Math equations can feel like puzzles, especially when they involve multiple operations. Do you add first, or multiply? What about division and subtraction?
The order you perform these actions matters. Getting it wrong can lead to a completely different answer.
This is where a clear understanding of the order of operations and our free MDAS calculator becomes essential.
What is an MDAS Calculator?
An MDAS calculator is a tool designed to solve mathematical expressions by following the correct order of operations. MDAS is an acronym that stands for:
- Multiplication
- Division
- Addition
- Subtraction
This specific order is a fundamental rule in mathematics that ensures anyone solving the same equation will arrive at the same, correct answer.
Our calculator automates this process, providing a step-by-step solution that shows how it reached the final result.
The Four Steps of the MDAS Rule
The MDAS rule provides a simple, two-step hierarchy for solving equations that involve multiplication, division, addition, and subtraction.
Here’s how it works:
Step 1: Multiplication and Division
First, solve all multiplication and division operations as they appear from left to right in the equation.
It’s a common misconception that multiplication always comes before division. In reality, they have equal priority. You simply work through the equation from left to right.
Example
In the expression 10 ÷ 2 × 5, you would first divide 10 by 2 (which equals 5) and then multiply the result by 5 to get 25.
Step 2: Addition and Subtraction
After completing all multiplication and division, you move on to addition and subtraction. Like the first step, these operations also have equal priority.
You solve them as they appear from left to right.
Example
In the expression 9 – 3 + 5, you would first subtract 3 from 9 (which equals 6) and then add 5 to get 11.
MDAS Example
Let’s walk through an example to see the MDAS rule in action.
Problem: Solve 50 + 10 ÷ 2 × 5 – 3
Solution:
- Start with multiplication and division (from left to right). The first of these operations is 10 ÷ 2.
- 10 ÷ 2 = 5
- The equation now becomes: 50 + 5 × 5 – 3
- Continue with the next multiplication. The next operation is 5 × 5.
- 5 × 5 = 25
- The equation is now: 50 + 25 – 3
- Move to addition and subtraction (from left to right). The first operation is 50 + 25.
- 50 + 25 = 75
- The equation is now: 75 – 3
- Perform the final subtraction.
- 75 – 3 = 72
The final answer is 72.
Expanding Beyond MDAS: PEMDAS and Other Acronyms
MDAS covers the four basic arithmetic operations, but many equations are more complex. They often include parentheses and exponents.
To handle these, we use a more comprehensive set of rules, commonly remembered by acronyms like PEMDAS.
| Acronym | Stands For | Common Use |
| PEMDAS | Parentheses, Exponents, Multiplication/Division, Addition/Subtraction | USA |
| BEDMAS | Brackets, Exponents, Division/Multiplication, Addition/Subtraction | Canada |
| BODMAS | Brackets, Orders, Division/Multiplication, Addition/Subtraction | UK |
| GEMDAS | Grouping, Exponents, Multiplication/Division, Addition/Subtraction | Varies |
All these acronyms describe the same fundamental order of operations. Let’s break down PEMDAS, the most common version in the United States.
The Full Order of Operations: PEMDAS
PEMDAS provides a complete framework for solving mathematical expressions.
P – Parentheses
Always solve the expressions inside parentheses () first. If there are nested parentheses (e.g., [ ( ) ]), work from the innermost set outward.
E – Exponents
Next, calculate all exponents (powers) and roots. For example, 3² would be solved at this stage.
MD – Multiplication and Division
Perform all multiplication and division from left to right.
AS – Addition and Subtraction
Finally, perform all addition and subtraction from left to right.
Notice that MDAS is simply the last two steps of the PEMDAS rule.
Why Does the Order of Operations Matter?
The order of operations acts as the universal grammar of mathematics.
Without it, a single mathematical expression could have multiple different answers, leading to confusion and errors.
Consider the simple expression 3 + 5 × 2.
- If you add first, you get 8 × 2 = 16.
- If you multiply first (following PEMDAS), you get 3 + 10 = 13.
Only one of these answers is correct.
By establishing a standard sequence, mathematicians, scientists, engineers, and programmers can all communicate and work with numbers consistently, knowing they will all arrive at the same result.
This consistency is crucial in fields where precise calculations can mean the difference between a successful space mission and a failed one.
A More Powerful Approach
Ready to put your skills to the test? Try solving a few equations by hand, then check your work with our online order of operations calculator. Practice is the key to making these rules second nature.