The Immediate Instinct: Why 16 Feels Right
Let’s start with the most common incorrect answer: 16.
People who arrive at 16 typically compute the expression like this:
(5 + 3) × 2 = 8 × 2 = 16
This approach feels natural. After all, we often read from left to right, and in everyday language, that’s how we process most information. So when people see “5 + 3 × 2,” they instinctively start with 5 + 3.
It’s quick. It’s intuitive. And it’s wrong.
This mistake doesn’t come from a lack of intelligence—it comes from relying on instinct rather than structure.
The Correct Answer: 11
The correct way to evaluate the expression is:
5 + (3 × 2) = 5 + 6 = 11
The key difference lies in understanding the order of operations, a foundational rule in arithmetic that determines the sequence in which operations should be performed.
Understanding the Rules: Order of Operations
To avoid ambiguity in mathematical expressions, we follow a standard convention known as the order of operations. In many English-speaking countries, this is often remembered by the acronym PEMDAS:
- Parentheses
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Another common mnemonic is “Please Excuse My Dear Aunt Sally.”
But here’s where many people misunderstand things: multiplication and division are not strictly separate steps, nor are addition and subtraction. They are performed from left to right within their respective levels.
So the real hierarchy looks like this:
- Parentheses
- Exponents
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
In the expression 5 + 3 × 2, there are no parentheses or exponents. That means we move directly to multiplication and division.
Since multiplication comes before addition, we evaluate:
3 × 2 = 6
Then:
5 + 6 = 11
Why This Rule Exists
You might wonder: why not just go left to right for everything?
The answer lies in consistency and clarity.
Without a standard order, expressions could have multiple valid interpretations. For example:
5 + 3 × 2
Could mean:
- (5 + 3) × 2 = 16
- 5 + (3 × 2) = 11
Both are mathematically valid structures—but without agreed-upon rules, there’s no way to know which one was intended.
The order of operations eliminates that ambiguity. It ensures that anyone, anywhere in the world, will interpret the same expression in the same way.
A Brief Historical Perspective
The concept of prioritizing certain operations over others didn’t appear overnight. It evolved gradually as mathematics became more formalized.
Early mathematicians often relied heavily on parentheses or wrote expressions in ways that avoided ambiguity altogether. As notation developed, especially between the 16th and 18th centuries, conventions like the order of operations became standardized.
Today, these rules are embedded in everything from basic arithmetic to advanced algebra and computer programming.
The Psychology of Getting It Wrong
So why do so many people still answer 16?
There are a few reasons:
1. Left-to-Right Bias
Humans naturally process information sequentially. When reading, we go from left to right (in most languages), and we often apply that same habit to math—even when it doesn’t apply.
2. Forgotten Rules
Many people learn the order of operations in school but don’t use it regularly. Over time, the rule fades, and instinct takes over.
3. Overconfidence
Because the problem looks simple, people don’t double-check their reasoning. They assume they’re right without revisiting the rules.
4. Misunderstood Mnemonics
Some learners interpret PEMDAS as a strict sequence where multiplication must always come before division, and addition before subtraction, rather than understanding the grouping of operations.
Real-World Consequences
You might think this kind of mistake only matters in classrooms, but misunderstandings about the order of operations can have real consequences.
In fields like engineering, finance, and computer science, a small error in calculation can lead to significant problems. Software bugs, structural miscalculations, and financial discrepancies can all stem from incorrect assumptions about how expressions are evaluated.
Even in everyday life—calculating discounts, splitting bills, or adjusting recipes—getting the order wrong can lead to inaccurate results.
Making It Intuitive
One way to better understand the rule is to think about multiplication as a form of repeated addition—but with higher priority.
In the expression:
5 + 3 × 2
You’re not just adding 3 and 2. You’re adding 5 to two groups of 3.
So it becomes:
5 + (3 + 3) = 5 + 6 = 11
This perspective helps reinforce why multiplication comes first—it defines a structure that addition then builds upon.
Practice Makes Permanent
Let’s look at a few similar examples:
-
7 + 4 × 3
→ 7 + 12 = 19 -
10 − 2 × 5
→ 10 − 10 = 0 -
6 + 8 ÷ 4
→ 6 + 2 = 8
Each time, multiplication or division is handled before addition or subtraction.
When Parentheses Change Everything
Parentheses override the default order of operations.
Compare:
- 5 + 3 × 2 = 11
- (5 + 3) × 2 = 16
Same numbers. Same operations. Completely different results.
This is why parentheses are so powerful—they allow you to explicitly control the sequence of operations.
The Role of Education
This simple question highlights a broader issue in education: the difference between memorization and understanding.
Many students learn PEMDAS as a rule to memorize, but without fully grasping why it exists. As a result, they may forget or misapply it later.
A deeper understanding—knowing the reasoning behind the rule—makes it easier to remember and apply correctly.
A Lesson Beyond Math
Interestingly, this problem also serves as a metaphor.
In life, as in math, the order in which you approach things matters. Priorities shape outcomes. Small misunderstandings can lead to big differences.
Taking a moment to pause, reconsider, and apply the right framework can change everything.
Final Answer
So, what is:
5 + 3 × 2?
The correct answer is:
11
Not 16. Not anything else.
Just 11.
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