In a certain code language, arithmetic symbols are used with different meanings as follows: "x" represents "+", "÷" represents "x", "-" represents "÷" and "+" represents "-". Using this code, find the value of the expression 5 ÷ 4 - 10 + 7 x 16.

Introduction / Context:
This is a classic operator coding problem. Each familiar arithmetic symbol has been reassigned a new meaning. We are given that "x" stands for addition, "÷" stands for multiplication, "-" stands for division and "+" stands for subtraction. The expression 5 ÷ 4 - 10 + 7 x 16 must therefore be translated into an ordinary arithmetic expression under these new meanings and then evaluated. The question checks precise translation and careful use of precedence.

Given Data / Assumptions:

• "x" represents addition (+).
• "÷" represents multiplication (×).
• "-" represents division (÷).
• "+" represents subtraction (-).
• Coded expression: 5 ÷ 4 - 10 + 7 x 16.
• Standard arithmetic precedence (× and ÷ before + and -) is used after substitution.

Concept / Approach:
The solution involves two main stages. First, replace each operator in the coded expression with the true operation it represents. This produces a standard expression involving multiplication, division, addition and subtraction in their usual sense. Second, evaluate this expression using standard order of operations, taking care not to mix up the original symbols with their coded meanings. Writing out the fully translated expression clearly before doing any calculations greatly reduces the risk of error.

Step-by-Step Solution:
Step 1: Start from the coded expression: 5 ÷ 4 - 10 + 7 x 16. Step 2: Replace each operator using the given meanings. "÷" represents multiplication, so 5 ÷ 4 becomes 5 × 4. "-" represents division, so - 10 becomes ÷ 10. "+" represents subtraction, so + 7 becomes - 7. "x" represents addition, so 7 x 16 becomes 7 + 16. Step 3: The fully translated expression is 5 × 4 ÷ 10 - 7 + 16. Step 4: Apply precedence. First handle multiplication and division from left to right. Compute 5 × 4 = 20. Step 5: Then compute 20 ÷ 10 = 2. Step 6: Substitute back: the expression becomes 2 - 7 + 16. Step 7: Now evaluate from left to right: 2 - 7 = -5. Step 8: Next, -5 + 16 = 11. Step 9: Therefore, the value of the coded expression is 11.

Verification / Alternative check:
To verify the result, you can insert brackets to emphasise precedence: (5 × 4 ÷ 10) - 7 + 16. Recalculating 5 × 4 ÷ 10 as 20 ÷ 10 = 2 and then performing the remaining operations leads again to 11. Because the mapping between coded and real operators is used consistently, and the arithmetic operations are straightforward, 11 is a secure and unambiguous result.

Why Other Options Are Wrong:
Option 50 might come from evaluating the original expression without applying the coding rules and treating ÷ and x in their usual sense. Option 5 or 9 can appear if someone misapplies operator precedence or mistranslates one of the symbols. Option 15 could arise from calculating 5 × 4 ÷ 10 as 5 instead of 2 and then doing the additions and subtractions incorrectly. None of these values match the correctly translated and evaluated expression, which yields 11.

Common Pitfalls:
A common mistake is blending the original and coded meanings, for example interpreting "x" as multiplication out of habit. Another frequent error is to skip the full translation step and attempt to compute directly on the mixed symbol expression, which almost guarantees confusion. To avoid these issues, always write a clear mapping list, perform substitution carefully and only then apply ordinary arithmetic rules with proper precedence.

Final Answer:
After substituting the coded meanings and evaluating correctly, the expression 5 ÷ 4 - 10 + 7 x 16 equals 11.