At first glance, the expression 8 − 2 × 3 + 6 looks like a simple arithmetic problem. It contains only four numbers and three basic mathematical operations. Most people learn how to solve expressions like this in elementary school, so the answer might seem obvious.
Yet this little puzzle can lead to surprisingly different answers. Some people say 24, others arrive at 8, and a few may come up with entirely different results depending on how they approach the calculation.
The image accompanying this puzzle makes the challenge even more interesting with the message, “Answer is not 24.”
But why do so many people arrive at 24? Is there a hidden trick, or does the solution depend on a mathematical rule that is easy to overlook?
The answer comes down to one essential concept: the order of operations.
Understanding this rule will not only help you solve this particular puzzle correctly, but will also make many other arithmetic problems easier to understand.
The Correct Answer Is 8
Let's begin with the expression:
8 − 2 × 3 + 6
The correct answer is 8.
To understand why, we need to remember that multiplication and division take priority over addition and subtraction. These operations are not all performed in the same order simply because they appear from left to right.
The expression contains multiplication, subtraction, and addition. Therefore, we must perform the multiplication first.
Here is the calculation:
Step 1: Multiply 2 by 3.
(2 \times 3 = 6)
Replace the multiplication in the original expression with the result:
(8 - 6 + 6)
Step 2: Perform subtraction and addition from left to right.
First:
(8 - 6 = 2)
Then:
(2 + 6 = 8)
The final result is therefore:
[
\boxed{8}
]
There is no hidden mathematical trick in this calculation. The answer follows the standard rules used in arithmetic.
Why Do So Many People Get 24?
The answer 24 usually comes from a common mistake: calculating the expression strictly from left to right without considering the priority of multiplication.
Someone might approach the expression this way:
(8 - 2 = 6)
Then multiply:
(6 \times 3 = 18)
Finally, add 6:
(18 + 6 = 24)
The result appears logical if every operation is treated as having equal priority. However, that is not how standard arithmetic works.
Multiplication must be completed before addition or subtraction, regardless of where it appears in the expression.
This is why the result of 24 is incorrect.
The mistake is understandable. When reading a sentence, people naturally move from left to right. It is tempting to apply the same habit to a mathematical expression. But arithmetic has its own rules, and those rules determine which operation must come first.
Once you understand this distinction, the puzzle becomes much easier.
What Is the Order of Operations?
The order of operations is a set of conventions that tells us how to evaluate mathematical expressions consistently.
Without these rules, two people could interpret the same expression differently and arrive at different answers. Mathematics would become confusing because there would be no shared method for determining which calculation should be performed first.
One common way to remember the rules is through the acronym PEMDAS:
-
P — Parentheses: Evaluate expressions inside parentheses first.
-
E — Exponents: Evaluate powers and roots as appropriate.
-
M — Multiplication: Perform multiplication.
-
D — Division: Perform division.
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A — Addition: Perform addition.
-
S — Subtraction: Perform subtraction.
Some countries and schools use BODMAS or BIDMAS instead. These acronyms refer to the same general principle, although the wording differs.
An important detail is that multiplication does not always take priority over division, and addition does not always take priority over subtraction.
Multiplication and division share the same level of precedence. When both appear, they are evaluated from left to right.
Similarly, addition and subtraction share the same level of precedence and are evaluated from left to right.
This distinction is essential for solving expressions correctly.
Why PEMDAS Can Be Misleading
Although PEMDAS is a useful memory aid, some people misunderstand the acronym.
They assume that multiplication must always be completed before division and that addition must always be completed before subtraction.
That interpretation is incorrect.
Consider the expression:
(12 \div 3 \times 2)
Division and multiplication have equal precedence, so we work from left to right:
(12 \div 3 = 4)
Then:
(4 \times 2 = 8)
The answer is 8, not 2.
If multiplication always came before division, someone might calculate (3 \times 2 = 6) first and then obtain (12 \div 6 = 2). But that would apply a rule that standard arithmetic does not use.
The same principle applies to addition and subtraction.
For example:
(10 - 4 + 2)
Work from left to right:
(10 - 4 = 6)
Then:
(6 + 2 = 8)
The correct answer is 8.
You should not add 4 and 2 first simply because addition appears earlier in the PEMDAS acronym.
The key lesson is that multiplication and division share priority, just as addition and subtraction do.
Let's Examine the Original Puzzle More Carefully
Return to the original expression:
(8 - 2 \times 3 + 6)
There are three operations:
-
Subtraction: (8 - 2)
-
Multiplication: (2 \times 3)
-
Addition: (3 + 6), depending on how the expression is interpreted during calculation.
The multiplication has higher precedence than the other two operations. That means we cannot begin by subtracting 2 from 8.
Instead, we calculate:
(2 \times 3 = 6)
The expression becomes:
(8 - 6 + 6)
Now only subtraction and addition remain. Since these operations have equal precedence, we evaluate them from left to right:
(8 - 6 = 2)
Then:
(2 + 6 = 8)
This produces the correct answer.
Notice that the order of operations does not mean that we should always perform every addition before subtraction or every subtraction before addition. It means we must respect the hierarchy of operations and then work from left to right when the remaining operations share the same priority.
Try These Similar Math Puzzles
The best way to become comfortable with the order of operations is to practice with a few examples.