In a certain coded arithmetic system, the usual meanings of the symbols are changed as follows: "+" represents multiplication (x), "-" represents addition (+), "x" represents division (÷) and "÷" represents subtraction (-). Using this code, what is the value of 56 ÷ 8 + 12 - 72 ?

Introduction / Context:
This question belongs to the coding–decoding type for arithmetic operators. The symbols +, -, x and ÷ no longer have their usual meanings. Instead, each symbol stands for a different basic operation. The task is to apply these new meanings correctly to the given expression 56 ÷ 8 + 12 - 72 and then evaluate it using normal arithmetic rules. This checks careful reading of symbol definitions and disciplined application of operator precedence.

Given Data / Assumptions:

• "+" represents multiplication.
• "-" represents addition.
• "x" represents division.
• "÷" represents subtraction.
• Expression to evaluate: 56 ÷ 8 + 12 - 72.
• After substituting the true operations, normal precedence (multiplication and division before addition and subtraction) is used.

Concept / Approach:
The key idea is that we must first translate the coded expression into an ordinary arithmetic expression by replacing each operator with its actual meaning. Only after this substitution do we use the usual order of operations. Directly performing calculations on the original symbols as if they still meant their usual operations will give a completely wrong answer. A systematic two step process (substitute, then evaluate) avoids confusion and typical exam mistakes.

Step-by-Step Solution:
Step 1: Start from the coded expression: 56 ÷ 8 + 12 - 72. Step 2: Replace each symbol using the code. "÷" represents subtraction, so 56 ÷ 8 becomes 56 - 8. "+" represents multiplication, so + 12 becomes × 12. "-" represents addition, so - 72 becomes + 72. Step 3: The correctly translated expression is 56 - 8 * 12 + 72. Step 4: Apply precedence. First calculate the multiplication: 8 * 12 = 96. Step 5: Substitute back: the expression becomes 56 - 96 + 72. Step 6: Now evaluate from left to right for addition and subtraction. First 56 - 96 = -40. Step 7: Then -40 + 72 = 32. Step 8: Therefore, the coded expression has the value 32.

Verification / Alternative check:
You can quickly recheck by writing the translated expression with brackets: 56 - (8 * 12) + 72. This shows clearly that only 8 and 12 are multiplied. Multiplying first gives 96, and then 56 - 96 + 72 leads again to 32. There is no alternative consistent interpretation of the given code that would change this result, so 32 is stable and reliable as the final answer.

Why Other Options Are Wrong:
Option 88 would come from treating the original + as addition and the ÷ as real division, which ignores the coding. Option 44 can appear if someone does part of the translation correctly but mishandles the order of operations. Option 82 and option 40 similarly reflect partial or incorrect substitutions or left to right evaluation without respecting precedence. None of these values match the carefully translated and correctly evaluated expression.

Common Pitfalls:
Typical mistakes include forgetting that + now means multiplication, or assuming that ÷ still means division. Another common error is to perform all operations from left to right even after the correct substitution, which breaks the precedence rule. To avoid such issues, always rewrite the coded expression clearly and then underline or bracket the multiplication and division operations before starting numeric calculations.

Final Answer:
After applying the coded meanings of the symbols and evaluating correctly, the value of the expression is 32.