If "+" means "minus", "x" means "divided by", "÷" means "plus" and "-" means "multiplied by", then what is the value of 126 x 14 + 7 - 3 ÷ 2 when interpreted with these coded operations?

Introduction:
This question presents an expression in which the usual arithmetic symbols have been reassigned new meanings. We must carefully decode the expression 126 x 14 + 7 - 3 ÷ 2 according to the given mappings and then compute the result. It is a test of attention to detail and correct application of operator precedence after decoding.

Given Data / Assumptions:
The mappings are as follows: the symbol "+" actually means minus, the symbol "x" actually means divided by, the symbol "÷" actually means plus, and the symbol "-" actually means multiplied by. The expression to evaluate is 126 x 14 + 7 - 3 ÷ 2. We assume standard precedence rules after decoding the operators.

Concept / Approach:
The key is to first translate the coded operators into their real arithmetic counterparts, turning the expression into one with standard symbols. Then we apply ordinary precedence: division and multiplication first, followed by addition and subtraction, proceeding from left to right within the same precedence level.

Step-by-Step Solution:
Step 1: Replace each operator with its true meaning. "x" becomes "/", "+" becomes "-", "-" becomes "*", and "÷" becomes "+".Step 2: The coded expression 126 x 14 + 7 - 3 ÷ 2 becomes 126 / 14 - 7 * 3 + 2.Step 3: Apply precedence. First handle division and multiplication from left to right.Step 4: Compute 126 / 14. Since 14 * 9 = 126, we get 126 / 14 = 9.Step 5: Compute 7 * 3 = 21.Step 6: Now the expression reduces to 9 - 21 + 2.Step 7: Perform the addition and subtraction from left to right. First, 9 - 21 = -12.Step 8: Then -12 + 2 = -10.Step 9: Therefore, the value of the decoded expression is -10.

Verification / Alternative check:
We can double check by re reading the mappings and confirming each replacement: x as division, + as minus, - as multiplication, and ÷ as plus. Re applying these mappings confirms that we correctly transformed the expression to 126 / 14 - 7 * 3 + 2. Recomputing yields the same result of -10.

Why Other Options Are Wrong:
The value -12 would occur if someone stopped after 9 - 21 and forgot to add 2. The value -17 might result from miscomputing either 126 / 14 or 7 * 3. The value -41 is far off and would require a serious misinterpretation of operators or order of operations. The value -14 does not match any consistent variation of the correct decoding.

Common Pitfalls:
One common mistake is to apply the original meanings of the symbols instead of their reassigned meanings. Another is to ignore order of operations and simply calculate from left to right. Writing out the fully translated expression before calculating helps prevent these errors.

Final Answer:
The correct value of the expression is -10.