If "#" means "subtraction", "&" means "division", "@" means "addition" and "%" means "multiplication", then what is the value of 217 & 7 # 3 @ 2 % 7 when it is evaluated using these coded operators?

Introduction:
This coded arithmetic problem assigns new meanings to four different symbols and asks us to evaluate a compound expression. The symbols #, &, @, and % are interpreted as subtraction, division, addition, and multiplication respectively. Our task is to decode the expression 217 & 7 # 3 @ 2 % 7 and compute its result, carefully respecting operator precedence.

Given Data / Assumptions:
The mappings are: # stands for subtraction, & stands for division, @ stands for addition, and % stands for multiplication. The expression is 217 & 7 # 3 @ 2 % 7. We assume that after decoding, we follow standard arithmetic precedence, performing division and multiplication before addition and subtraction, and evaluating left to right within each level.

Concept / Approach:
The first step is to translate the coded expression into standard arithmetic symbols by replacing each symbol with its true operation. After we obtain a normal looking expression, we apply the usual BODMAS or PEMDAS order of operations. Being systematic in substitution and evaluation reduces the chance of error.

Step-by-Step Solution:
Step 1: Apply the mappings: & is division, # is subtraction, @ is addition, and % is multiplication.Step 2: Rewrite the expression 217 & 7 # 3 @ 2 % 7 as 217 / 7 - 3 + 2 * 7.Step 3: According to precedence rules, handle division and multiplication first from left to right.Step 4: Compute 217 / 7. Since 7 * 31 = 217, we get 217 / 7 = 31.Step 5: Compute 2 * 7 = 14.Step 6: Substitute these values back into the expression, which becomes 31 - 3 + 14.Step 7: Now evaluate the addition and subtraction from left to right. First, 31 - 3 = 28.Step 8: Then compute 28 + 14 = 42.Step 9: Therefore, the value of the coded expression is 42.

Verification / Alternative check:
We can recheck the transformation by confirming each mapping: & as division gives 217 / 7, # as subtraction gives - 3, @ as addition gives + 2, and % as multiplication gives * 7. Recomputing 217 / 7 = 31 and 2 * 7 = 14 and then 31 - 3 + 14 indeed results in 42, confirming the correctness of the calculation.

Why Other Options Are Wrong:
The value 21 would arise if someone mistakenly divided 217 by 7 and then divided again by 3 or halved the final answer. The value 19 could result from subtracting 14 instead of adding it. The value 22 or 28 might appear if the multiplication was omitted or the addition and subtraction were done before resolving all products. None of these alternative results respect both the correct mappings and proper precedence.

Common Pitfalls:
Students may forget that multiplication and division must be completed before addition and subtraction or may accidentally replace symbols with their original meanings instead of the reassigned ones. Another error is to miscalculate 217 / 7 or 2 * 7, which can propagate into a wrong final answer even if the decoding is correct.

Final Answer:
The computed value of the expression is 42.