Why is there so much debate over the math question

Why is there so much debate over the math question 12÷2(2+4) =?

The debate over the math question 12÷2(2+4) arises from different interpretations of the order of operations, specifically involving multiplication and division.

According to the PEMDAS/BODMAS (Parentheses/Brackets, Orders (exponents and roots), Division and Multiplication, Addition and Subtraction) rules followed in mathematics, you’d solve the expression in the following order:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Multiplication and Division: Perform multiplication and division from left to right.
3. Addition and Subtraction: Perform addition and subtraction from left to right.

So, applying these rules:

1. Inside the parentheses: 2 + 4 = 6.
2. Now, the expression becomes 12 ÷ 2 * 6.
3. Following PEMDAS/BODMAS, multiplication and division are on the same level of precedence, so you perform them from left to right.
   • First, 12 ÷ 2 = 6.
   • Then, 6 * 6 = 36.

Thus, according to this interpretation, the answer would be 36.

However, the confusion arises because some people argue that multiplication and division should be performed from left to right as they appear in the expression. So, they would interpret the expression as (12 ÷ 2) * (2 + 4), resulting in a different answer.

To avoid ambiguity, it’s often recommended to use parentheses to clearly indicate the intended order of operations.

The confusion often stems from the interpretation of how implicit multiplication interacts with explicit multiplication and division. In mathematics, when a number is placed next to parentheses, it’s understood as multiplication. However, this implicit multiplication can sometimes lead to ambiguity, especially when combined with explicit multiplication and division.

In the expression 12÷2(2+4), the presence of both implicit and explicit multiplication can cause different interpretations:

1. Interpretation 1 (Implicit multiplication first): Here, we interpret the expression as 12÷2×(2+4)12÷2×(2+4).
   • First, we solve the parentheses: 2+4=62+4=6.
   • Then, we divide: 12÷2=612÷2=6.
   • Finally, we multiply: 6×6=366×6=36.
2. Interpretation 2 (Explicit multiplication and division performed from left to right): Here, we interpret the expression as (12÷2)×(2+4)(12÷2)×(2+4).
   • First, we divide: 12÷2=612÷2=6.
   • Then, we solve the parentheses: 2+4=62+4=6.
   • Finally, we multiply: 6×6=366×6=36.

Both interpretations follow standard mathematical rules, but they lead to different results due to the lack of clarity in the expression. To avoid such ambiguity, it’s advisable to use parentheses to explicitly indicate the intended order of operations.

When we see numbers and symbols together in math, we follow a set of rules to solve them in the correct order. These rules ensure that everyone gets the same answer. But sometimes, expressions like this one can be tricky because they could be interpreted in different ways.

In this expression, we have to figure out what to do first: divide, multiply, or add the numbers. The confusion comes from whether we should multiply first because of the numbers next to the parentheses (which means multiply) or if we should do the division first because it’s written explicitly.

Some people say we should do the division first, then the multiplication inside the parentheses, and finally the multiplication outside the parentheses. Others say we should do the multiplication first (including the one next to the parentheses), then the addition inside the parentheses, and finally the division.

So, the debate boils down to the order in which we should follow these rules. That’s why you might see different answers from different people. To avoid this confusion, it’s best to use parentheses to clearly show which operations should be done first.

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