are really multi-step arithmetic wearing different clothes. Once the order-of-operations habit is automatic, those word problems become the same mechanical process: translate to an expression, then work it from the inside out without shortcuts.
Key terms
- Order of operations
- The fixed sequence — parentheses, exponents, multiplication/division, addition/subtraction — that gives one correct value for an expression.
- Reciprocal
- The flipped version of a fraction, used to turn division by a fraction into multiplication.
- Grouping symbol
- A parenthesis, bracket, or fraction bar that forces everything inside it to be simplified before anything outside touches it.
- Exponent
- A small raised number showing how many times a value is multiplied by itself, resolved after parentheses but before multiplication.
Watch it explained
- How it worksHow does the order of operations resolve a nested expression from the inside out?
- Worked exampleFind the value of -3 + 4 times (2 - 7) squared, all divided by 5.
- Worked exampleFind the value of two thirds divided by five sixths, minus one fourth.
- Common mistakeWhy isn't a negative number squared the same as the negative of that number squared?
Check yourself
End of topic.