Order of Operations

Order of operations questions are pure procedure. Nobody finds the concept hard and everybody makes the mistake, because the expression is designed so that doing it in reading order gives a plausible wrong answer that is sitting in the choices.

The four stages

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1. Grouping - parentheses, brackets, braces, and the implicit groupings below.
2. Exponents and roots.
3. Multiplication and division, together, left to right.
4. Addition and subtraction, together, left to right.

PEMDAS is a useful word and a misleading one. It reads as six ranked operations and it is four stages, with the last two stages holding two operations apiece.

The two stages people rank wrongly

Multiplication does not beat division. They are equal, so you go left to right.

24 / 4 x 3

Left to right: 24 divided by 4 is 6, times 3 is 18. Doing the multiplication first gives 24 divided by 12, which is 2, and 2 will be a choice.

Addition does not beat subtraction. Also equal, also left to right.

10 - 4 + 3

Left to right: 10 minus 4 is 6, plus 3 is 9. Doing the addition first gives 10 minus 7, which is 3, and 3 will be a choice.

Those two expressions are the entire trick, and a test writer who wants a hard order-of-operations question writes one of them into a longer expression.

Invisible grouping

Three things group without brackets, and the test uses all three.

A fraction bar. Everything above it is one group and everything below it is another.

The expression with 12 plus 8 on top and 4 plus 1 on the bottom is (12 plus 8) divided by (4 plus 1), which is 20 divided by 5, or 4. It is not 12 plus 2 plus 1.

A radical sign. Everything under it is grouped. The square root of (9 plus 16) is the square root of 25, which is 5. It is not 3 plus 4.

That one is worth dwelling on, because the square root of a sum is not the sum of the square roots, and 7 is always a choice.

Absolute value bars, which group exactly like parentheses.

Nested grouping

Work from the inside out.

Evaluate 3 x [ 8 - 2 x ( 5 - 3 ) ] + 4

• Innermost parentheses: 5 minus 3 is 2.
• Inside the brackets now: 8 minus 2 times 2. Multiplication before subtraction, so 8 minus 4 is 4.
• Brackets done: 3 times 4 is 12.
• Finally: 12 plus 4 is 16.

Four lines, one operation each. Writing it that way is slower to read and faster to get right, and on a test with no calculator that trade is always worth taking.

A longer one, worked

Evaluate 5 + 2 x ( 6 - 4 ) squared / 2 - 7

• Grouping: 6 minus 4 is 2. The expression is 5 + 2 x 2 squared / 2 - 7.
• Exponent: 2 squared is 4. Now 5 + 2 x 4 / 2 - 7.
• Multiplication and division, left to right: 2 times 4 is 8, then 8 divided by 2 is 4. Now 5 + 4 - 7.
• Addition and subtraction, left to right: 5 plus 4 is 9, minus 7 is 2.

The trap is squaring the 2 that is being multiplied rather than the parenthesis, or dividing before multiplying. Both produce clean-looking numbers.

What you can skip

Across the 40 questions on this topic:

• Deeply nested brackets. Square brackets come up once and braces three times; work from the innermost outward and nothing more is needed.
• Mnemonic variants. Whether you learned PEMDAS or another order, the rule is the one in this lesson: multiplication and division, then addition and subtraction, each left to right.

Where people lose points

Doing multiplication before division, or addition before subtraction, when the division or subtraction came first.

Splitting a fraction bar and applying operations across it.

Distributing a root or an exponent across a sum. The square root of a sum is not the sum of the roots, and (a plus b) squared is not a squared plus b squared.

Applying an exponent to the wrong base, particularly when a coefficient sits in front of a parenthesis.

Doing two steps in one line and losing a sign in the process.

Ignoring an implicit multiplication. 2(3 plus 5) means 2 times 8, which is 16.

Work one in under a minute

Evaluate 40 / 2 / 4 x 3

Every operation is in the same stage, so it is purely left to right.

40 divided by 2 is 20. Twenty divided by 4 is 5. Five times 3 is 15.

Grouping it as 40 divided by (2 divided by 4) times 3, or as 40 divided by (2 times 4 times 3), gives 240 and 1.67 respectively, and at least one of those will be a choice. When every operation has equal rank, the only rule is direction.

Where this leads

Every algebraic manipulation in this subtest is an application of these four stages, and the grouping rule is what makes distributing and factoring legal.