Order of operations (PEMDAS) and the traps tests use

One of the three example math questions in NEIEP's EIAT study guide is a multi-step expression that has to be worked in the right order. These are easy marks to lose, because a common wrong answer is usually among the options. The rules are simple; the traps are in the details.

PEMDAS

Work through an expression in this order:

1. Parentheses — anything in parentheses first.
2. Exponents — powers and roots.
3. Multiplication and Division — together, working left to right.
4. Addition and Subtraction — together, working left to right.

Trap 1: multiplication does not always come before division

PEMDAS lists M before D, which makes people think multiplication goes first. It does not. Multiplication and division have equal priority, so you do them in the order they appear, left to right.

8 ÷ 2 × 4: left to right, 8 ÷ 2 = 4, then 4 × 4 = 16. Doing the multiplication first gives 8 ÷ 8 = 1, which is wrong.

Trap 2: the same for addition and subtraction

10 − 4 + 3: left to right, 10 − 4 = 6, then 6 + 3 = 9. Adding first gives 10 − 7 = 3, which is wrong.

Trap 3: multiplication before addition

2 + 3 × 4 = 2 + 12 = 14, not 20. Only parentheses change that: (2 + 3) × 4 = 20.

Trap 4: exponents and negatives

An exponent applies only to what it is directly attached to. −3² means −(3²) = −9, whereas (−3)² = 9.

Trap 5: a fraction bar groups things

A long fraction bar acts like parentheses around the top and around the bottom. Work each out fully before dividing: (6 + 4) ÷ (3 − 1) = 10 ÷ 2 = 5.

A full worked example

18 − 6 ÷ 3 × 2 + 4²

1. Exponent: 4² = 16. Now 18 − 6 ÷ 3 × 2 + 16.
2. Division and multiplication, left to right: 6 ÷ 3 = 2, then 2 × 2 = 4. Now 18 − 4 + 16.
3. Subtraction and addition, left to right: 18 − 4 = 14, then 14 + 16 = 30.

How to avoid the traps on test day

• Rewrite the expression after each step, as in the example above. It is slower, and it catches nearly every mistake.
• When a line has both × and ÷ (or + and −), underline them and work left to right.
• Check your answer against the options exactly — and remember that NEIEP's example questions include a "No answer" option.

Practice

The EIAT math practice includes order-of-operations questions where the most common wrong answer is always among the options.