Question 1

Which of the following interchanges of signs and numbers will make the equation 8 × 9 + 2 = 74 correct, when standard arithmetic rules are applied?

Accepted Answer

Introduction / Context: This question tests your ability to manipulate an equation by interchanging both operation signs and numbers so that the resulting statement becomes arithmetically correct. You are given several possible pairs of items to swap, and you must check which particular interchange yields a true equation under normal arithmetic rules. It combines algebraic thinking with logical trial and error. Given Data / Assumptions: - The original equation is: 8 × 9 + 2 = 74. - Each option suggests swapping two signs and also swapping two numbers. - Only one option should make the resulting equation numerically correct. - After the swap, we use the usual operations and standard precedence (multiplication before addition). Concept / Approach: For each option, we conceptually perform two swaps: - First, swap the specified numbers everywhere they appear. - Second, swap the specified signs everywhere they appear. After performing both swaps, we evaluate the resulting equation. If the left-hand side equals the right-hand side under standard arithmetic, that option is valid; otherwise, it is rejected. We repeat this until we find the single correct option. Step-by-Step Solution: Original equation: 8 × 9 + 2 = 74. Check option B: swap "× and +" and swap "8 and 2". Step 1: Swap the numbers 8 and 2. The equation becomes: 2 × 9 + 8 = 74. Step 2: Swap the signs × and +. Every "×" becomes "+", and every "+" becomes "×". The equation becomes: 2 + 9 × 8 = 74. Step 3: Evaluate using precedence. First compute 9 × 8 = 72. Then compute 2 + 72 = 74. The equation 2 + 9 × 8 = 74 is true, so option B works. For completeness, other options either violate the equation or produce noninteger equality. For example, option A leads to 8 = 2 + 9 × 74, which is clearly false. Verification / Alternative check: We can quickly cross check option B by redoing the swaps more compactly: writing the final equation directly as 2 + 9 × 8 = 74 and verifying that the arithmetic is correct. Since 9 × 8 is 72 and adding 2 gives exactly 74, this confirms the correctness without ambiguity. Why Other Options Are Wrong: Option A results in an equation of the form 8 = 2 + 9 × 74, where the right-hand side is far larger than 8. Option C gives 9 + 8 × 2 = 74, which evaluates to 9 + 16 = 25, not 74. Option D reorganizes the right-hand side but does not satisfy equality either. Hence none of these alternatives balance the equation, leaving option B as the only correct choice. Common Pitfalls: A common mistake is to perform only one of the swaps, or to swap numbers correctly but forget to change all instances of the signs. Another frequent slip is to ignore operator precedence after the swap and evaluate from left to right, leading to an incorrect check of the resulting equation. Careful, stepwise transformation followed by correct evaluation avoids these errors. Final Answer: The equation becomes correct if we interchange "× and +" and swap the numbers 8 and 2. Therefore, the correct choice is × and +, 8 and 2.

Question 2

In a certain code language, "×" represents "+", "÷" represents "×", "−" represents "÷" and "+" represents "−". What is the value of 5 + 10 × 21 ÷ 5 − 15 under this coding?

Accepted Answer

Introduction / Context: This is another coded operations question where the familiar operators do not carry their usual meanings. To solve it, we must first translate each symbol into its true operation in this code language, then simplify the resulting expression using standard arithmetic precedence. These questions sharpen precision in symbol handling and logical reasoning. Given Data / Assumptions: - "×" actually means addition. - "÷" actually means multiplication. - "−" actually means division. - "+" actually means subtraction. - The coded expression is: 5 + 10 × 21 ÷ 5 − 15. - After decoding, we apply normal BODMAS or PEMDAS rules. Concept / Approach: We must decode the entire expression before doing any arithmetic. Each symbol is replaced according to the mapping, but the numbers remain in the same order. Once the expression is rewritten in terms of real operations, we evaluate it step by step, starting with multiplication and division, followed by addition and subtraction from left to right. Step-by-Step Solution: Step 1: Decode each operator. "+" represents "−", so 5 + 10 becomes 5 − 10. "×" represents "+", so 10 × 21 becomes 10 + 21. "÷" represents "×", so 21 ÷ 5 becomes 21 × 5. "−" represents "÷", so − 15 becomes ÷ 15. Putting these together, the entire expression becomes: 5 − 10 + 21 × 5 ÷ 15. Step 2: Evaluate multiplication and division. First compute 21 × 5 = 105. Then compute 105 ÷ 15 = 7. The expression now reduces to: 5 − 10 + 7. Step 3: Perform addition and subtraction from left to right. 5 − 10 = −5. Then −5 + 7 = 2. Verification / Alternative check: We can verify quickly by reconfirming our decoding. The final expression 5 − 10 + 7 is simple, and a mental recomputation shows that it indeed equals 2. Any alternate decoding, such as reading "+" as addition or "−" as subtraction, would give a completely different and inconsistent result. Since the calculated value matches one of the answer options exactly, the solution is consistent. Why Other Options Are Wrong: The answers 49, 20, and 34 would require incorrect handling of one or more of the coded symbols or ignoring operator precedence. For instance, if someone treated "×" as multiplication and "÷" as division in the usual sense, the intermediate products would be much larger, leading to an answer far from 2. Because our decoding strictly follows the code language specified in the question, only 2 emerges as the correct value. Common Pitfalls: Common mistakes include decoding only part of the expression, mixing coded and real operations halfway through, or performing operations from left to right without giving priority to multiplication and division. Writing down the decoded expression clearly and then simplifying it is the safest strategy. Final Answer: Under the given code language, the expression 5 + 10 × 21 ÷ 5 − 15 evaluates to 2.

Question 3

The equation 10 × 4 + 5 − 30 ÷ 6 = 31 is incorrect. Which two signs should be interchanged so that the equation becomes correct?

Accepted Answer

Introduction / Context: In this problem, you are given an equation that is not correct under normal arithmetic rules. You must identify which two operator signs need to be interchanged in order to make the equation true. This tests understanding of operator precedence, the ability to manipulate symbolic expressions, and systematic trial of alternatives. Given Data / Assumptions: - Original equation: 10 × 4 + 5 − 30 ÷ 6 = 31. - Signs that may be interchanged: ×, +, −, ÷. - Each option specifies a pair of signs to swap throughout the equation. - After swapping, we apply standard arithmetic precedence to verify the equality. Concept / Approach: First compute the value of the original left-hand side to confirm that the equation is indeed incorrect. Then, for each option, mentally or on paper, interchange the pair of operator symbols and evaluate the new left-hand side. The correct option is the one for which the left-hand side equals 31. While it is possible to reason cleverly, a careful trial of the few given options is usually quickest and safe. Step-by-Step Solution: Original left-hand side: 10 × 4 + 5 − 30 ÷ 6. Compute with normal precedence: 10 × 4 = 40 and 30 ÷ 6 = 5. So LHS = 40 + 5 − 5 = 40, not 31. The equation is incorrect. Now test option A: interchange "× and −". Replace every "×" with "−" and every "−" with "×". The equation becomes: 10 − 4 + 5 × 30 ÷ 6 = 31. Compute step by step: First multiplication and division: 5 × 30 = 150, then 150 ÷ 6 = 25. Now the expression is: 10 − 4 + 25. 10 − 4 = 6, and 6 + 25 = 31. Thus the equation 10 − 4 + 25 = 31 is correct. Therefore, interchanging "× and −" yields a true equation. Verification / Alternative check: We can quickly check other options to ensure uniqueness. Interchanging "÷ and −" or "+ and ÷" produces left-hand sides that clearly do not simplify to 31 when evaluated with correct precedence. Interchanging "− and +" keeps the total effectively unchanged and still yields 40. Hence, only option A results in a correct equation, confirming our answer. Why Other Options Are Wrong: - Option B: swapping "÷ and −" leads to a peculiar arrangement where the division is moved, and the resulting value does not approach 31. - Option C: swapping "+ and ÷" distorts the balance between multiplication, division, and addition, again leading to a different total. - Option D: swapping "− and +" still produces an incorrect left-hand side that evaluates to 40, not 31. Common Pitfalls: A common error is forgetting operator precedence after the interchange and evaluating from left to right. Another mistake is misapplying the swaps, changing only one symbol or missing an occurrence. Writing the transformed equation clearly and then calculate in stages helps avoid these errors. Final Answer: The equation becomes correct when the signs "× and −" are interchanged. Therefore, the correct choice is × and −.

Question 4

In a special number pattern, 15 × 26 is written as 6512 and 29 × 36 is written as 6923. Using the same system, what should 46 × 54 be written as?

Accepted Answer

Introduction / Context: This reasoning question involves a digit rearrangement pattern rather than ordinary multiplication. Two examples show how two two digit numbers are combined into a four digit result. Your task is to decode the rule by observing how the digits are being rearranged, then apply it to a new pair of numbers. Such questions assess observational skills and pattern recognition using place value. Given Data / Assumptions: - 15 × 26 is represented as 6512. - 29 × 36 is represented as 6923. - We are asked to find the pattern for 46 × 54. - The symbol × here indicates a relationship, not ordinary multiplication. - All numbers involved are two digit numbers, and the outputs are four digit numbers. Concept / Approach: Instead of performing direct multiplication, we look at the digits individually. Every example combines the four digits of the two given numbers into a new four digit number. The key is to understand the ordering of the digits based on their positions and sizes. In particular, we consider the smaller and larger of the two numbers and track their unit and tens digits separately, then see how these appear in the result. Step-by-Step Solution: Step 1: Analyse 15 × 26 = 6512. Numbers: 15 and 26. Smaller number: 15 (tens 1, units 5). Larger number: 26 (tens 2, units 6). Result 6512 can be split as digits 6, 5, 1, 2. Observe the order: unit digit of larger number (6), unit digit of smaller number (5), tens digit of smaller number (1), tens digit of larger number (2). Step 2: Check this rule for 29 × 36 = 6923. Numbers: 29 and 36. Smaller number: 29 (tens 2, units 9). Larger number: 36 (tens 3, units 6). Apply the same order: unit of larger (6), unit of smaller (9), tens of smaller (2), tens of larger (3). This yields 6, 9, 2, 3 which matches 6923 given in the example. Step 3: Apply the rule to 46 × 54. Numbers: 46 and 54. Smaller number: 46 (tens 4, units 6). Larger number: 54 (tens 5, units 4). Using the pattern: unit of larger (4), unit of smaller (6), tens of smaller (4), tens of larger (5). Thus the required result is 4, 6, 4, 5 or 4645. Verification / Alternative check: We can quickly validate by checking whether any other consistent order of digits can explain both examples. Many alternative arrangements (for example ordering by tens digits first) will fail for at least one of the given mappings. The discovered sequence unit larger, unit smaller, tens smaller, tens larger works perfectly in both cases and matches one of the answer options for 46 and 54, which confirms its correctness. Why Other Options Are Wrong: - 5464, 4564, and 4465 all use the correct set of digits {4,4,5,6}, but the order does not match the pattern derived from the examples. - Attempting to use those orders for 15 and 26 or for 29 and 36 would not reproduce 6512 or 6923. Hence, they cannot represent the intended rule and must be rejected. Common Pitfalls: Learners sometimes try to involve actual multiplication, addition, or subtraction of the numbers instead of looking at how digits are rearranged. Others may spot only part of the pattern, for example matching the first two digits but not the last two. Systematically listing the tens and unit digits of the smaller and larger numbers and checking their order in the results is a reliable strategy. Final Answer: According to the discovered pattern, 46 × 54 should be written as 4645.

Question 5

In the following problem, "=" stands for "÷", "+" stands for "-", "×" stands for "=", "−" stands for "+" and "÷" stands for "×". Which of the following coded equations is actually arithmetically correct?

Accepted Answer

Introduction / Context: Here you are given a code for mathematical signs and must determine which one of the given equations becomes a true statement after decoding. Each symbol in the options does not carry its usual meaning; instead, there is a mapping given in the question. This type of problem evaluates accuracy in symbol substitution and careful checking of several alternatives. Given Data / Assumptions: - "=" stands for "÷" (division). - "+" stands for "-" (subtraction). - "×" stands for "=" (equality). - "−" stands for "+" (addition). - "÷" stands for "×" (multiplication). - We must test each option by translating all symbols according to this mapping and then checking if the resulting arithmetic equation is correct. Concept / Approach: For each option, replace every symbol with its true operation. Where "×" appears, it should become "=", meaning that the location of the equality sign may shift. After decoding, the left-hand side and right-hand side must be evaluated using standard arithmetic precedence. The correct option is the one where both sides are equal numerically. Step-by-Step Solution: Consider option C: 5 ÷ 3 − 25 + 20 = 20 × 39. Apply the code mapping symbol by symbol. "÷" stands for "×", so 5 ÷ 3 becomes 5 × 3. "−" stands for "+", so − 25 becomes + 25. "+" stands for "-", so + 20 becomes − 20. "=" stands for "÷", so = 20 becomes ÷ 20. "×" stands for "=", so 20 × 39 becomes 20 = 39. Thus the decoded equation is: 5 × 3 + 25 − 20 ÷ 20 = 39. Step 1: Evaluate the left-hand side with precedence. Compute 5 × 3 = 15. Compute 20 ÷ 20 = 1. So the expression becomes: 15 + 25 − 1. Now compute 15 + 25 = 40. Then 40 − 1 = 39. So the left-hand side equals 39, which matches the right-hand side. Verification / Alternative check: We can quickly test the other options similarly and see that none of them produce valid equalities once decoded. Option A, for instance, does not balance after the conversions and arithmetic. Option B likewise leads to left and right sides that differ. Option D is incomplete as an equation. Hence, option C is the only valid decoded equation. Why Other Options Are Wrong: - Option A yields a left-hand side that does not simplify to the right-hand value after decoding and evaluation. - Option B produces a mismatch between left and right sides, even though at a glance it seems simple. - Option D does not even form a proper equality once symbols are decoded. In all these cases, the decoded equations are not true. Common Pitfalls: A typical error is to change only some symbols or to forget that "×" actually produces an equality sign, not a multiplication. Another common mistake is to ignore operator precedence when simplifying the left-hand side after decoding. Writing out each transformed expression clearly and then simplifying step by step avoids these traps. Final Answer: The only equation that becomes correct after decoding the signs is 5 ÷ 3 − 25 + 20 = 20 × 39 (option C).

Question 6

If (−) stands for division, (+) stands for multiplication, (÷) stands for subtraction and (×) stands for addition, which one of the following equations is actually correct?

Accepted Answer

Introduction / Context: This problem uses a code for arithmetic operations. Instead of using the usual meanings of +, −, ×, and ÷, each symbol is redefined. You must determine which of the provided statements becomes a numerically correct equality after decoding each symbol and applying standard arithmetic rules. This tests careful interpretation and systematic checking. Given Data / Assumptions: - "−" stands for division. - "+" stands for multiplication. - "÷" stands for subtraction. - "×" stands for addition. - We are to choose from the given options which equation, after decoding, is actually true. - Once decoded, standard precedence of operations (multiplication and division before addition and subtraction) must be followed. Concept / Approach: For each option, replace the coded operators with their actual meanings, then simplify the left-hand side and compare it with the right-hand side. Because there are only four options, a direct check is efficient. The equation that results in equal values on both sides is the correct choice. Step-by-Step Solution: Consider option A: 100+5−10×250÷200 = 100. Decode each symbol using the given mapping. "+" stands for multiplication, so 100+5 becomes 100 × 5. "−" stands for division, so −10 becomes ÷ 10. "×" stands for addition, so 10×250 becomes 10 + 250. "÷" stands for subtraction, so 250÷200 becomes 250 − 200. Putting this together, the left-hand side becomes: 100 × 5 ÷ 10 + 250 − 200. Step 1: Apply multiplication and division from left to right. 100 × 5 = 500. 500 ÷ 10 = 50. The expression now is: 50 + 250 − 200. Step 2: Perform addition and subtraction. 50 + 250 = 300. 300 − 200 = 100. Thus the left-hand side simplifies to 100, which equals the right-hand side 100. Verification / Alternative check: We can briefly check one of the other options to see that it fails. For example, option B decodes to 200 × 10 ÷ 20 + 200 − 100. Simplifying gives 200 × 10 = 2000, 2000 ÷ 20 = 100, and then 100 + 200 − 100 = 200, which does not equal 150. Similar checks show that options C and D also do not balance, so option A is uniquely correct. Why Other Options Are Wrong: - Option B simplifies to a value different from the right-hand side, as shown above. - Option C gives a left-hand side that does not equal 50 after correct decoding. - Option D also leads to a mismatch between the two sides of the equation. In every incorrect option, either the left-hand side is not equal to the right-hand side, or a misinterpretation of symbols would be required to force equality, which is not allowed. Common Pitfalls: Learners may forget to decode every symbol, or they may revert to the original meanings of the operators out of habit. Another source of errors is disregarding the precedence of multiplication and division, which can alter the final result. Working slowly and rewriting the decoded expression clearly makes it easier to avoid such mistakes. Final Answer: The only equation that becomes numerically correct after decoding the signs is 100+5−10×250÷200 = 100 (option A).

Question 7

If 4 × 3 = 14, 5 × 4 = 18 and 6 × 5 = 22 under a certain pattern, then what is the value of 7 × 6 following the same rule?

Accepted Answer

Introduction / Context: This is a number pattern question where the symbol × does not represent ordinary multiplication. Instead, it denotes a special rule illustrated by the first three examples. After identifying this rule, you are asked to apply it to a new pair of numbers. Such questions check pattern recognition and basic arithmetic reasoning. Given Data / Assumptions: - 4 × 3 = 14. - 5 × 4 = 18. - 6 × 5 = 22. - We are asked to find 7 × 6 under the same pattern. - The symbol × indicates a custom operation, not the usual product. Concept / Approach: Observe the relation between the numbers on the left and the results on the right. One simple and consistent pattern is: Result = 2 × (first number + second number). If this holds for all the given examples, we can confidently apply it to 7 and 6. The pattern is simple and uses only basic addition and multiplication by 2. Step-by-Step Solution: Check the pattern for 4 × 3. First number = 4, second number = 3. Compute the sum: 4 + 3 = 7. Multiply by 2: 2 × 7 = 14, which matches the given result. Check the pattern for 5 × 4. First number = 5, second number = 4. 5 + 4 = 9. 2 × 9 = 18, which matches the second result. Check the pattern for 6 × 5. First number = 6, second number = 5. 6 + 5 = 11. 2 × 11 = 22, which matches the third result. Apply the same rule to 7 × 6. 7 + 6 = 13. 2 × 13 = 26. Verification / Alternative check: Since the same simple rule works perfectly for all three given examples, it is reasonable to trust that it is the intended pattern. More complicated patterns would be unnecessary. The computed value 26 is present among the options, reinforcing that this interpretation is correct. Why Other Options Are Wrong: - 20, 30, and 42 do not equal 2 × (7 + 6) and have no natural connection to the established rule. - To obtain those values, one would have to change the pattern or apply inconsistent operations, which would contradict the examples provided. Thus, they cannot be the correct continuation of the pattern. Common Pitfalls: Sometimes students may try to involve actual multiplication 4 × 3 = 12 and then add or subtract small adjustments without checking consistency across all examples. Another mistake is to invent a complicated rule when a very simple one, like doubling the sum, fits all cases. Always verify any proposed rule against every given example. Final Answer: Using the pattern Result = 2 × (first + second), the value of 7 × 6 is 26.

Question 8

By using which sequence of mathematical operators will the expression 18 ? 6 ? 9 ? 27 become a correct equation?

Accepted Answer

Introduction / Context: In this question, you are given an expression with three blanks for operators and four numerical options for which operators should go in which position. The goal is to select the sequence of operators so that the resulting mathematical sentence becomes a correct equation. This tests understanding of operator precedence and basic arithmetic. Given Data / Assumptions: - The incomplete expression is: 18 ? 6 ? 9 ? 27. - Each option provides three operators in order, to fill the three question marks. - We must form a true equation by using one of the sequences. - Standard precedence rules (multiplication and division before addition and subtraction) apply after substitution. Concept / Approach: We treat each option as a proposed pattern for the three blanks. For each option, we substitute the given operators in order and evaluate the left-hand side. If the expression simplifies to 27, and the final symbol is "=", we have a valid equation. Systematic trial of each option with careful arithmetic is efficient here. Step-by-Step Solution: Test option B: "÷, × and =". Substitute the operators into 18 ? 6 ? 9 ? 27. We get: 18 ÷ 6 × 9 = 27. Now evaluate the left-hand side using precedence. First compute 18 ÷ 6 = 3. Then compute 3 × 9 = 27. So the left-hand side equals 27, which matches the right-hand side 27. Thus, 18 ÷ 6 × 9 = 27 is a correct equation. Verification / Alternative check: Check another option briefly to confirm it fails. For example, option A suggests "×, ÷ and =". That would give 18 × 6 ÷ 9 = 27. Evaluating, 18 × 6 = 108 and 108 ÷ 9 = 12, so we would have 12 = 27, which is false. Similar issues arise for options C and D, where the left-hand sides do not simplify to 27. Why Other Options Are Wrong: - Option A produces 18 × 6 ÷ 9 = 27, which simplifies to 12 = 27, not correct. - Option C yields 18 × 6 + 9 = 27, which becomes 108 + 9 = 117, again false. - Option D gives 18 + 6 − 9 = 27, which simplifies to 15 = 27, also incorrect. Thus, only option B satisfies the equation. Common Pitfalls: Some learners might ignore operator precedence and evaluate strictly from left to right, which could temporarily make wrong options seem plausible. Others may misplace the equality sign, forgetting that it must compare two sides, not act as an operator. Remember to apply division and multiplication before addition or subtraction when checking each possibility. Final Answer: The correct sequence of operators is "÷, × and =", so the valid equation is 18 ÷ 6 × 9 = 27. The correct choice is ÷, × and =.

Question 9

If 18 (9) 3 and 36 (30) 5 follow the same numerical rule, then what value of A will satisfy 19 (A) 18?

Accepted Answer

Introduction / Context: This is a number puzzle where the expression a (b) c represents a certain relationship between the outer numbers and the inner number. You are given two examples where the middle value is already supplied, and you must determine the rule that connects the three numbers. Once that rule is known, you apply it to a new pair of outer numbers to find the missing middle value A. Given Data / Assumptions: - 18 (9) 3 satisfies the hidden rule. - 36 (30) 5 satisfies the same rule. - 19 (A) 18 must satisfy that rule as well. - The outer numbers play a symmetric role in the pattern, and the middle number is derived from them through some arithmetic relation. Concept / Approach: We look for a simple function of the outer numbers that yields the middle one. Trying with the first example: - Outer numbers: 18 and 3. - Middle number: 9. One neat observation is that 18 × 3 = 54, and 54 ÷ 6 = 9. For the second example, 36 × 5 = 180, and 180 ÷ 6 = 30. Thus in both cases, the middle number equals the product of the outer numbers divided by 6. We assume the same constant factor applies in the third case. Step-by-Step Solution: Step 1: Verify the pattern for 18 (9) 3. Compute the product of the outer numbers: 18 × 3 = 54. Divide by 6: 54 ÷ 6 = 9, which matches the middle number. Step 2: Verify the pattern for 36 (30) 5. Compute 36 × 5 = 180. Divide by 6: 180 ÷ 6 = 30, which again matches the middle number. Step 3: Apply the pattern to 19 (A) 18. Outer numbers: 19 and 18. Product: 19 × 18 = 342. Divide by 6: 342 ÷ 6 = 57. Therefore, A must be 57. Verification / Alternative check: The factor 6 is consistent across both given examples, so there is no need to change it for the third case. Also, 57 is present among the options, which confirms that the puzzle is well aligned with this rule. Any attempt to change the divisor or to use a more complicated formula would either fail for one of the examples or not match any answer choice. Why Other Options Are Wrong: - 33, 75, and 96 cannot be obtained by multiplying the outer numbers 19 and 18 and dividing by any constant that also works for both examples. - For instance, 19 × 18 ÷ 6 is exactly 57 and not 33, 75, or 96. Hence those values are inconsistent with the established pattern. Common Pitfalls: Some students may try to combine the outer numbers using addition or subtraction only, or may attempt to involve digits separately in a complicated way. While such methods might incidentally fit one example, they will fail when checked against both. Always validate any discovered pattern against all provided examples before applying it to the unknown case. Final Answer: Using the rule middle = (first × third) ÷ 6, the value of A in 19 (A) 18 is 57.

Question 10

If 13 * 45 = 29, 24 * 58 = 41 and 74 * 32 = 53 under a certain rule, what should 97 * 47 be equal to?

Accepted Answer

Introduction / Context: This is a number pattern question where the symbol * does not mean ordinary multiplication. Instead, it represents a special operation that combines two numbers into a new result. You are given three examples and then asked to apply the same rule to a fourth pair. This tests your ability to recognize patterns involving sums and averages of numbers. Given Data / Assumptions: - 13 * 45 = 29. - 24 * 58 = 41. - 74 * 32 = 53. - We must find 97 * 47 using the same rule. - All given results are between the two original numbers, suggesting an average or midpoint pattern. Concept / Approach: Notice that 13 and 45 have the result 29, which is exactly the average of 13 and 45. Similarly, 24 and 58 have 41 between them, and 74 and 32 have 53 between them. This suggests a simple rule: a * b = (a + b) ÷ 2. If this holds for all the given examples, we can confidently apply it to 97 and 47. Step-by-Step Solution: Check the pattern for 13 * 45. Compute the sum: 13 + 45 = 58. Compute the average: 58 ÷ 2 = 29. This matches the given result. Check the pattern for 24 * 58. 24 + 58 = 82. 82 ÷ 2 = 41, which matches the second result. Check the pattern for 74 * 32. 74 + 32 = 106. 106 ÷ 2 = 53, which matches the third result. Apply the pattern to 97 * 47. 97 + 47 = 144. 144 ÷ 2 = 72. Verification / Alternative check: Since the rule a * b = (a + b) ÷ 2 is valid for all three examples, it is very likely the intended pattern. The computed result 72 appears as one of the options, reinforcing our conclusion. A quick mental check confirms that 72 indeed lies midway between 97 and 47, which is consistent with the idea of an average. Why Other Options Are Wrong: - 73, 63, and 64 are not equal to (97 + 47) ÷ 2. - If we tried to fit those results into the same rule for the earlier examples, the pattern would break. Thus, they do not represent a consistent continuation of the given relationship. Common Pitfalls: A common error is to attempt actual multiplication or do arbitrary operations on digits rather than looking for something as simple as the average. Others may miscalculate the sum or divide incorrectly. Carefully computing the sum first and then halving it helps avoid these mistakes. Final Answer: Using the rule a * b = (a + b) ÷ 2, the value of 97 * 47 is 72.

Question 11

In the following question, correct the equation 8 × 3 ÷ 4 + 9 − 5 = 16 by interchanging two numbers. Which pair of numbers must be swapped?

Accepted Answer

Introduction / Context: This problem presents a numerical equation that is currently incorrect under normal arithmetic rules. You are allowed to fix it by interchanging (swapping) two numbers. The challenge is to determine which pair of numbers must be swapped so that the left-hand side evaluates exactly to the right-hand side when standard rules of arithmetic are applied. Given Data / Assumptions: - Original equation: 8 × 3 ÷ 4 + 9 − 5 = 16. - Candidate pairs to swap: (3,4), (4,8), (5,3), (5,9). - Only two numbers are to be swapped, and all operators stay as they are. - Standard precedence rules apply: multiplication and division before addition and subtraction. Concept / Approach: First, evaluate the original left-hand side to confirm that the equation is indeed incorrect. Then, for each possible pair from the options, swap the two numbers in the expression and recalculate the left-hand side. The correct pair is the one that produces 16. Because there are only four options, a direct trial is simple and reliable. Step-by-Step Solution: Step 1: Evaluate the original left-hand side. Expression: 8 × 3 ÷ 4 + 9 − 5. Compute 8 × 3 = 24. Compute 24 ÷ 4 = 6. Then 6 + 9 = 15. 15 − 5 = 10. Thus LHS = 10, which is not 16. Step 2: Test swapping 5 and 3 (option C). After swapping 5 and 3, the expression becomes 8 × 5 ÷ 4 + 9 − 3. Now evaluate step by step: 8 × 5 = 40. 40 ÷ 4 = 10. 10 + 9 = 19. 19 − 3 = 16. Now LHS = 16, which matches the right-hand side. Verification / Alternative check: We can quickly verify that other swaps do not work. Swapping 3 and 4, or 4 and 8, or 5 and 9 each leads to a new expression whose value is not 16 when evaluated correctly. Hence, option C is the only pair that fixes the equation. Why Other Options Are Wrong: - Swapping 3 and 4 leads to 8 × 4 ÷ 3 + 9 − 5, which does not evaluate to 16. - Swapping 4 and 8 gives 4 × 3 ÷ 8 + 9 − 5, whose value is also not 16. - Swapping 5 and 9 yields 8 × 3 ÷ 4 + 5 − 9, a much smaller result. Only swapping 5 and 3 produces the required value of 16. Common Pitfalls: Students sometimes swap the numbers but forget to recompute with correct precedence, or they might mistakenly swap more than two numbers. Another error is to try to swap operators instead of numbers, which is not allowed in this question. Careful, methodical testing of each candidate pair avoids these issues. Final Answer: The equation is corrected by interchanging the numbers 5 and 3. Therefore, the correct pair to swap is 5 and 3.

Question 12

If "+" denotes "divided by", "×" denotes "added to", "÷" denotes "subtracted from" and "−" denotes "multiplied by", then what is the value of 54 + 162 − 18 × 12 ÷ 6?

Accepted Answer

Introduction / Context: This question involves coded arithmetic operations where each familiar symbol stands for a different operation. You must decode the expression to reveal the true arithmetic being performed and then evaluate it using standard rules. This improves precision with operator substitution and cautious stepwise calculation. Given Data / Assumptions: - "+" denotes "divided by". - "×" denotes "added to". - "÷" denotes "subtracted from". - "−" denotes "multiplied by". - The coded expression is: 54 + 162 − 18 × 12 ÷ 6. - After decoding, standard arithmetic precedence (division and multiplication before addition and subtraction) is applied. Concept / Approach: We translate each symbol into its true operation while keeping the numbers in the same order. In many exam contexts, when a symbol denotes an operation like subtraction or division, it is treated as a normal binary operator with the same left to right orientation as usual. After this translation, we simplify the resulting expression carefully, respecting operator precedence. Step-by-Step Solution: Step 1: Decode each symbol. "+" denotes "divided by", so 54 + 162 becomes 54 ÷ 162. "−" denotes "multiplied by", so − 18 becomes × 18. "×" denotes "added to", so 18 × 12 becomes 18 + 12. "÷" denotes "subtracted from", and in many such coded questions it is treated as subtraction, so we interpret the last operation as minus 6. Putting these together, the decoded expression becomes: 54 ÷ 162 × 18 + 12 − 6. Step 2: Evaluate division and multiplication first. 54 ÷ 162 = 1 ÷ 3 (since 54 ÷ 162 simplifies to 1 ÷ 3). (1 ÷ 3) × 18 = 18 ÷ 3 = 6. Now the expression reduces to: 6 + 12 − 6. Step 3: Perform addition and subtraction from left to right. 6 + 12 = 18. 18 − 6 = 12. Verification / Alternative check: A quick check of the arithmetic shows no step was miscalculated. The fraction 54 ÷ 162 correctly simplifies to 1 ÷ 3, and multiplying by 18 indeed gives 6. Adding 12 and subtracting 6 gives 12, which is one of the provided options. Reconfirming the symbol mapping ensures we have not mistaken any operation. Why Other Options Are Wrong: Options 4, 16, and 10 could only arise if: - The coded operations were misread, for example treating "+" as real addition or "−" as real subtraction, or - Operator precedence was ignored and the expression was evaluated strictly left to right. Because our calculation adheres to the exact mapping and correct arithmetic rules, those alternate results are not acceptable. Common Pitfalls: It is easy to revert unconsciously to the usual meanings of +, −, ×, and ÷, or to forget to simplify fractions fully before multiplying. Another frequent error is to misinterpret "subtracted from" and reverse the order of subtraction. Reading the mapping carefully and rewriting the entire expression with standard symbols before evaluating helps avoid these issues. Final Answer: Under the given code language, the value of 54 + 162 − 18 × 12 ÷ 6 is 12.

Question 13

In a certain code language, the arithmetic symbols are replaced in the following way:  The plus sign (+) represents subtraction, the minus sign (–) represents addition, the multiplication sign (x) represents division and the division sign (÷) repres…

Accepted Answer

Introduction / Context: This question tests your ability to decode a symbolic representation of arithmetic operations and then verify which complete equation is correct after applying the hidden rules. Such coded operator problems are common in verbal reasoning and help check conceptual clarity about basic operations and precedence. Given Data / Assumptions: In the code language: + means subtraction, – means addition, x means division and ÷ means multiplication. We assume normal arithmetic precedence: division and multiplication before addition and subtraction. We must check each option by first decoding the symbols and then simplifying the left-hand side to see if it matches the right-hand side. Concept / Approach: The approach is to translate each coded expression into a normal arithmetic expression using the given mapping of symbols, then evaluate the left-hand side. If the evaluated result equals the right-hand side, that option is correct. Pay attention to the order of operations, because division and multiplication must be carried out before addition and subtraction to avoid incorrect results. Step-by-Step Solution: Step 1: Use the mapping + → minus, – → plus, x → division, ÷ → multiplication. Step 2: Decode option (d): 18 x 6 ÷ 8 – 12. Step 3: Replace x by division and ÷ by multiplication, and – by plus: 18 x 6 ÷ 8 – 12 becomes 18 ÷ 6 × 8 + 12. Step 4: Apply precedence: first 18 ÷ 6 = 3. Step 5: Then 3 × 8 = 24. Step 6: Finally 24 + 12 = 36, which matches the right-hand side 36. Verification / Alternative check: We can quickly re-calc option (d) to verify: 18 ÷ 6 = 3, 3 × 8 = 24 and 24 + 12 = 36. The equality holds, so option (d) is consistent with the given code. This confirms our decoding and evaluation are correct. Why Other Options Are Wrong: For option (a), decoding gives 16 ÷ 19 × 21 + 5, which does not equal 201 after calculation. Option (b) turns into 5 ÷ 6 - 4 × 3, which is a negative value, not 37/6. Option (c) becomes 6 ÷ 3 - 12 × 3, which is also not equal to 21. Therefore, only option (d) gives a correct equality. Common Pitfalls: A common mistake is to treat the coded symbol as if it still had its usual meaning, or to ignore operator precedence and simply calculate from left to right. Another frequent error is to misapply the mapping to only part of the expression instead of every occurrence of that symbol. Carefully substituting every coded symbol and respecting precedence avoids these issues. Final Answer: After proper decoding and evaluation, the only equation that becomes numerically correct is 18 x 6 ÷ 8 – 12 = 36, which corresponds to option (d).

Question 14

The following equation is not correct when the usual arithmetic symbols are used:  6 ÷ 17 x 51 + 6 – 12 = – 4  By interchanging exactly two of the operation signs, the equation can be made correct. Which one of the following pairs of signs must be i…

Accepted Answer

Introduction / Context: This problem belongs to the mathematical operations segment of verbal reasoning, where you must repair an incorrect equation by interchanging two operation signs. The key skill is to systematically test possible sign swaps and check which one makes the numerical equality valid. Given Data / Assumptions: Original equation: 6 ÷ 17 x 51 + 6 – 12 = -4. We are allowed to interchange only two operation signs from among ÷, x, + and –. After the swap, we evaluate the left-hand side using normal operator precedence. Concept / Approach: The basic idea is to pick each suggested pair of signs, swap all their occurrences in the equation, and then evaluate the new left-hand side. If the resulting numerical value is -4, that choice is correct. Because there are only four options, a structured trial approach is reasonable and efficient. Step-by-Step Solution: Step 1: Consider option (a), swapping x and ÷ everywhere. Step 2: The equation becomes 6 x 17 ÷ 51 + 6 - 12. Step 3: Replace x with multiplication and ÷ with division: 6 × 17 ÷ 51 + 6 - 12. Step 4: Evaluate: 6 × 17 = 102. Step 5: Then 102 ÷ 51 = 2. Step 6: Now compute 2 + 6 - 12 = 8 - 12 = -4. Step 7: The left-hand side equals -4, matching the right-hand side, so option (a) works. Verification / Alternative check: We can briefly check another option to see that it fails. For instance, if we swap + and ÷, the expression becomes 6 + 17 x 51 ÷ 6 - 12, which evaluates to a large positive number, not -4. This confirms that not all swaps work and that the successful one is special. Why Other Options Are Wrong: Swapping + and ÷, or + and –, or – and ÷ leads to left-hand sides that are far from -4 when calculated with correct precedence. They either yield positive numbers much larger than -4 or other unrelated values, so those options cannot repair the equation. Common Pitfalls: Many learners try to mentally evaluate several swaps at once and lose track of the replaced symbols. Another pitfall is ignoring operator precedence and performing operations strictly from left to right, which produces incorrect results. Writing the modified expression clearly and simplifying carefully avoids these mistakes. Final Answer: The equation becomes correct only when we interchange the division and multiplication signs, that is, swap x and ÷, so the correct choice is x and ÷ (option (a)).

Question 15

In the following pattern, the usual arithmetic symbols do not keep their ordinary meanings:  If 25 + 5 ÷ 2 = 40 and 35 + 5 ÷ 2 = 60, then using the same hidden rule, what should be the value of 45 + 5 ÷ 2?

Accepted Answer

Introduction / Context: This is a number analogy question in which the symbols do not represent normal operations. Instead, the entire expression 25 + 5 ÷ 2 produces 40 according to some hidden rule, and 35 + 5 ÷ 2 produces 60. We must discover that consistent rule and then apply it to 45 + 5 ÷ 2 to obtain the missing value. Given Data / Assumptions: Expression 1: 25 + 5 ÷ 2 results in 40. Expression 2: 35 + 5 ÷ 2 results in 60. The small numbers 5 and 2 are the same in all expressions. The same rule must apply to the case 45 + 5 ÷ 2. Concept / Approach: Rather than treating + and ÷ as normal operations, we look at how the first number is transformed into the result. If we compare 25 and 40, and 35 and 60, both results are close to twice the first number. This suggests a rule of the form result = 2 × (first number) – constant. Once we identify the constant using the two given examples, we can apply the same pattern to the third case. Step-by-Step Solution: Step 1: From 25 to 40, the increase is 40 - 25 = 15. Step 2: From 35 to 60, the increase is 60 - 35 = 25. Step 3: Check whether the expression 2 × first number - 10 fits both results. Step 4: For 25: 2 × 25 - 10 = 50 - 10 = 40, which matches the given value. Step 5: For 35: 2 × 35 - 10 = 70 - 10 = 60, again matching the given value. Step 6: Apply the same rule to 45: 2 × 45 - 10 = 90 - 10 = 80. Step 7: Therefore, 45 + 5 ÷ 2 should give 80 according to the established pattern. Verification / Alternative check: We might try other simple forms like result = first number + some multiple of 5, but these do not consistently fit both initial expressions. The linear rule 2 × first number - 10 is the simplest pattern that works perfectly for both 25 and 35 and so is the most reasonable hidden rule. Why Other Options Are Wrong: If we choose 70, 60, or 90, none of these follow from the rule 2 × first number - 10. For 45, this rule uniquely gives 80. Other answers would require inventing a different rule that breaks the consistency across all three expressions, so they must be rejected. Common Pitfalls: One common mistake is to assume that the + and ÷ symbols must be used directly and start doing mixed arithmetic with 25, 5 and 2. Another error is to look only at the differences 15 and 25 without seeing the underlying doubling pattern. Focusing on how the first number alone transforms into the result usually makes such puzzles easier to decode. Final Answer: Using the consistent rule result = 2 × first number - 10, we get 45 + 5 ÷ 2 = 80, so option (d) is correct.

Question 16

In a certain coded arithmetic, the symbols do not have their usual meanings:  The minus sign (–) means division, the plus sign (+) means multiplication, the division sign (÷) means addition and the multiplication sign (x) means subtraction.  Using t…

Accepted Answer

Introduction / Context: This problem is a classic coded-operations question in verbal reasoning. The usual arithmetic symbols have been reassigned new meanings, and you must evaluate a given expression by first decoding each symbol and then applying standard arithmetic rules like precedence of operations. Given Data / Assumptions: – stands for division. + stands for multiplication. ÷ stands for addition. x stands for subtraction. Expression to evaluate: 11 ÷ 6 – 2 + 5 x 3. Normal operator precedence (division and multiplication before addition and subtraction) still applies after decoding. Concept / Approach: We begin by carefully substituting each coded symbol with its true operation. After rewriting the entire expression with ordinary symbols, we simplify it step by step. Keeping track of precedence is crucial, because performing operations in the wrong order will produce the wrong final value and lead to an incorrect option choice. Step-by-Step Solution: Step 1: Rewrite each symbol according to the code. Step 2: The division sign ÷ stands for addition, so 11 ÷ 6 becomes 11 + 6. Step 3: The minus sign – stands for division, so – 2 becomes ÷ 2. Step 4: The plus sign + stands for multiplication, so + 5 becomes × 5. Step 5: The multiplication sign x stands for subtraction, so x 3 becomes - 3. Step 6: Thus the coded expression 11 ÷ 6 – 2 + 5 x 3 translates to 11 + 6 ÷ 2 × 5 - 3. Step 7: Apply precedence: first 6 ÷ 2 = 3. Step 8: Then 3 × 5 = 15. Step 9: Now compute 11 + 15 - 3 = 26 - 3 = 23. Verification / Alternative check: We can recompute quickly: 6 ÷ 2 = 3, 3 × 5 = 15, and 11 + 15 = 26, then 26 - 3 = 23. No other option equals 23, confirming that our decoding and calculation are consistent and accurate. Why Other Options Are Wrong: The options 17, 21 and 26 would occur if we mistakenly used the usual meanings of the symbols or ignored the proper order of operations. For example, treating x as multiplication instead of subtraction or performing additions before division and multiplication would change the numerical result, making these values inappropriate for the coded rules given. Common Pitfalls: Students sometimes replace only some of the symbols and inadvertently mix coded and normal meanings, or they simply read ÷ and x as they look instead of according to the mapping. Another common issue is computing from left to right without observing precedence. A disciplined substitution step followed by a clear order of operations avoids these errors. Final Answer: After decoding the symbols and simplifying correctly, the value of the expression is 23, so option (c) is correct.

Question 17

Consider the following incorrect equation involving the standard arithmetic symbols:  15 x 12 + 40 ÷ 40 - 6 = 21  By interchanging exactly two of the operation signs, the equation can be made correct. Which pair of signs should be interchanged so th…

Accepted Answer

Introduction / Context: This question checks your ability to manipulate arithmetic equations by interchanging operation signs. The original equation is numerically false, but if we swap the correct pair of signs everywhere they occur, we can restore the equality. This type of puzzle evaluates logical reasoning with basic arithmetic skills. Given Data / Assumptions: Original equation: 15 x 12 + 40 ÷ 40 - 6 = 21. We may interchange only the two signs mentioned in an option. After the swap, all other signs keep their original meanings. We use normal operator precedence for multiplication, division, addition and subtraction. Concept / Approach: The systematic method is to test each option one by one. For each suggested pair of signs, we swap all their occurrences in the expression, rewrite the equation with ordinary symbols, and then simplify the left-hand side. If the result equals 21, that option is the required answer. Step-by-Step Solution: Step 1: Try option (a): interchange + and x. Step 2: Every x becomes + and every + becomes x in the left-hand side. Step 3: The new equation becomes 15 + 12 x 40 ÷ 40 - 6 = 21. Step 4: Evaluate the left-hand side with standard precedence. Step 5: Compute 12 x 40 = 480. Step 6: Then 480 ÷ 40 = 12. Step 7: Now compute 15 + 12 - 6 = 27 - 6 = 21. Step 8: The left-hand side equals 21, matching the right-hand side. So option (a) works. Verification / Alternative check: We can quickly inspect another option to ensure it fails. If we swapped + and ÷ instead, the equation would become 15 x 12 ÷ 40 + 40 - 6, which simplifies to a number different from 21. This confirms that the successful repair is unique among the given choices. Why Other Options Are Wrong: Swapping + and ÷, or - and +, or ÷ and x leads to left-hand sides that are clearly not equal to 21 after correct evaluation. Either the result is a fraction or a much larger integer, so these options do not fix the equation and must be discarded. Common Pitfalls: A typical mistake is to interchange signs only at one place instead of every occurrence, or to recompute without respecting operator precedence. Another pitfall is performing mental arithmetic too quickly and miscalculating intermediate products or quotients. Writing each intermediate step clearly helps avoid such errors. Final Answer: The equation becomes correct only when we interchange the plus and multiplication signs, so the correct pair is + and x, corresponding to option (a).

Question 18

In a certain code language, each letter symbol in an arithmetic statement represents a different operation:  K denotes multiplication, M denotes subtraction ('subtracted from' is treated as ordinary minus), J denotes addition and L denotes division.…

Accepted Answer

Introduction / Context: This coded-arithmetic problem replaces the usual symbols +, -, × and ÷ by letters. To solve it, we must first decode each letter into the correct basic operation, then rewrite the entire expression with normal symbols and finally evaluate it carefully using the standard order of operations. Given Data / Assumptions: K denotes multiplication. M denotes subtraction. J denotes addition. L denotes division. Expression: 44 M 24 K 56 L 14 J 60. We apply usual precedence: division and multiplication before addition and subtraction. Concept / Approach: The strategy is straightforward: substitute each letter with its corresponding arithmetic symbol, simplify the expression step by step and ensure that we do not mix up the order of operations. Once we compute the numerical value of the left-hand side, we match it with the closest option provided. Step-by-Step Solution: Step 1: Replace M by minus, K by multiplication, L by division and J by plus. Step 2: The coded expression 44 M 24 K 56 L 14 J 60 becomes 44 - 24 × 56 ÷ 14 + 60. Step 3: First handle multiplication and division from left to right. Step 4: Compute 24 × 56 = 1344. Step 5: Then 1344 ÷ 14 = 96. Step 6: Substitute back to get 44 - 96 + 60. Step 7: Now perform the additions and subtractions from left to right: 44 - 96 = -52. Step 8: Finally, -52 + 60 = 8. Verification / Alternative check: We can verify quickly by recomputing the core product and quotient: 24 × 56 = 1344 and 1344 ÷ 14 = 96, which is correct. This leads consistently to 44 - 96 + 60 = 8. No other arithmetic interpretation consistent with the given code gives a different value, so 8 is reliable. Why Other Options Are Wrong: The alternatives 16, 72 and 40 would require an error in decoding the letters or misapplying precedence, for example by adding 44 and 60 before subtracting 96, or by dividing in the wrong order. Any such mistake changes the intermediate steps and leads away from the correct value of 8. Common Pitfalls: Students sometimes misread 'subtracted from' and reverse the order of terms or forget that division and multiplication must be done before addition and subtraction. Another common error is to carry out division or multiplication mentally and slip on basic calculations. Writing down each intermediate result greatly reduces such mistakes. Final Answer: After decoding the letters and simplifying correctly, the value of 44 M 24 K 56 L 14 J 60 is 8, so option (c) is correct.

Question 19

In a coded arithmetic notation, the following replacements are used:  A denotes 'subtracted from', B denotes 'added to', C denotes 'divided by' and D denotes 'multiplied by'.  Using this code, which one of the following statements is numerically cor…

Accepted Answer

Introduction / Context: Here the usual symbols +, -, × and ÷ are replaced by letter codes A, B, C and D. We must decode each option, translate it into an ordinary numerical expression and then check which equality holds. This tests both symbol-decoding skills and comfort with mixed operations. Given Data / Assumptions: A means subtracted from (we treat it as ordinary minus in left to right form). B means added to (plus). C means divided by. D means multiplied by. We must test each of the four coded equations and identify the one that is numerically true. Concept / Approach: For each option we decode the letter symbols into their arithmetic counterparts and then evaluate the left-hand side step by step. We then compare the computed value with the number written on the right-hand side of the equality sign. The option where left and right match is the correct statement. Step-by-Step Solution: Step 1: Decode option (c): 13 C 13 A 13 B 13 D 13. Step 2: Replace C by ÷, A by -, B by + and D by ×. Step 3: This gives 13 ÷ 13 - 13 + 13 × 13. Step 4: Apply precedence: division and multiplication first. Step 5: Compute 13 ÷ 13 = 1. Step 6: Compute 13 × 13 = 169. Step 7: Substitute: 1 - 13 + 169. Step 8: Now perform addition and subtraction from left to right: 1 - 13 = -12, -12 + 169 = 157. Step 9: The left-hand side equals 157, which matches the right-hand side given in option (c). Verification / Alternative check: We can quickly test another option to see that it fails. For example, option (a) becomes 3 - 12 + 16 × 17 ÷ 1. This simplifies to 3 - 12 + 272, which is 263, not 163. Similar checks show that the other statements do not produce the claimed right-hand side values. Why Other Options Are Wrong: In option (a), the computed value is different from 163. In option (b), after decoding and calculating, the left-hand side is far from 294. In option (d), a correct decoding also leads to a number that is not 200. Therefore none of these equations is numerically consistent, and they must be rejected. Common Pitfalls: Learners may mishandle 'subtracted from' and reverse the order of numbers, or forget to apply multiplication and division before addition and subtraction. Another common pitfall is to decode correctly but then commit a small arithmetic mistake in squaring or dividing, which leads to rejecting the right answer. Stepwise evaluation and checking squares like 13 × 13 carefully helps avoid these issues. Final Answer: Only the decoded equation 13 ÷ 13 - 13 + 13 × 13 gives 157, so the correct coded statement is 13 C 13 A 13 B 13 D 13 = 157, corresponding to option (c).

Question 20

In a certain code language for arithmetic operations, the following substitutions are used:  The plus sign (+) represents multiplication, the minus sign (-) represents addition, the multiplication sign (x) represents division and the division sign (…

Accepted Answer

Introduction / Context: This is a coded-operations question where each familiar arithmetic symbol has been assigned a different meaning. To obtain the correct numerical value, we must first decode the expression according to the given mapping and then evaluate it using standard arithmetic precedence rules. Given Data / Assumptions: + means multiplication. - means addition. x means division. ÷ means subtraction. Expression to decode: 642 x 6 ÷ 25 + 4. We use the usual precedence of operations: division and multiplication before addition and subtraction. Concept / Approach: The method is to translate each coded symbol into its true operation and rewrite the expression entirely in normal arithmetic form. After that, we simplify step by step, following the correct order of operations. The final result is then compared with the answer options. Step-by-Step Solution: Step 1: Replace x with ÷, because x represents division. Step 2: Replace ÷ with -, because ÷ represents subtraction. Step 3: Replace + with ×, because + represents multiplication. Step 4: The coded expression 642 x 6 ÷ 25 + 4 becomes 642 ÷ 6 - 25 × 4. Step 5: Evaluate division and multiplication first. Step 6: Compute 642 ÷ 6 = 107. Step 7: Compute 25 × 4 = 100. Step 8: Now perform the subtraction: 107 - 100 = 7. Verification / Alternative check: We can recompute the division 642 ÷ 6 by checking 6 × 107 = 642, which confirms the quotient is correct. The multiplication 25 × 4 = 100 is straightforward. Since 107 - 100 = 7 exactly, the decoded and evaluated expression is reliable. Why Other Options Are Wrong: The answers 11, 64 and 18 would arise only if we misinterpreted one of the coded symbols or changed the order of operations, such as treating + as ordinary addition or performing addition before multiplication. Because our decoding is consistent with the rules given, these alternative values are not correct. Common Pitfalls: Students often forget to apply the mapping to every occurrence of the symbol, or accidentally mix original meanings with coded meanings. Another pitfall is ignoring precedence, for example subtracting 25 from 642 before dividing. Being systematic in substitution and evaluation is essential in such coded-operator questions. Final Answer: After decoding and simplifying, the value of the expression 642 x 6 ÷ 25 + 4 is 7, so option (d) is correct.

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