In a certain coded arithmetic system, the usual operators are replaced by new meanings as follows: the symbol + stands for multiplication, the symbol - stands for addition, and the symbol x stands for division. Under this coding scheme, what is the numerical value of the expression 128 + 9 - 16 x 4?

Introduction / Context:
This question belongs to the category of coded arithmetic operations, where familiar mathematical symbols are reassigned different meanings. The challenge is not simply to evaluate the expression mechanically, but to correctly interpret each symbol according to the new code and then apply normal rules of arithmetic, including the usual precedence of multiplication and division before addition and subtraction. This tests both attention to detail and conceptual understanding of operator precedence.

Given Data / Assumptions:

• The code specifies: + means multiplication. • The code specifies: - means addition. • The code specifies: x means division. • The written expression is 128 + 9 - 16 x 4. • After replacing operators by their coded meanings, we then use standard arithmetic rules.

Concept / Approach:
The method has two main stages. First, translate the coded expression into its real mathematical form by replacing each symbol with its coded meaning. Second, evaluate the resulting expression using the usual operator precedence rules: multiplication and division first (from left to right), followed by addition and subtraction. Carefully following both stages ensures that the final value is correct.

Step-by-Step Solution:
Step 1: Rewrite each operator with its real meaning based on the code. Step 2: In the expression 128 + 9 - 16 x 4, replace + with multiplication to get 128 * 9. Step 3: Replace - with addition to get + 16. Step 4: Replace x with division to get ÷ 4. Step 5: The coded expression now becomes 128 * 9 + 16 ÷ 4. Step 6: Apply operator precedence. First compute 128 * 9 = 1152. Step 7: Compute 16 ÷ 4 = 4. Step 8: Now perform the remaining addition: 1152 + 4 = 1156.

Verification / Alternative check:
A quick verification is to check whether any alternate interpretation could give a different result. However, once the mapping of each symbol is fixed, there is only one consistent way to write the real expression. Evaluating it again with the same precedence rules will always give 1156. This confirms that the answer is stable and not dependent on any arbitrary choice of grouping.

Why Other Options Are Wrong:
Option 73 is much smaller and would correspond to treating some operations incorrectly or ignoring precedence. Option 256 might arise if someone performs 128 * 2, misreading part of the expression and not interpreting the code properly. Option 1352 could come from incorrect multiplication such as 128 * 10 or a mistaken additional step. Option 1024 resembles 32 * 32 or 2^10 and is unrelated to the coded calculation here.

Common Pitfalls:
Common errors include assuming that after decoding the symbols, all operations are done strictly from left to right, ignoring precedence. Another frequent mistake is to replace the numbers instead of only the operators, or to forget which symbol stands for which operation. Writing down the real expression clearly after decoding is a very helpful way to avoid such confusion.

Final Answer:
After decoding the operators and evaluating the expression with correct precedence, the value of the expression is 1156.