In a certain code language, the symbols are redefined as follows: "+" represents multiplication, "-" represents addition, "x" represents division and "÷" represents subtraction. Using these coded operators, what is the value of 9 + 3 - 72 x 6 ÷ 3?

Introduction:
Here we again encounter an expression where familiar arithmetic symbols have been assigned new meanings. The challenge is to decode each symbol according to the new definitions and then evaluate the resulting numerical expression correctly. This tests attention to detail and understanding of operation precedence.

Given Data / Assumptions:
The special meanings are:
1) "+" represents multiplication.
2) "-" represents addition.
3) "x" represents division.
4) "÷" represents subtraction.
The coded expression to evaluate is 9 + 3 - 72 x 6 ÷ 3. After decoding, we apply the usual arithmetic precedence rules: division and multiplication before addition and subtraction.

Concept / Approach:
The safe strategy is to ignore the usual meanings of the symbols and wholly rely on the given code. We rewrite the entire expression with the true operations first, and only then proceed to calculate. This avoids mixing interpretations and reduces logical mistakes, especially in more complex expressions that contain several different coded operators.

Step-by-Step Solution:
Step 1: Replace "+" with its coded meaning multiplication. Thus 9 + 3 becomes 9 * 3.Step 2: Replace "-" with addition. Therefore "- 72" becomes + 72.Step 3: Replace "x" with division. So "72 x 6" becomes 72 / 6.Step 4: Replace "÷" with subtraction. So "6 ÷ 3" means 6 - 3, but note it is part of the coded structure in the original expression as 72 x 6 ÷ 3. After full decoding, the expression becomes 9 * 3 + 72 / 6 - 3.Step 5: Apply operation precedence. First compute 9 * 3 = 27.Step 6: Compute 72 / 6 = 12.Step 7: Substitute these results into the expression, giving 27 + 12 - 3.Step 8: Evaluate addition and subtraction from left to right: 27 + 12 = 39, then 39 - 3 = 36.Step 9: Hence, the value of the coded expression is 36.

Verification / Alternative check:
We can rephrase the decoded expression verbally: "nine multiplied by three plus seventy two divided by six minus three." Clearly this is 27 plus 12 minus 3, which equals 36. There are no additional brackets or special rules that modify this order, so the computation is stable.

Why Other Options Are Wrong:
The value 46 or 30 could appear if someone miscalculates 72 / 6 or fails to perform the final subtraction. The value 21 or 9 might come from treating one or more operators with their original meanings or from evaluating strictly left to right. Such interpretations contradict the given symbol definitions and correct precedence rules.

Common Pitfalls:
Learners may forget to translate all operators before starting the computation, especially when many different symbols are present together. Others ignore precedence and compute 9 * 3 + 72 then divide by 6 and subtract 3, which is not intended. Carefully translating and then working step by step prevents these common mistakes.

Final Answer:
According to the coded definitions of the operators, the expression evaluates to 36.