If "-" means "plus", "x" means "divide", "÷" means "multiply" and "+" means "minus", then how should the expression 26 + 400 x 20 - 21 ÷ 12 be evaluated and what is its final value?

Introduction:
This problem provides an arithmetic expression in which each arithmetic symbol has been assigned a new meaning. The goal is to decode the expression 26 + 400 x 20 - 21 ÷ 12 according to the specified symbol mappings and then compute its value. This tests understanding of operator substitution and the correct application of precedence once decoding is done.

Given Data / Assumptions:
The mappings are: "-" means plus, "x" means divide, "÷" means multiply, and "+" means minus. The expression is 26 + 400 x 20 - 21 ÷ 12. We assume normal precedence rules after decoding, with division and multiplication evaluated before addition and subtraction. We also treat the minus symbol in the expression as the one whose meaning has been reassigned to plus.

Concept / Approach:
The correct approach is to rewrite the entire expression using the true operations that each symbol stands for. Once the expression has standard operators, we evaluate it using the standard order of operations. Care must be taken not to confuse the original meanings of the symbols with their reassigned roles in this question.

Step-by-Step Solution:
Step 1: Replace each symbol with its true meaning based on the mappings.Step 2: The symbol "+" stands for minus, so 26 + 400 becomes 26 - 400.Step 3: The symbol "x" stands for divide, so 400 x 20 becomes 400 / 20.Step 4: The symbol "-" stands for plus, so the segment - 21 becomes + 21.Step 5: The symbol "÷" stands for multiply, so 21 ÷ 12 becomes 21 * 12.Step 6: Putting it all together, the expression becomes 26 - 400 / 20 + 21 * 12.Step 7: Apply precedence rules: first evaluate division and multiplication from left to right.Step 8: Compute 400 / 20 = 20.Step 9: Compute 21 * 12 = 252.Step 10: The expression now simplifies to 26 - 20 + 252.Step 11: Perform addition and subtraction from left to right. First, 26 - 20 = 6.Step 12: Then 6 + 252 = 258.Step 13: Therefore, the value of the decoded expression is 258.

Verification / Alternative check:
We can verify by carefully reconfirming each mapping and the transformed expression. Using "-" as plus and "+" as minus, and substituting "x" and "÷" with divide and multiply respectively, we arrive at the same decoded form. Recomputing the numerical operations confirms the final result of 258.

Why Other Options Are Wrong:
The value 219 would require miscomputing one of the products or sums. The value 216 might be obtained by erroneously performing 26 - 400 first and then combining without respecting precedence. The value 230 or 240 could result from mixing up plus and minus or handling the division incorrectly. None of these alternate values emerge from a consistent application of the correct mappings and precedence rules.

Common Pitfalls:
Students may mistakenly apply the original meanings of the symbols, turning the expression into a completely different calculation. Another common error is performing arithmetic strictly left to right without handling multiplication and division first. Writing the fully decoded expression and then following standard rules step by step avoids such mistakes.

Final Answer:
The correctly evaluated value of the expression is 258.