Order of Operations Problems- Practice and Solutions

What the Order of Operations Actually Is

PEMDAS. You've seen it. You've probably forgotten what it means halfway through a problem. The order of operations is just the agreed-upon sequence for solving math expressions. Without it, everyone gets different answers.

Math works because we all follow the same rules. That's it. Nothing mystical about it.

The Acronyms (Pick One and Stick With It)

PEMDAS is the US version. Other countries use BODMAS or BIDMAS. They mean the same thing.

That last point trips people up constantly. Multiplication doesn't always come before division. Neither does addition before subtraction. You work left to right through each pair.

Why People Get These Wrong

Most errors come from two sources:

Here's the brutal truth: if you're getting different answers than everyone else, you're skipping a step or doing steps in the wrong order. The math isn't broken.

Practice Problems with Solutions

Work through each one. Cover the solutions. Try it yourself first.

Problem 1: The Basics

8 + 2 × 5

Step 1: Multiplication first. 2 × 5 = 10

Step 2: Addition. 8 + 10 = 18

Answer: 18

If you added first (8 + 2 = 10, then 10 × 5 = 50), you got the wrong answer. That's the most common mistake on easy problems like this.

Problem 2: Parentheses Change Everything

(8 + 2) × 5

Step 1: Parentheses first. 8 + 2 = 10

Step 2: Multiplication. 10 × 5 = 50

Answer: 50

Same numbers, completely different result. Parentheses override the normal order. That's their job.

Problem 3: Exponents Included

3 + 2² × 4

Step 1: Exponents first. 2² = 4

Step 2: Multiplication. 4 × 4 = 16

Step 3: Addition. 3 + 16 = 19

Answer: 19

Problem 4: Division and Multiplication Mixed

12 ÷ 3 × 2

Left to right. That's it.

Step 1: 12 ÷ 3 = 4

Step 2: 4 × 2 = 8

Answer: 8

Don't multiply first. Don't divide first. Left. To. Right.

Problem 5: Nested Parentheses

2 × (3 + (4 - 2)²)

Step 1: Inner parentheses first. 4 - 2 = 2

Step 2: Exponent. 2² = 4

Step 3: Outer parentheses. 3 + 4 = 7

Step 4: Multiplication. 2 × 7 = 14

Answer: 14

Problem 6: The Full Monty

10 - 2 × 3 + 8 ÷ 4

Step 1: Multiplication and division left to right.

2 × 3 = 6

8 ÷ 4 = 2

Step 2: Rewrite with results. 10 - 6 + 2

Step 3: Addition and subtraction left to right.

10 - 6 = 4

4 + 2 = 6

Answer: 6

Common Mistakes Table

Mistake What People Do What They Should Do
MD before DM Multiply before dividing Work left to right through both
AS before SA Subtract before adding Work left to right through both
Skip inner parentheses Ignore nested grouping Solve innermost first
Exponents last Calculate after everything else Second step, right after parentheses
Ignore the order entirely Left to right always Follow PEMDAS sequence

How to Get Better at These

You don't need talent. You need repetition.

  1. Write out every step. Don't do it in your head. Write 8 + 2 × 5 = 8 + 10 = 18. The act of writing forces you to acknowledge the order.
  2. Circle or highlight the operation you're doing next. Keeps you from jumping ahead.
  3. Check your work by re-reading the problem — did you actually do parentheses first? Did you do exponents before multiplication?
  4. Use a calculator to verify — but only after you've tried by hand. The goal is to build the mental habit.

Quick Reference Cheat Sheet

That's the whole system. It looks complicated until it isn't. A few practice problems a day for a week and you'll stop thinking about the rules entirely — you'll just see the right answer.