How to Identify Errors Involving the Order of Operations?

Order of operations errors are among the most frequent mistakes on the GED Math test. A single misstep — like adding before multiplying — can turn a correct setup into a wrong answer. This lesson teaches you to identify, understand, and correct those errors using the PEMDAS framework, so you can catch mistakes in your own work and in test questions that ask you to spot someone else’s error.

What Is the Order of Operations?

The order of operations is a set of rules that tells you which calculations to perform first in a mathematical expression. The standard acronym is PEMDAS:

1. P — Parentheses (and all grouping symbols: brackets, braces, absolute value)
2. E — Exponents (powers and roots)
3. \(\color{blue}{\frac{M}{D}}\) — Multiplication and Division (left to right)
4. \(\color{blue}{\frac{A}{S}}\) — Addition and Subtraction (left to right)

When two operations share the same priority (multiplication and division, or addition and subtraction), you work from left to right.

Common Errors and How to Identify Them

Error 1: Adding or subtracting before multiplying or dividing

Wrong: \(\color{blue}{3 + 4 \times 2 = 7 \times 2 = 14}\)
Correct: Multiply first: \(\color{blue}{4 \times 2 = 8}\), then add: \(\color{blue}{3 + 8 = 11}\).
The error is performing addition before multiplication.

Error 2: Ignoring exponents before other operations

Wrong: \(\color{blue}{2^{3} + 4 \div 2 = 2^{7} = 128}\) (adding the exponent to the next number)
Correct: Evaluate exponent first: \(\color{blue}{2^{3} = 8}\); then divide: \(\color{blue}{4 \div 2 = 2}\); then add: \(\color{blue}{8 + 2 = 10}\).

Error 3: Performing left-to-right operations out of order

Wrong: \(\color{blue}{18 \div (2 + 1) \times 3 = 18 \div 3 = 6}\) (divides all of 18 by 3 then ignores the ×3)
Correct: Parentheses first: \(\color{blue}{2 + 1 = 3}\); then left to right: \(\color{blue}{18 \div 3 = 6}\), \(\color{blue}{6 \times 3 = 18}\).

Error 4: Subtracting before multiplying

Wrong: \(\color{blue}{5^{2} – 3 \times (4 – 1) = 25 – 3 \times 3}\) but then computed as \(\color{blue}{22 \times 3 = 66}\)
Correct: Parentheses: \(\color{blue}{4 – 1 = 3}\); Exponent: \(\color{blue}{5^{2} = 25}\); Multiply: \(\color{blue}{3 \times 3 = 9}\); Subtract: \(\color{blue}{25 – 9 = 16}\).

Step-by-Step Summary

1. Scan the expression for grouping symbols (parentheses, brackets). Evaluate innermost first.
2. Evaluate all exponents.
3. Perform all multiplication and division, working left to right.
4. Perform all addition and subtraction, working left to right.
5. When checking someone else’s work, redo each step yourself and identify the first step that differs.

Watch: Order of Operations Explained (Math Antics)

Math Antics provides a clear, visual explanation of PEMDAS and why the rules exist:

Worked Examples

Example 1: Find and correct the error: a student wrote \(\color{blue}{10 – 2 \times 3 + 1 = 25}\).

The student likely computed \(\color{blue}{(10 – 2) \times (3 + 1) = 8 \times 4 = 32}\) or some other wrong grouping.
Correct: Multiply first: \(\color{blue}{2 \times 3 = 6}\). Then left to right: \(\color{blue}{10 – 6 + 1 = 5}\).
The error: performing subtraction and addition before multiplication.

Example 2: A student simplified \(\color{blue}{(5 + 3)^{2} \div 4}\) as follows: \(\color{blue}{25 + 9 \div 4 = 25 + 2.25 = 27.25}\). What is the error?

The student incorrectly expanded \(\color{blue}{(5 + 3)^{2}}\) as \(\color{blue}{5^{2} + 3^{2}}\).
Correct: Parentheses first: \(\color{blue}{5 + 3 = 8}\); Exponent: \(\color{blue}{8^{2} = 64}\); Divide: \(\color{blue}{64 \div 4 = 16}\).

Example 3: Evaluate correctly: \(\color{blue}{6 + 2^{3} \times 5 – 4}\).

Exponent: \(\color{blue}{2^{3} = 8}\). Multiply: \(\color{blue}{8 \times 5 = 40}\). Left to right: \(\color{blue}{6 + 40 – 4 = 42}\).

Example 4: Evaluate: \(\color{blue}{4 \times (3 + 2^{2}) – 6}\).

Inside parentheses, exponent first: \(\color{blue}{2^{2} = 4}\), then \(\color{blue}{3 + 4 = 7}\). Multiply: \(\color{blue}{4 \times 7 = 28}\). Subtract: \(\color{blue}{28 – 6 = 22}\).

More Practice: Order of Operations with 4 Examples (Math with Mr. J)

Math with Mr. J demonstrates PEMDAS with four complete worked examples:

Exercises

Evaluate each expression correctly using PEMDAS.

1. \(\color{blue}{3 + 4 \times 2}\)
2. \(\color{blue}{2^{3} + 4 \div 2}\)
3. \(\color{blue}{(5 + 3)^{2} \div 4}\)
4. \(\color{blue}{10 – 2 \times 3 + 1}\)
5. \(\color{blue}{18 \div (2 + 1) \times 3}\)
6. \(\color{blue}{5^{2} – 3 \times (4 – 1)}\)

Answers

1. \(\color{blue}{11}\)  (multiply first: \(\color{blue}{4\times 2=8}\), then \(\color{blue}{3+8}\))
2. \(\color{blue}{10}\)  (\(\color{blue}{2^{3}=8}\), \(\color{blue}{4\div 2=2}\), then \(\color{blue}{8+2}\))
3. \(\color{blue}{16}\)  (\(\color{blue}{(5+3)=8}\), \(\color{blue}{8^{2}=64}\), \(\color{blue}{64\div 4}\))
4. \(\color{blue}{5}\)  (\(\color{blue}{2\times 3=6}\), then \(\color{blue}{10-6+1}\))
5. \(\color{blue}{18}\)  (\(\color{blue}{2+1=3}\), \(\color{blue}{18\div 3=6}\), \(\color{blue}{6\times 3}\))
6. \(\color{blue}{16}\)  (\(\color{blue}{4-1=3}\), \(\color{blue}{5^{2}=25}\), \(\color{blue}{3\times 3=9}\), \(\color{blue}{25-9}\))

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A 4-Step Check for Finding the Mistake

1. Solve it yourself first. Work the original expression on your own so you know the correct answer.
2. Compare line by line. Find the first line where the student’s work differs from yours. The error is there or just before it.
3. Name the error. Was it the wrong operation first, a dropped parenthesis, an exponent applied to the wrong number, or a copying slip?
4. Fix and explain. Redo the work from that line and write one sentence about what went wrong.

Practice: Find the Mistake

Each student answer below is wrong. Find the error before you read the answer.

1. 20 − 8 ÷ 4 = 3
2. 12 ÷ 2 × 3 = 2
3. (3 + 1)² = 10
4. 2 × 3² = 36
5. −3² = 9

Answers:

1. The student subtracted first (12 ÷ 4). Divide first: 20 − 2 = 18.
2. The student multiplied first. Work left to right: 6 × 3 = 18.
3. The student squared each number (9 + 1). Add inside the parentheses first: 4² = 16.
4. The student multiplied first (6²). The exponent comes first: 2 × 9 = 18.
5. The exponent applies only to 3, not to the negative sign: −3² = −(3 × 3) = −9. Only (−3)² equals 9.

Frequently Asked Questions

What acronym is used to describe the order of operations?

In the United States the usual acronym is PEMDAS: parentheses, exponents, multiplication and division, addition and subtraction. Other countries use BODMAS or BIDMAS, with brackets and orders or indices. All mean the same order, and in each one multiplication and division share a rank and go left to right, as do addition and subtraction.

How do you identify the error in an order of operations problem?

Redo the problem yourself, then compare it with the worked solution one step at a time. At each step, ask whether a higher-priority operation was skipped. For 20 − 4 × 3, a student who writes 16 × 3 = 48 subtracted first. The correct work is 20 − 12 = 8, so the error is in the first step.

What is order of operations error analysis?

Error analysis means studying a solution that someone else worked out and finding where it went wrong. In order of operations problems, you check whether each step followed parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right. You then explain the mistake in words and give the correct answer.

What are examples of order of operations errors?

Common ones include adding before multiplying, as in 5 + 2 × 4 = 28 instead of 13; dividing and multiplying out of order, as in 12 ÷ 2 × 3 = 2 instead of 18; and squaring before handling parentheses, as in (1 + 3)² = 10 instead of 16. Each one breaks a specific rule in PEMDAS.

Related Topics

• Identifying Expressions and Equations
• Use Properties to Write Equivalent Expressions
• Evaluate Integers Raised to Rational Exponents
• Use Exponents to Write Multiplication Expressions
• Identify Equivalent Expressions