In a coded arithmetic system, the operators are redefined as follows: "+" means "×", "-" means "÷", "x" means "-" and "÷" means "+". Under these meanings, what is the value of the expression 16 ÷ 4 x 10 - 5 + 8 ?

Introduction / Context:
This coded arithmetic question reassigns new meanings to the familiar operators plus, minus, multiplication and division. The expression 16 ÷ 4 x 10 - 5 + 8 must be evaluated after replacing each symbol by the operation it now represents. The question checks whether you can correctly interpret each symbol and then apply conventional precedence rules to compute the final value.

Given Data / Assumptions:

• "+" means "×" (multiplication).
• "-" means "÷" (division).
• "x" means "-" (subtraction).
• "÷" means "+" (addition).
• Expression to interpret: 16 ÷ 4 x 10 - 5 + 8.
• Standard arithmetic precedence applies after decoding: division and multiplication are evaluated before addition and subtraction, working left to right within each level.

Concept / Approach:
The main idea is to translate the coded expression into a normal one where each symbol has its usual meaning. This is done by writing a new expression in terms of actual plus, minus, multiplication and division based on the given mapping. Once you have the decoded expression, ignore the original symbols and evaluate the new expression using standard precedence. It is essential not to mix the old and new meanings during calculations.

Step-by-Step Solution:
Step 1: Start from the coded expression: 16 ÷ 4 x 10 - 5 + 8. Step 2: Replace each symbol by the operation it represents. "÷" means "+", so 16 ÷ 4 becomes 16 + 4. "x" means "-", so 4 x 10 becomes 4 - 10. "-" means "÷", so 10 - 5 becomes 10 ÷ 5. "+" means "×", so + 8 becomes × 8. Step 3: Writing in standard form, the entire decoded expression is 16 + 4 - 10 ÷ 5 × 8. Step 4: Apply the usual precedence, handling division and multiplication first. 10 ÷ 5 = 2. 2 × 8 = 16. Step 5: Substitute this back: 16 + 4 - 16. Step 6: Evaluate from left to right: 16 + 4 = 20, then 20 - 16 = 4.

Verification / Alternative check:
To confirm, you can redo the translation more slowly to be sure each operator has been replaced correctly and that no code meaning has been confused. Checking the critical part 10 ÷ 5 × 8 after decoding shows that it equals 16, which balances out the 16 + 4 at the start to give 4 overall. Any misinterpretation, such as swapping the meanings of plus and minus, will result in a different answer that does not match the consistent rule described in the question statement.

Why Other Options Are Wrong:
Options 12, 8, 2 and 16 correspond to typical mistakes like ignoring operator precedence, computing from left to right, or misassigning one of the code meanings. For example, if you treat x as multiplication instead of subtraction, you will obtain a larger number. Because the decoded and correctly evaluated expression yields exactly 4, none of the other options can be considered correct.

Common Pitfalls:
Students sometimes attempt to compute directly on the original expression while mentally remembering the new meanings, which increases the chance of mixing semantics mid calculation. Another pitfall is not rewriting the full decoded expression before starting to compute, leading to an error in one of the replacements. The safest strategy is always to translate the entire expression into standard operators first, then apply normal arithmetic rules step by step.

Final Answer:
Under the given operator definitions, the value of 16 ÷ 4 x 10 - 5 + 8 is 4.