In a coded arithmetic scheme, the meanings of the standard operators are changed as follows: the symbol plus (+) stands for division (÷), the symbol division (÷) stands for subtraction (-), the symbol minus (-) stands for multiplication (x), and the symbol multiplication (x) stands for addition (+). Using this coding, what is the value of the expression 48 + 16 x 4 - 2 ÷ 8?

Introduction / Context:
This problem requires you to reinterpret an arithmetic expression in which the operator symbols are reassigned different meanings. Rather than treating the symbols in their usual sense, you must decode each one using the given mapping and then perform the calculation with standard precedence rules. Such problems help develop flexibility in understanding how symbolic rules can be changed and still interpreted consistently.

Given Data / Assumptions:

• + means division (÷). • ÷ means subtraction (-). • - means multiplication (x). • x means addition (+). • Expression: 48 + 16 x 4 - 2 ÷ 8.

Concept / Approach:
The steps are to decode each operator and then evaluate the resulting expression with normal rules. We must not mix the coded meaning with the usual meaning while decoding. Once the translation is complete, we treat the decoded expression like an ordinary arithmetic calculation, performing multiplication and division first, followed by addition and subtraction from left to right.

Step-by-Step Solution:
Step 1: Start with the expression 48 + 16 x 4 - 2 ÷ 8. Step 2: Decode each operator: + becomes ÷, x becomes +, - becomes x, and ÷ becomes -. Step 3: Replace in order: 48 + 16 becomes 48 ÷ 16. Step 4: 16 x 4 becomes 16 + 4. Step 5: The minus sign before 2 becomes a multiplication sign, so - 2 becomes x 2. Step 6: Finally, ÷ 8 becomes - 8. Step 7: Putting this together, the decoded expression is 48 ÷ 16 + 4 x 2 - 8. Step 8: Apply precedence. First compute 48 ÷ 16 = 3. Step 9: Compute 4 x 2 = 8. Step 10: Now evaluate 3 + 8 - 8 from left to right. First 3 + 8 = 11. Step 11: Then 11 - 8 = 3.

Verification / Alternative check:
If you repeat the entire translation and calculation carefully, you will again obtain 3, indicating that there is no arithmetic slip. Approximate reasoning also supports this result: the division 48 ÷ 16 gives a small number 3, the middle product gives 8, and then subtracting 8 at the end neutralizes that contribution, leaving the value from the first term, which is 3.

Why Other Options Are Wrong:
Option 6 would require the net effect of the operations to double the initial 3, which is not the case. Option 112 is far too large and could only arise from misinterpreting several operators as multiplication. Option −28 would require a large negative net effect that does not match the steps of the decoded expression. Option 10 also does not match the exact arithmetic sequence after correct decoding.

Common Pitfalls:
Learners often forget to apply operator precedence after decoding and simply calculate from left to right. Another frequent error is to only partially decode the expression or to accidentally revert to the original meaning of a symbol midway through the solution. Writing the translated expression clearly in one line before starting arithmetic is a very effective way to prevent confusion.

Final Answer:
After decoding the operators and evaluating step by step, the value of the expression is 3.