In a coded arithmetic system, the meanings of the operation symbols are changed as follows: the symbol x represents addition (+), the symbol + represents division (÷), the symbol minus (-) represents multiplication (x), and the symbol division (÷) represents subtraction (-). Using this coding, what is the value of the expression 8 x 7 - 8 + 40 ÷ 2?

Introduction / Context:
This coded arithmetic question replaces standard operation symbols with different meanings. Instead of interpreting x, +, -, and ÷ in their usual roles, we must translate each symbol into the corresponding actual operation, and then evaluate the expression. Questions of this type test your ability to carefully track symbolic meanings and to respect standard precedence of operations in the decoded expression.

Given Data / Assumptions:

• x means + (addition). • + means ÷ (division). • - means x (multiplication). • ÷ means - (subtraction). • Expression given: 8 x 7 - 8 + 40 ÷ 2.

Concept / Approach:
The method is to first decode each coded operator into its real meaning, thereby rewriting the expression in standard arithmetic notation. After that, we calculate the value by applying the usual precedence rules: perform multiplication and division before addition and subtraction, moving from left to right among operations of the same precedence. It is crucial not to confuse the symbolic appearance of the operators with their coded definitions during the evaluation.

Step-by-Step Solution:
Step 1: Start from 8 x 7 - 8 + 40 ÷ 2. Step 2: Decode x as +, so 8 x 7 becomes 8 + 7. Step 3: Decode - as multiplication x, so - 8 becomes x 8. Step 4: Decode + as division ÷, so + 40 becomes ÷ 40. Step 5: Decode ÷ as subtraction -, so ÷ 2 becomes - 2. Step 6: Putting it together, the decoded expression is 8 + 7 x 8 ÷ 40 - 2. Step 7: Apply precedence. First handle multiplication and division from left to right. Step 8: Compute 7 x 8 = 56. Step 9: Then compute 56 ÷ 40 = 56 / 40 = 7 / 5 after simplifying. Step 10: Now the expression becomes 8 + 7/5 - 2. Step 11: Convert 8 and 2 into fractions with denominator 5 to add easily: 8 = 40/5 and 2 = 10/5. Step 12: Compute 40/5 + 7/5 = 47/5. Step 13: Subtract 10/5: 47/5 − 10/5 = 37/5.

Verification / Alternative check:
Merge the integer operations as decimals as a cross check: 7 x 8 ÷ 40 = 1.4, so the expression becomes 8 + 1.4 − 2. Then 8 + 1.4 = 9.4, and 9.4 − 2 = 7.4. Converting 7.4 to a fraction gives 7.4 = 74 / 10 = 37 / 5, which matches our earlier result. This confirms that the arithmetic is correct.

Why Other Options Are Wrong:
Option 1 is much smaller than the true value and would come from ignoring the contribution of the fraction 7/5. Option 83/5 is larger and does not follow from the correct sequence of operations. Option 44 corresponds to a simple integer calculation rather than a mixed fraction result and shows a misinterpretation of division. Option 9 is close numerically to 8 plus a small positive correction, but the final subtraction of 2 makes the correct value significantly smaller than 9.

Common Pitfalls:
Many learners forget to apply the precedence correctly after decoding and instead perform addition and subtraction first. Others misread one of the operator mappings, for example thinking that + still means addition instead of division in this system. Carefully rewriting the entire expression after decoding and only then doing the arithmetic helps avoid such errors.

Final Answer:
Under the given coded operator scheme, the value of 8 x 7 - 8 + 40 ÷ 2 is 37/5.