Question 1

In a certain code language for arithmetic symbols, the following substitutions are made:  The plus sign (+) stands for multiplication, the minus sign (-) stands for addition, the multiplication sign (x) stands for division and the division sign (÷) …

Accepted Answer

Introduction / Context: This problem again tests your understanding of coded arithmetic, where familiar symbols are reassigned different meanings. You must carefully decode the expression 24 x 6 - 8 + 2 and then evaluate it according to standard arithmetic precedence to find the correct numerical result. Given Data / Assumptions: + represents multiplication. - represents addition. x represents division. ÷ represents subtraction (though it does not appear in this expression). Coded expression: 24 x 6 - 8 + 2. Normal precedence of division and multiplication over addition and subtraction applies after decoding. Concept / Approach: We will rewrite the coded expression by substituting each symbol with its real operation and then carry out the calculations in the correct order. Because the mapping is simple and only three different operations are present, the process is straightforward if we are careful and systematic. Step-by-Step Solution: Step 1: Replace x with ÷, since x stands for division. Step 2: Replace - with +, since - stands for addition. Step 3: Replace + with ×, since + stands for multiplication. Step 4: Thus, 24 x 6 - 8 + 2 becomes 24 ÷ 6 + 8 × 2. Step 5: Evaluate division and multiplication first. Step 6: Compute 24 ÷ 6 = 4. Step 7: Compute 8 × 2 = 16. Step 8: Now perform the addition: 4 + 16 = 20. Verification / Alternative check: We can quickly recalculate: 24 ÷ 6 is clearly 4 and 8 × 2 is 16; their sum is 20. No other operation ordering is consistent with the rules, so the value 20 is stable and reliable. This matches option (d) exactly. Why Other Options Are Wrong: Values such as 25, 40 or 50 arise only if the mappings are misapplied, such as treating the symbols according to their appearance instead of the code, or performing addition before multiplication and division. Since the coded meanings are explicit, any such interpretation is incorrect and leads to the wrong options. Common Pitfalls: A frequent error is to forget that the minus sign now means addition, and the plus sign now means multiplication, which drastically changes the structure of the expression. Another pitfall is to ignore that division must be carried out before addition. Carefully performing substitutions and observing precedence avoids these mistakes. Final Answer: After decoding and simplifying correctly, the coded expression 24 x 6 - 8 + 2 evaluates to 20, so option (d) is correct.

Question 2

In a certain coded system of arithmetic operations, the symbols are used as follows:  The minus sign (-) represents addition, the plus sign (+) represents multiplication, the multiplication sign (x) represents division and the division sign (÷) repr…

Accepted Answer

Introduction / Context: This question again modifies the meanings of the familiar arithmetic symbols. The expression 2 - 12 + 10 x 5 ÷ 12 must be evaluated using these coded meanings. Your task is to decode each symbol correctly and then simplify the resulting ordinary arithmetic expression. Given Data / Assumptions: - stands for addition. + stands for multiplication. x stands for division. ÷ stands for subtraction. Coded expression: 2 - 12 + 10 x 5 ÷ 12. We apply the normal precedence rules of arithmetic after decoding. Concept / Approach: The method is to translate the whole expression into one that uses the usual symbols but with the operations indicated by the code. Once rewritten, we simplify step by step, taking care to perform division and multiplication before addition and subtraction. The final result must match one of the answer options exactly. Step-by-Step Solution: Step 1: Replace - with +, because - now means addition. Step 2: Replace + with ×, because + now means multiplication. Step 3: Replace x with ÷, because x means division. Step 4: Replace ÷ with -, because ÷ means subtraction. Step 5: The coded expression 2 - 12 + 10 x 5 ÷ 12 becomes 2 + 12 × 10 ÷ 5 - 12. Step 6: Evaluate multiplication and division first. Step 7: Compute 12 × 10 = 120. Step 8: Then 120 ÷ 5 = 24. Step 9: Substitute back to get 2 + 24 - 12. Step 10: Now perform the additions and subtractions: 2 + 24 = 26 and 26 - 12 = 14. Verification / Alternative check: Rechecking, 12 × 10 is clearly 120, and 120 ÷ 5 is 24. Adding 2 gives 26, subtracting 12 gives 14. This is unique among the options, so our decoding and evaluation are correct. Why Other Options Are Wrong: The values 32, 8 and 4 would arise if we misinterpreted one or more symbols, for instance treating - as subtraction or + as addition. Another source of error is ignoring the precedence of multiplication and division, which changes intermediate results. Since the code and precedence are explicit, such alternatives are not acceptable. Common Pitfalls: Many students rush through the substitution step and end up mixing coded and normal meanings. Others treat all operations with the same priority and work strictly left to right. Writing down the fully decoded expression and then simplifying carefully step by step is the safest strategy for coded-operator problems. Final Answer: After decoding the symbols and simplifying correctly, the expression evaluates to 14, so option (d) is the correct answer.

Mathematical Operations Questions