In a coded arithmetic system, the symbols are redefined as follows: x means addition, the minus sign means division, the division sign means subtraction and the plus sign means multiplication. Under these rules, which one of the following coded equations is actually arithmetically correct?

Introduction / Context:
This coding and decoding question involves operator substitution. The usual arithmetic symbols are redefined so that each symbol now represents a different operation. To identify the correct equation, you must translate each coded equation into a standard arithmetic expression using the new meanings, then check which one yields a true statement.

Given Data / Assumptions:

• x means addition.
• The minus sign (−) means division.
• The division sign (÷) means subtraction.
• The plus sign (+) means multiplication.
• Four or more coded equations are given, and exactly one of them should evaluate correctly when decoded.
• Normal precedence rules apply after decoding: multiplication and division before addition and subtraction.

Concept / Approach:
The key is to systematically decode each option. Every time you see x, you replace it with plus, every minus with division, every division sign with subtraction and every plus with multiplication. After decoding, you simplify each expression step by step. Only the decoded equation that matches the stated result on the right hand side is considered correct.

Step-by-Step Solution:
Step 1: Decode option (c) because it contains all the symbols and often reveals the pattern well: 16 + 5 - 10 x 4 ÷ 3 = 9.Step 2: Apply the replacements. The plus sign (+) means multiplication, so 16 + 5 becomes 16 * 5. The minus sign (−) means division, so the symbol after 5 becomes division by 10. The x sign means addition, so 10 x 4 becomes 10 + 4. The division sign (÷) means subtraction, so 4 ÷ 3 becomes 4 − 3.Step 3: Combining these, the decoded expression is 16 * 5 / 10 + 4 − 3.Step 4: Now evaluate with the standard order of operations. First compute 16 * 5 = 80.Step 5: Then divide by 10 to get 80 / 10 = 8.Step 6: Now handle addition and subtraction from left to right: 8 + 4 = 12 and 12 − 3 = 9.Step 7: The left hand side evaluates to 9, which exactly matches the right hand side of option (c), so option (c) is a true equation under the given coding.

Verification / Alternative check:
To ensure that option (c) is uniquely correct, you can similarly decode options (a), (b), (d) and (e). When decoded, each of those yields a left hand side value that does not match the number given on the right hand side, or produces a noninteger result that clearly differs from the stated value. Therefore, option (c) is the only coded equation that becomes a correct arithmetic statement once the operators are translated.

Why Other Options Are Wrong:
In option (a), after decoding, the left hand side evaluates to a noninteger value around 7.67, not 19. In option (b), the decoded operations lead to a much larger number than 9. In option (d), the decoded expression gives a value close to 1.2, not 12. In option (e), the substitutions again produce a value that does not match the stated result 11. Only for option (c) does the decoding yield exactly 9.

Common Pitfalls:
Students often forget to apply all replacements correctly or they unintentionally treat some symbols as if they still had their usual meanings. Another frequent mistake is ignoring the precedence rules and performing operations strictly from left to right. Carefully decoding each symbol and then applying the correct order of operations is essential to avoid these errors.

Final Answer:
The only coded equation that becomes a true arithmetic statement is 16 + 5 - 10 x 4 ÷ 3 = 9.