Question 1

In a certain code language, each letter in a word is replaced by the letter two places before it in the English alphabet (with wrap around from A to Z). Under this rule, BOULDER is coded as ZMSJBCP. Using the same pattern, how is ELK written in this…

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Introduction / Context: This letter coding question uses a fixed backward shift in the alphabet. You are told that BOULDER becomes ZMSJBCP under the code, and you must infer the pattern and apply it to the word ELK. Such questions check whether you can detect a consistent shift and handle wrap around correctly for letters at the beginning of the alphabet. Given Data / Assumptions: BOULDER is coded as ZMSJBCP. The code is letter by letter and deterministic. The pattern described is that each letter is replaced by the letter two positions earlier in the alphabet, with A mapped back to Y and B to Z. We must code ELK using this same rule. Concept / Approach: To confirm the rule, we compare each letter in BOULDER with its coded partner in ZMSJBCP and compute the shift in terms of alphabetical positions. If each coded letter is exactly two positions before the original letter, with B mapping to Z to account for wrapping, then the pattern is simply a uniform backward shift of two. Once verified, we apply this shift to E, L and K to obtain the code for ELK. Step-by-Step Solution: Step 1: Write the mapping from BOULDER to ZMSJBCP. B → Z, O → M, U → S, L → J, D → B, E → C, R → P. Step 2: Convert letters to their alphabetical positions. B(2) → Z(26). Treating the alphabet cyclically, 2 - 2 = 0 and 0 corresponds to 26, so this is effectively a shift of -2. O(15) → M(13): 15 - 2 = 13. U(21) → S(19): 21 - 2 = 19. L(12) → J(10): 12 - 2 = 10. D(4) → B(2): 4 - 2 = 2. E(5) → C(3): 5 - 2 = 3. R(18) → P(16): 18 - 2 = 16. Step 3: All mappings are consistent with a shift of two positions backward, wrapping as needed. Step 4: Now apply this rule to ELK. E is the 5th letter; 5 - 2 = 3, which is C. L is the 12th letter; 12 - 2 = 10, which is J. K is the 11th letter; 11 - 2 = 9, which is I. Step 5: Combine the coded letters in order: C J I → CJI. Verification / Alternative check: As a quick consistency check, you can decode CJI back to ELK by shifting forward by two positions: C to E, J to L and I to K. This recovers ELK exactly, which confirms that the coding process is reversible and the shift amount has been chosen correctly. Rechecking the BOULDER example with the same backward shift of two for each letter further confirms the general rule. Why Other Options Are Wrong: Options XIG, EOC, BXM and ZIG require different shifts or mixtures of forward and backward movement that do not match the -2 pattern verified with BOULDER. For example, mapping E to X would involve a large backward shift, not just two positions. Because the coding rule must be uniform and consistent across all words, only CJI respects the shift by two positions backward for every letter of ELK. Common Pitfalls: A common mistake is to assume a forward shift because many code questions use that pattern. Another is to ignore wrapping and think that letters at the start of the alphabet cannot map to letters near the end. Remember that with wrap around, B shifted back by two becomes Z. Writing down the alphabet in a circle or noting pairs explicitly helps visualise this. Once the shift is confirmed, applying it to any given word is straightforward. Final Answer: Therefore, using the same shift of two letters backward, ELK is written as CJI in the code language.

Question 2

In a certain coded arithmetic system, the usual meanings of the symbols are changed as follows: "+" represents multiplication (x), "-" represents addition (+), "x" represents division (÷) and "÷" represents subtraction (-). Using this code, what is …

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Introduction / Context: This question belongs to the coding–decoding type for arithmetic operators. The symbols +, -, x and ÷ no longer have their usual meanings. Instead, each symbol stands for a different basic operation. The task is to apply these new meanings correctly to the given expression 56 ÷ 8 + 12 - 72 and then evaluate it using normal arithmetic rules. This checks careful reading of symbol definitions and disciplined application of operator precedence. Given Data / Assumptions: "+" represents multiplication. "-" represents addition. "x" represents division. "÷" represents subtraction. Expression to evaluate: 56 ÷ 8 + 12 - 72. After substituting the true operations, normal precedence (multiplication and division before addition and subtraction) is used. Concept / Approach: The key idea is that we must first translate the coded expression into an ordinary arithmetic expression by replacing each operator with its actual meaning. Only after this substitution do we use the usual order of operations. Directly performing calculations on the original symbols as if they still meant their usual operations will give a completely wrong answer. A systematic two step process (substitute, then evaluate) avoids confusion and typical exam mistakes. Step-by-Step Solution: Step 1: Start from the coded expression: 56 ÷ 8 + 12 - 72. Step 2: Replace each symbol using the code. "÷" represents subtraction, so 56 ÷ 8 becomes 56 - 8. "+" represents multiplication, so + 12 becomes × 12. "-" represents addition, so - 72 becomes + 72. Step 3: The correctly translated expression is 56 - 8 * 12 + 72. Step 4: Apply precedence. First calculate the multiplication: 8 * 12 = 96. Step 5: Substitute back: the expression becomes 56 - 96 + 72. Step 6: Now evaluate from left to right for addition and subtraction. First 56 - 96 = -40. Step 7: Then -40 + 72 = 32. Step 8: Therefore, the coded expression has the value 32. Verification / Alternative check: You can quickly recheck by writing the translated expression with brackets: 56 - (8 * 12) + 72. This shows clearly that only 8 and 12 are multiplied. Multiplying first gives 96, and then 56 - 96 + 72 leads again to 32. There is no alternative consistent interpretation of the given code that would change this result, so 32 is stable and reliable as the final answer. Why Other Options Are Wrong: Option 88 would come from treating the original + as addition and the ÷ as real division, which ignores the coding. Option 44 can appear if someone does part of the translation correctly but mishandles the order of operations. Option 82 and option 40 similarly reflect partial or incorrect substitutions or left to right evaluation without respecting precedence. None of these values match the carefully translated and correctly evaluated expression. Common Pitfalls: Typical mistakes include forgetting that + now means multiplication, or assuming that ÷ still means division. Another common error is to perform all operations from left to right even after the correct substitution, which breaks the precedence rule. To avoid such issues, always rewrite the coded expression clearly and then underline or bracket the multiplication and division operations before starting numeric calculations. Final Answer: After applying the coded meanings of the symbols and evaluating correctly, the value of the expression is 32.

Question 3

In a certain numerical puzzle, the symbol # denotes a special operation between two numbers. It is given that 19 # 13 = 3, 25 # 3 = 11 and 36 # 10 = 13. Using the same rule, what is the value of 7 # 3 ?

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Introduction / Context: This problem is based on a coded numerical operation where the symbol # represents some hidden rule instead of a standard arithmetic operation. We are given three examples of input and output values involving this symbol and then asked to apply the same rule to a new pair of numbers. Such questions test the ability to identify simple patterns and relationships between numbers and then generalise that pattern to an unseen case without guessing randomly. Given Data / Assumptions: 19 # 13 = 3. 25 # 3 = 11. 36 # 10 = 13. The same rule for # is used in all three examples. We are required to find the value of 7 # 3. Only basic arithmetic operations are assumed to underlie the pattern. Concept / Approach: The outputs 3, 11 and 13 are much smaller than the original numbers, which suggests that the operation is not simple multiplication or addition. A natural first idea is to look at the difference between the two numbers in each pair and then see how that difference might be transformed. For example, 19 - 13 is 6, and halving this gives 3, which matches the first result. We check whether halving the difference explains all given examples. If it does, then this becomes the working rule for the # operation and can be applied confidently to the new pair 7 and 3. Step-by-Step Solution: Step 1: For 19 # 13, compute the difference 19 - 13 = 6. Step 2: Halve this difference: 6 / 2 = 3, which matches the given result 3. Step 3: For 25 # 3, compute the difference 25 - 3 = 22. Step 4: Halve this difference: 22 / 2 = 11, which matches the second result 11. Step 5: For 36 # 10, compute the difference 36 - 10 = 26. Step 6: Again halving gives 26 / 2 = 13, which matches the third result 13. Step 7: The pattern is therefore clearly a # b = (a - b) / 2. Step 8: Apply the same rule to 7 # 3. The difference is 7 - 3 = 4. Step 9: Halve this difference: 4 / 2 = 2. Verification / Alternative check: All three original examples are perfectly explained by the single formula a # b = (a - b) / 2. There is no need to invent a more complicated rule involving digit sums or squares. If we tried such alternatives, they would quickly fail for at least one of the given pairs. Because a simple, consistent formula exists and matches every known case, it is overwhelmingly likely to be the intended rule, and using it for 7 and 3 is fully justified. Why Other Options Are Wrong: Option 21 corresponds to simply taking the product 7 * 3 and ignores the pattern of halving the difference. Option 26 and option 39 reflect other uncontrolled operations such as adding or multiplying the numbers without checking consistency with the examples. Option 10 neither comes from halving the difference nor matches any pattern demonstrated in the given data. Only 2 agrees with the clearly established formula. Common Pitfalls: A common error is to look at just one example and assume a rule like a - b or a / b without verifying against all given equations. Another frequent mistake is to jump to a complex rule involving digit sums or factorials, which is rarely required for simple exam questions. Always derive a candidate rule from one example and then rigorously test it against all other examples before trusting it. In this way, you avoid patterns that only fit partially and you arrive at the correct operation more reliably. Final Answer: Using the rule a # b = (a - b) / 2, the value of 7 # 3 is 2.

Question 4

In a certain numerical code, the symbol $ represents a special operation between two numbers. It is given that 75 $ 26 = 4 and 69 $ 53 = 7. Using the same rule, what is the value of 82 $ 46 ?

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Introduction / Context: This is a coded operation question where the symbol $ stands for a specific rule rather than ordinary arithmetic. We are given two examples of how this operation behaves on pairs of two digit numbers, and then we must determine the result for a new pair. The challenge is to recognise that the operation is based on the digits of the numbers rather than their whole values, which is a common trick in reasoning questions of this type. Given Data / Assumptions: 75 $ 26 = 4. 69 $ 53 = 7. The same definition of $ applies to all pairs. We must evaluate 82 $ 46. Numbers are two digit, so digit based operations like digit sums are natural candidates. Concept / Approach: Because the outputs 4 and 7 are much smaller than the inputs, it is likely that the operation uses digit sums rather than direct multiplication or subtraction of the full numbers. A simple and effective hypothesis is that we take the sum of the digits of each number separately, and then combine those sums in some way such as taking their difference. We test this idea on the provided examples. If a consistent rule emerges, we can apply it to the new pair 82 and 46 with confidence. Step-by-Step Solution: Step 1: For 75 $ 26, find the sum of digits of 75: 7 + 5 = 12. Step 2: Find the sum of digits of 26: 2 + 6 = 8. Step 3: Take the difference of these sums: 12 - 8 = 4, which matches the given result. Step 4: For 69 $ 53, sum of digits of 69 is 6 + 9 = 15, and for 53 it is 5 + 3 = 8. Step 5: The difference 15 - 8 = 7 matches the second given result. Step 6: The pattern is therefore clear: a $ b = (sum of digits of a) - (sum of digits of b). Step 7: Now apply the rule to 82 $ 46. Sum of digits of 82 is 8 + 2 = 10. Step 8: Sum of digits of 46 is 4 + 6 = 10. Step 9: The difference is 10 - 10 = 0. Verification / Alternative check: Both original equations are perfectly explained by this simple digit sum difference. Alternative ideas, such as subtracting the second full number from the first or adding all digits together, do not consistently give 4 and 7. Because the same neat pattern holds in multiple examples, it is very likely to be the intended rule. Using that rule for the third pair naturally leads to the result 0, which fits the coding logic precisely. Why Other Options Are Wrong: Options 62, 56 and 91 might appear if someone misreads the operation as ordinary subtraction or miscalculates with the full numbers. These values do not relate to the digit sums in the way the examples demonstrate. Option 4 repeats the first output but does not follow from the digit sum difference of the third pair, where the sums are equal. Only 0 correctly reflects the rule applied to 82 and 46. Common Pitfalls: Students often assume the symbol $ means something like ordinary subtraction or division and do not consider that the digits themselves may be involved. Another frequent mistake is calculating one digit sum correctly but forgetting to subtract in the right order. To avoid this, always keep the pattern simple: compute digit sums separately for each number and then subtract the second sum from the first in a consistent way for every example and the target pair. Final Answer: Using the pattern of subtracting digit sums, the value of 82 $ 46 is 0.

Question 5

In a numerical coding pattern, expressions of the form a θ b δ c are evaluated using the same hidden rule. It is given that 8 θ 12 δ 6 = 60 and 13 θ 15 δ 11 = 74. Using the same rule, what is the value of 18 θ 21 δ 15 ?

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Introduction / Context: This question involves a coded operation that takes three numbers at a time. The symbols θ and δ together represent a fixed numerical rule which combines the three numbers into a single result. We are given two examples of this pattern and must identify the underlying formula so that we can apply it to a third triple of numbers. Recognising how products and squares might be involved is the central reasoning skill tested here. Given Data / Assumptions: 8 θ 12 δ 6 = 60. 13 θ 15 δ 11 = 74. The symbols θ and δ are part of a single fixed pattern applied to the three numbers. We must find 18 θ 21 δ 15 under the same rule. Only standard arithmetic operations such as addition, subtraction and multiplication are assumed to be used inside the pattern. Concept / Approach: Because we have three numbers and a relatively small result, a good starting point is to try simple formulas built from multiplication and subtraction. In the first example, 8 * 12 is 96 and 6^2 is 36. The difference 96 - 36 equals 60, which matches the result perfectly. For the second example, 13 * 15 is 195 and 11^2 is 121. The difference 195 - 121 equals 74, which again matches the given result. This strongly suggests that the coded pattern is: a θ b δ c = (a * b) - (c^2). Once this structure is identified and verified, we can safely use it for the third triple (18, 21, 15). Step-by-Step Solution: Step 1: From the first example, compute 8 * 12 = 96 and 6^2 = 36, then 96 - 36 = 60, which matches 8 θ 12 δ 6. Step 2: From the second example, compute 13 * 15 = 195 and 11^2 = 121, then 195 - 121 = 74, which matches 13 θ 15 δ 11. Step 3: Conclude that the pattern is a θ b δ c = (a * b) - (c^2). Step 4: For 18 θ 21 δ 15, compute the product of the first two numbers: 18 * 21. Step 5: 18 * 21 = 378. Step 6: Compute the square of the third number: 15^2 = 225. Step 7: Subtract: 378 - 225 = 153. Step 8: Therefore, 18 θ 21 δ 15 = 153. Verification / Alternative check: The same formula works perfectly for both given examples without any adjustment or extra terms, which is strong evidence that it is the intended rule. If we tried alternative expressions such as (a + b) * c or a * c + b, they would not produce both 60 and 74 from the given data. The use of a product minus a square is neat and matches typical exam patterns. Recomputing 378 - 225 carefully confirms the final result 153, and no other simple formula fits all the information so cleanly. Why Other Options Are Wrong: Option 161, 139, 147 and 172 do not equal 378 - 225 and would require changing the rule in a way that breaks the first two examples. For instance, adding instead of subtracting the square, or using the square of a different number, will change the value substantially and make it inconsistent with the given equations. These options act as distractors for students who misidentify which number is being squared or who add instead of subtracting. Common Pitfalls: A frequent mistake is to assume that θ and δ each correspond to simple operations such as + or × and to try to evaluate the expression directly without noticing the three number structure. Another error is to square the wrong number (for example, squaring b instead of c) or to forget the square entirely. To avoid this, always check that any proposed pattern fits every given example exactly before using it for the final calculation. Final Answer: Using the rule (a * b) - (c^2), the value of 18 θ 21 δ 15 is 153.

Question 6

In the following expression, each question mark must be replaced by one mathematical operator so that the entire inequality becomes correct: 27 ? 9 ? 18 ? 24 ? 12. Which of the following sets of operators, when placed in order in the blanks from lef…

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Introduction / Context: This problem asks you to choose a sequence of standard arithmetic operators that will make a given expression correct. The expression contains four blanks, each to be filled with one operator from the chosen option. Some options use equality, others use inequality signs like greater than or less than. The question tests how well you can apply operator precedence and compare the values of expressions on different sides of a relational symbol such as > or <. Given Data / Assumptions: Base expression with blanks: 27 ? 9 ? 18 ? 24 ? 12. Each option gives four operators to insert from left to right. Option A: ÷, +, = and +. Option B: ÷, x, < and +. Option C: ÷, x, > and +. Option D: +, ÷, = and ÷. Standard operator precedence (multiplication and division before addition) applies once operators are inserted. Concept / Approach: The strategy is to test each option by inserting its operators into the blanks and then simplifying the resulting expression. Because some options use equality and others use inequality, we must calculate the left and right sides of any relation and check whether the relation holds. Rather than doing complex algebra, we simply follow arithmetic rules carefully. The correct option will be the one where the entire statement is true. If more than one looks possible, rechecking calculations usually reveals where a mistake has occurred. Step-by-Step Solution: Step 1: Test option C because it uses both ÷ and x, which seem natural candidates. Inserting these gives 27 ÷ 9 x 18 > 24 + 12. Step 2: Evaluate the left side with precedence. First compute 27 ÷ 9 = 3. Step 3: Next compute 3 x 18 = 54. So the left side is 54. Step 4: Now evaluate the right side: 24 + 12 = 36. Step 5: Compare the two sides: 54 > 36 is a true statement. Step 6: Therefore, option C produces a correct inequality. Step 7: Briefly check that the other options fail. For example, option B would give 27 ÷ 9 x 18 < 24 + 12, which simplifies to 54 < 36, a false statement. Step 8: Option A results in 27 ÷ 9 + 18 = 24 + 12, or 3 + 18 = 36, that is 21 = 36, which is false. Step 9: Option D gives 27 + 9 ÷ 18 = 24 ÷ 12, or 27 + 0.5 = 2, which is 27.5 = 2, again false. Verification / Alternative check: Once option C is shown to work, it is good practice to double check the arithmetic. Recomputing 27 ÷ 9 as 3, then 3 x 18 as 54, and 24 + 12 as 36, again confirms that 54 is greater than 36. Since none of the other options produce a true equality or inequality, option C is uniquely correct. There is no hidden trick or additional rule beyond ordinary arithmetic precedence here. Why Other Options Are Wrong: Option A fails because the two sides of the equality are 21 and 36, which are not equal. Option B fails because 54 is not less than 36. Option D fails because 27.5 is not equal to 2. These incorrect options highlight the importance of applying precedence rules correctly and not simply inserting operators that look appealing without checking the resulting statement thoroughly. Common Pitfalls: A frequent pitfall is evaluating from left to right without giving priority to multiplication and division. For instance, if someone treated 27 ÷ 9 x 18 as (27 ÷ 9) x 18 they are actually still applying correct precedence, but mistakenly doing 27 ÷ (9 x 18) would give a wrong result. Another common mistake is misreading the inequality signs and assuming < when > is required. Carefully rewriting each candidate expression and simplifying step by step helps avoid these issues. Final Answer: The set of operators that makes the expression correct is ÷, x, > and + (option C).

Question 7

In a certain coded pattern, expressions of the form a + b x c + d are not evaluated normally. Instead, 19 + 2 x 2 + 3 is equal to 369 and 23 + 2 x 6 + 2 is equal to 565. Using the same hidden rule, what is the value of 7 + 3 x 11 + 3 ?

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Introduction / Context: This question presents equations where normal arithmetic does not apply. Instead of evaluating 19 + 2 x 2 + 3 in the usual way, the result is given as 369. Similarly, 23 + 2 x 6 + 2 is given as 565. This clearly indicates that the symbols are being used as part of a coding pattern. The goal is to identify the hidden relationship between the four numbers in each expression and the three digit result, and then use this relationship to find the value corresponding to 7 + 3 x 11 + 3. This tests deeper pattern recognition and comfort with powers and squares. Given Data / Assumptions: 19 + 2 x 2 + 3 = 369 (coded form). 23 + 2 x 6 + 2 = 565 (coded form). The same rule applies for all such expressions. The expression to decode is 7 + 3 x 11 + 3. The plus and multiplication signs are part of the pattern rather than ordinary operations. Concept / Approach: A good starting point is to look at the first number and see whether its square is related to the coded result. For the first expression, 19^2 = 361, which is very close to 369. The difference is 369 - 361 = 8. Now observe that 2^3 = 8, and 2 and 3 are the second and fourth numbers in the expression. For the second expression, 23^2 = 529. The difference 565 - 529 = 36, and 6^2 = 36 using the third number 6 and the fourth number 2 as the exponent (6^2). These observations suggest that the coded result is of the form (first number)^2 + (third number)^(fourth number). Once this rule is confirmed for both given equations, we can apply it to 7, 3, 11 and 3. Step-by-Step Solution: Step 1: For 19 + 2 x 2 + 3, treat the numbers as a = 19, b = 2, c = 2, d = 3. Step 2: Compute a^2 = 19^2 = 361. Step 3: Compute c^d = 2^3 = 8. Step 4: Add these: 361 + 8 = 369, matching the coded result. Step 5: For 23 + 2 x 6 + 2, a = 23, c = 6, d = 2. Compute a^2 = 23^2 = 529. Step 6: Compute c^d = 6^2 = 36. Step 7: Sum: 529 + 36 = 565, again matching the coded result exactly. Step 8: Now apply the rule to 7 + 3 x 11 + 3, which gives a = 7, c = 11, d = 3. Step 9: Compute a^2 = 7^2 = 49. Step 10: Compute c^d = 11^3 = 1331. Step 11: Add these: 49 + 1331 = 1380. Verification / Alternative check: The rule coded result = a^2 + c^d fits both original examples without any adjustment, which makes it a very strong candidate for the intended pattern. Alternative constructions, such as using the second number as the base or adding all four numbers together, do not reproduce both 369 and 565. Rechecking the cube of 11 (11 * 11 * 11 = 1331) and adding 49 confirms that 1380 is accurate. Because the pattern is consistent and exact, 1380 is the correct coded value for the third expression. Why Other Options Are Wrong: Options 1674, 1268, 1496 and 1532 do not arise from a^2 + c^d for the numbers 7 and 11. They would require changing the pattern in a way that breaks the earlier examples, for instance by using different exponents or summing all four numbers. These distractions are meant to catch those who attempt ad hoc calculations or partial observations rather than establishing a single rule that works in every given case. Common Pitfalls: One common mistake is to treat the plus and multiplication signs as normal and compute 19 + 2 x 2 + 3 directly, which yields 26 instead of 369. Another error is to suspect a pattern involving sums of digits or concatenation without checking how the first number's square naturally appears in the result. Because the coded outputs are relatively large, recognising squares and powers is crucial. Always verify that your proposed rule explains every given example before applying it to the target expression. Final Answer: Using the pattern coded value = (first number)^2 + (third number)^(fourth number), the value of 7 + 3 x 11 + 3 is 1380.

Question 8

In a certain coded arithmetic language, the letters represent operations as follows: "S" denotes multiplied by, "V" denotes subtracted from, "M" denotes added to and "L" denotes divided by. Using this code, what is the value of 5 V 9 M 63 L 9 S 7 ?

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Introduction / Context: Here, each letter stands for a basic arithmetic operator. Instead of the usual symbols +, -, x and ÷, the problem uses S, V, M and L. We must translate the coded expression 5 V 9 M 63 L 9 S 7 into an ordinary arithmetic expression and then evaluate it. Questions like this test careful reading of symbol definitions and precise use of operator precedence, since multiplication and division still take priority after substitution. Given Data / Assumptions: S denotes multiplied by (×). V denotes subtracted from (used here as the minus operation). M denotes added to (+). L denotes divided by (÷). Expression to evaluate: 5 V 9 M 63 L 9 S 7. After substitution, normal arithmetic precedence (× and ÷ before + and -) is applied. Concept / Approach: The solution method is to perform a two stage process. First, substitute each coded letter by its true arithmetic operation to obtain a standard numerical expression. Second, carefully evaluate that expression by performing multiplication and division first, then addition and subtraction from left to right. It is important not to be distracted by the phrase subtracted from in natural language and to treat V consistently as the subtraction operator between the adjacent numbers in the given coded format. Step-by-Step Solution: Step 1: Rewrite the coded expression with real operators. V is subtraction, M is addition, L is division and S is multiplication. Step 2: Therefore, 5 V 9 M 63 L 9 S 7 becomes 5 - 9 + 63 ÷ 9 × 7. Step 3: Apply precedence by handling division and multiplication first. Compute 63 ÷ 9 = 7. Step 4: Then multiply 7 × 7 = 49. Step 5: Substitute these results back: the expression simplifies to 5 - 9 + 49. Step 6: Now evaluate from left to right: 5 - 9 = -4. Step 7: Then -4 + 49 = 45. Step 8: Hence the value of 5 V 9 M 63 L 9 S 7 is 45. Verification / Alternative check: To verify, you can bracket the expression after substitution: (5 - 9) + (63 ÷ 9 × 7). This makes it clear that all operations in the second bracket are evaluated before adding to the first bracket. Recomputing 63 ÷ 9 × 7 as 7 × 7 = 49 and then adding -4 confirms the final result 45. No alternative reading of the symbol meanings is consistent with the definitions given, so the answer is robust. Why Other Options Are Wrong: Option 50 could arise if someone mistakenly treated 5 V 9 as 9 - 5 and then mismanaged the subsequent operations. Option 40 and option 35 usually come from doing the operations strictly from left to right without respecting the precedence of division and multiplication. Option 30 might reflect an arithmetic slip in the final addition step. None of these match the correct, carefully evaluated result of 45. Common Pitfalls: A major source of error in this type of question is forgetting the order of operations and calculating 5 - 9 + 63 ÷ 9 × 7 sequentially from left to right. Another common pitfall is misinterpreting subtracted from as reversing the order of numbers in every context rather than simply reading it as subtraction between the given terms in this compact symbolic form. Writing the fully substituted expression and then methodically applying precedence rules helps avoid these mistakes. Final Answer: After decoding the operators and evaluating correctly, the value of the expression is 45.

Question 9

In a coded arithmetic system, the symbols represent operations as follows: "$" means addition, "@" means division, "#" means multiplication and "%" means subtraction. Using this code, what is the value of 84 @ 12 # 3 $ 24 % 35 ?

Accepted Answer

Introduction / Context: This question uses special symbols to stand for the four basic arithmetic operations. The expression 84 @ 12 # 3 $ 24 % 35 must be translated from the symbolic language into a normal arithmetic expression and then evaluated. The primary skills tested are remembering the operator mappings correctly and applying the correct order of operations once translation is done. Given Data / Assumptions: "$" means addition (+). "@" means division (÷). "#" means multiplication (×). "%" means subtraction (-). Coded expression: 84 @ 12 # 3 $ 24 % 35. Standard arithmetic precedence (× and ÷ before + and -) is applied after substitution. Concept / Approach: To solve the problem, we first convert each symbol into its corresponding arithmetic operator. This yields an ordinary expression involving division, multiplication, addition and subtraction. Then we evaluate the expression by performing division and multiplication from left to right before handling addition and subtraction. Carefully following these steps ensures that the calculation is accurate and consistent with standard mathematical rules. Step-by-Step Solution: Step 1: Translate each symbol. "@" is division, "#" is multiplication, "$" is addition and "%" is subtraction. Step 2: Replace the coded expression: 84 @ 12 # 3 $ 24 % 35 becomes 84 ÷ 12 × 3 + 24 - 35. Step 3: Apply precedence. First calculate 84 ÷ 12 = 7. Step 4: Then multiply 7 × 3 = 21. Step 5: Substitute this back into the expression to get 21 + 24 - 35. Step 6: Now perform addition and subtraction from left to right. First 21 + 24 = 45. Step 7: Then 45 - 35 = 10. Step 8: Therefore, the value of the coded expression is 10. Verification / Alternative check: We can quickly verify by bracketing the higher precedence operations: (84 ÷ 12 × 3) + 24 - 35. Evaluating inside the bracket gives 7 × 3 = 21, and the rest of the calculation proceeds exactly as before, resulting in 10. Because the mapping of symbols to operations is straightforward and consistently applied, there is no ambiguity, and 10 is clearly the correct result. Why Other Options Are Wrong: Options 9, 8, 6 and 12 are typical of arithmetic mistakes or precedence errors. For example, treating the operations from left to right without giving priority to division and multiplication can shift intermediate results, leading to incorrect final values. Similarly, misreading "%" as a modulus operator rather than subtraction could give different wrong answers. None of these values match the carefully computed result of 10 after correct translation and evaluation. Common Pitfalls: A common error is to misremember which symbol stands for which operation, particularly confusing "$" and "%", or "#" and "@". Another frequent mistake is to ignore operator precedence and evaluate the expression strictly in reading order. Writing the fully translated expression and underlining the division and multiplication parts before calculation helps reduce these mistakes. Always double check the mapping of each symbol to its true operation before starting the arithmetic. Final Answer: After decoding the symbols and computing correctly, the value of 84 @ 12 # 3 $ 24 % 35 is 10.

Question 10

In a certain code language, arithmetic symbols are used with different meanings as follows: "x" represents "+", "÷" represents "x", "-" represents "÷" and "+" represents "-". Using this code, find the value of the expression 5 ÷ 4 - 10 + 7 x 16.

Accepted Answer

Introduction / Context: This is a classic operator coding problem. Each familiar arithmetic symbol has been reassigned a new meaning. We are given that "x" stands for addition, "÷" stands for multiplication, "-" stands for division and "+" stands for subtraction. The expression 5 ÷ 4 - 10 + 7 x 16 must therefore be translated into an ordinary arithmetic expression under these new meanings and then evaluated. The question checks precise translation and careful use of precedence. Given Data / Assumptions: "x" represents addition (+). "÷" represents multiplication (×). "-" represents division (÷). "+" represents subtraction (-). Coded expression: 5 ÷ 4 - 10 + 7 x 16. Standard arithmetic precedence (× and ÷ before + and -) is used after substitution. Concept / Approach: The solution involves two main stages. First, replace each operator in the coded expression with the true operation it represents. This produces a standard expression involving multiplication, division, addition and subtraction in their usual sense. Second, evaluate this expression using standard order of operations, taking care not to mix up the original symbols with their coded meanings. Writing out the fully translated expression clearly before doing any calculations greatly reduces the risk of error. Step-by-Step Solution: Step 1: Start from the coded expression: 5 ÷ 4 - 10 + 7 x 16. Step 2: Replace each operator using the given meanings. "÷" represents multiplication, so 5 ÷ 4 becomes 5 × 4. "-" represents division, so - 10 becomes ÷ 10. "+" represents subtraction, so + 7 becomes - 7. "x" represents addition, so 7 x 16 becomes 7 + 16. Step 3: The fully translated expression is 5 × 4 ÷ 10 - 7 + 16. Step 4: Apply precedence. First handle multiplication and division from left to right. Compute 5 × 4 = 20. Step 5: Then compute 20 ÷ 10 = 2. Step 6: Substitute back: the expression becomes 2 - 7 + 16. Step 7: Now evaluate from left to right: 2 - 7 = -5. Step 8: Next, -5 + 16 = 11. Step 9: Therefore, the value of the coded expression is 11. Verification / Alternative check: To verify the result, you can insert brackets to emphasise precedence: (5 × 4 ÷ 10) - 7 + 16. Recalculating 5 × 4 ÷ 10 as 20 ÷ 10 = 2 and then performing the remaining operations leads again to 11. Because the mapping between coded and real operators is used consistently, and the arithmetic operations are straightforward, 11 is a secure and unambiguous result. Why Other Options Are Wrong: Option 50 might come from evaluating the original expression without applying the coding rules and treating ÷ and x in their usual sense. Option 5 or 9 can appear if someone misapplies operator precedence or mistranslates one of the symbols. Option 15 could arise from calculating 5 × 4 ÷ 10 as 5 instead of 2 and then doing the additions and subtractions incorrectly. None of these values match the correctly translated and evaluated expression, which yields 11. Common Pitfalls: A common mistake is blending the original and coded meanings, for example interpreting "x" as multiplication out of habit. Another frequent error is to skip the full translation step and attempt to compute directly on the mixed symbol expression, which almost guarantees confusion. To avoid these issues, always write a clear mapping list, perform substitution carefully and only then apply ordinary arithmetic rules with proper precedence. Final Answer: After substituting the coded meanings and evaluating correctly, the expression 5 ÷ 4 - 10 + 7 x 16 equals 11.

Question 11

In a certain code language, the word "MATERIAL" is written as "RIALMATE" by rearranging its letters in a fixed way. Using the same coding rule, how is the word "REMEMBER" written in that code language?

Accepted Answer

Introduction / Context: This is a word coding question based on rearranging letters according to a fixed pattern. We are told that MATERIAL is written as RIALMATE in the code language. Our task is to discover the rearrangement rule and then apply the same rearrangement to the word REMEMBER. These questions test visual pattern recognition and consistent application of a transformation to a new word of the same length. Given Data / Assumptions: Original word: MATERIAL. Coded form: RIALMATE. Target word: REMEMBER. The same rearrangement rule is applied to both words. All letters in each word are used exactly once; there is no substitution with new letters. Concept / Approach: To identify the pattern, we compare MATERIAL and RIALMATE position by position. MATERIAL has eight letters: M A T E R I A L. The coded word RIALMATE also has eight letters: R I A L M A T E. Observing this, we see that the coded word starts with the last four letters of the original (R I A L) followed by the first four letters (M A T E). So the rule is: take the last half of the word and write it first, then write the first half after it. We then apply this same last four then first four rule to REMEMBER. Step-by-Step Solution: Step 1: Write MATERIAL with positions: 1 M, 2 A, 3 T, 4 E, 5 R, 6 I, 7 A, 8 L. Step 2: Note that RIALMATE in the code is R I A L M A T E. Step 3: Recognise that R I A L correspond to positions 5, 6, 7 and 8 (R, I, A, L). Step 4: The remaining part M A T E corresponds to positions 1, 2, 3 and 4. Step 5: Conclude that the rule is: coded word = letters 5–8 followed by letters 1–4. Step 6: Now apply this rule to REMEMBER. Write REMEMBER with positions: 1 R, 2 E, 3 M, 4 E, 5 M, 6 B, 7 E, 8 R. Step 7: Select letters 5–8: M, B, E, R, which form "MBER". Step 8: Then append letters 1–4: R, E, M, E, which form "REME". Step 9: Combine them to get the coded form MBERREME. Verification / Alternative check: To verify, we can go back to MATERIAL and reapply the rule: letters 5–8 are R I A L, and letters 1–4 are M A T E. Combining them yields RIALMATE, exactly matching the example. Applying the same steps again to REMEMBER reproduces MBERREME, consistent with the rule. Checking the answer choices shows that only option C matches this pattern and uses all letters in their correct transformed positions. Why Other Options Are Wrong: Options A, B, D and E involve different orders of the letters. For example, REMEREBM moves letters in a different pattern and does not follow the strict last four then first four rule. Similarly, MBEREMER and MBERREEM alter the order of some letters in the second half. Because the coding is a specific rotation of halves, only the arrangement where letters 5–8 come first and letters 1–4 follow is valid, which is MBERREME. Common Pitfalls: A common mistake is to assume that letters are being reversed or alternately picked from the beginning and end, rather than splitting the word exactly in half. Another is to miscount positions when copying letters, especially for longer words. Carefully writing the original word with position numbers and then explicitly selecting the second half followed by the first half prevents such errors. Always check that the pattern used on one example works identically on the other words in the question. Final Answer: Using the same rearrangement that converts MATERIAL to RIALMATE, the word REMEMBER is written as MBERREME in the code language.

Question 12

In a certain code language, the word "RAINBOW" is written as "1987645" and the word "SNAP" is written as "3790". Using the same code, how is the word "PIANO" written?

Accepted Answer

Introduction / Context: This coding question is based on converting letters to digits according to a fixed mapping. We are given the codes for two words, RAINBOW and SNAP, and asked to determine the code for PIANO. The underlying mapping must be consistent for all letters, and once it is deduced from the examples, it can be applied to the new word. Recognising and reusing this mapping is the core skill tested here. Given Data / Assumptions: RAINBOW is coded as 1987645. SNAP is coded as 3790. The same letter always corresponds to the same digit. We need the code for PIANO. Leading zeros in the numerical code may be omitted in the final representation. Concept / Approach: First, we match each letter in RAINBOW with the corresponding digit. Then we do the same for SNAP and check for consistency. This gives us a mapping from letters to digits. Finally, we apply this mapping to PIANO. If the first letter is coded as 0, the resulting number will start with zero, which is often left out when writing the final code as a plain number. The provided answer options are four digit numbers, which suggests that such a leading zero might indeed be dropped. Step-by-Step Solution: Step 1: Write RAINBOW and its code together: R A I N B O W → 1 9 8 7 6 4 5. Step 2: From this, deduce the mapping: R → 1, A → 9, I → 8, N → 7, B → 6, O → 4, W → 5. Step 3: Now check SNAP → 3 7 9 0. Letters S, N, A, P map as S → 3, N → 7, A → 9, P → 0. Step 4: N and A already appeared in RAINBOW and match the earlier mappings N → 7 and A → 9, so the code is consistent. Step 5: We now map PIANO. Letters are P, I, A, N, O. Step 6: Use the mapping: P → 0, I → 8, A → 9, N → 7, O → 4. Step 7: The raw code for PIANO is therefore 0 8 9 7 4, that is 08974. Step 8: Since a leading zero is typically omitted in numerical notation, the code is written as 8974. Verification / Alternative check: You can quickly double check by reapplying the mapping to RAINBOW and SNAP and ensuring that you get 1987645 and 3790 respectively. Every letter keeps the same digit value across words. For PIANO, there is no new letter beyond P, I, A, N and O, all of which have already been assigned digits. Among the options, only 8974 matches the digit sequence obtained after omitting the leading zero, so the answer is unique and consistent. Why Other Options Are Wrong: Options 8976, 8947, 8977 and 8975 change one or more digits in ways that contradict the established mapping. For example, any code that does not end in 4 would misrepresent the letter O, which must map to 4. Similarly, any code that does not contain the digit 9 in the correct position would misrepresent A. These options therefore cannot be correct under the consistent letter digit mapping required by the code language. Common Pitfalls: A common mistake is to assume that each letter's digit is based on its alphabetical position instead of using the explicit mapping from the examples. Another error is to misread the digits of RAINBOW or SNAP, leading to an incorrect mapping for one or more letters. It is also easy to overlook a leading zero and attempt to keep five digits when the options are four digit numbers. Always derive the mapping carefully from the given data and make sure every letter is handled consistently before encoding a new word. Final Answer: Using the same code that maps RAINBOW and SNAP, the word PIANO is written as 8974.

Question 13

In a certain code language, the word "TERMITE" is written as "UDSLJSF". Using the same pattern of letter shifting, how is the word "MINISTER" written in that code language?

Accepted Answer

Introduction / Context: This is a letter shifting code question. The word TERMITE becomes UDSLJSF in the code language. These transformations are not random; each letter is shifted forward or backward in the alphabet according to a fixed alternating pattern. Once that pattern is found, it must be applied to the word MINISTER to obtain its coded form. The exercise tests precision in tracking letter positions and consistent application of a multi step pattern. Given Data / Assumptions: TERMITE is written as UDSLJSF. Target word: MINISTER. English alphabet positions are used implicitly: A = 1, B = 2, ..., Z = 26. The transformation is applied position wise across the word, not by rearranging the order of letters. Concept / Approach: To detect the pattern, each letter of TERMITE is compared with the corresponding letter of UDSLJSF. This reveals whether the letter is shifted forward or backward and by how many positions. Observing each pair shows an alternating scheme: first letter shifted forward by one, second letter shifted backward by one, third forward by one, fourth backward by one, and so on. More precisely: T→U (+1), E→D (-1), R→S (+1), M→L (-1), I→J (+1), T→S (-1), E→F (+1). We then apply the same +1, -1, +1, -1 alternating pattern to the letters of MINISTER in order. Step-by-Step Solution: Step 1: Confirm the pattern for TERMITE: T (20) → U (21): +1. E (5) → D (4): -1. R (18) → S (19): +1. M (13) → L (12): -1. I (9) → J (10): +1. T (20) → S (19): -1. E (5) → F (6): +1. Step 2: The rule is: position 1 +1, position 2 -1, position 3 +1, position 4 -1, position 5 +1, position 6 -1, position 7 +1. Step 3: Apply this to MINISTER (8 letters): M, I, N, I, S, T, E, R. Step 4: Position 1: M (13) +1 → N (14). Step 5: Position 2: I (9) -1 → H (8). Step 6: Position 3: N (14) +1 → O (15). Step 7: Position 4: I (9) -1 → H (8). Step 8: Position 5: S (19) +1 → T (20). Step 9: Position 6: T (20) -1 → S (19). Step 10: Position 7: E (5) +1 → F (6). Step 11: Position 8: continue the alternation, so this is a -1 step: R (18) -1 → Q (17). Step 12: Combine the coded letters: N H O H T S F Q. Verification / Alternative check: To verify, we can reapply the pattern to TERMITE and ensure that UDSLJSF is obtained, then check each step for MINISTER again. All shifts for MINISTER follow the same alternating +1, -1 pattern with no exception. Comparing with answer choices shows that only NHOHTSFQ matches the exact coded sequence we derived, so the answer is uniquely determined by the pattern. Why Other Options Are Wrong: Options A, B, D and E differ from NHOHTSFQ by one or more letters. For example, some have extra H or misplaced T and S, meaning the +1 and -1 shifts were not applied consistently. Because the pattern is strictly defined by the example TERMITE → UDSLJSF, any deviation from that alternating shift must be incorrect. Only option C maintains the correct positions and shifts for every letter. Common Pitfalls: A frequent mistake is to assume every letter is shifted by the same amount in the same direction, such as always +1 or always -1. Another is to miscount the alphabet positions or to forget to continue the alternation for the additional eighth letter in MINISTER. Writing out a small alphabet reference and carefully tracking plus or minus moves for each position helps avoid these errors. Always verify your pattern against the example before applying it to the new word. Final Answer: Following the same alternating +1 and -1 shifts as for TERMITE, the word MINISTER is written as NHOHTSFQ in the code language.

Question 14

In a certain code language, the word "MOTHER" is written as "NPUGDQ". Using the same letter shifting rule, how is the word "ORANGE" written in that code language?

Accepted Answer

Introduction / Context: This is another letter shifting code question. The word MOTHER becomes NPUGDQ in the code language. Our goal is to discover the rule used to shift each letter of MOTHER and then apply the same rule to the word ORANGE. This tests the ability to recognise position wise transformations and to apply them consistently to a different six letter word. Given Data / Assumptions: MOTHER is coded as NPUGDQ. Target word: ORANGE. Both words contain six letters. Each position in the original word is shifted by a fixed amount forward or backward in the alphabet to obtain the corresponding coded letter. Concept / Approach: We compare MOTHER and NPUGDQ letter by letter. M → N, O → P, T → U, H → G, E → D, R → Q. The first three letters are each shifted forward by one place: M to N, O to P, T to U. The last three letters are each shifted backward by one place: H to G, E to D, R to Q. So the rule is: for a six letter word, add 1 to the first three letters and subtract 1 from the last three letters. We then apply this rule directly to O R A N G E. Step-by-Step Solution: Step 1: Confirm the pattern using MOTHER: M (13) → N (14): +1. O (15) → P (16): +1. T (20) → U (21): +1. H (8) → G (7): -1. E (5) → D (4): -1. R (18) → Q (17): -1. Step 2: Now apply the same pattern to ORANGE: letters O, R, A, N, G, E. Step 3: First three letters +1: O (15) +1 → P (16). R (18) +1 → S (19). A (1) +1 → B (2). Step 4: Last three letters -1: N (14) -1 → M (13). G (7) -1 → F (6). E (5) -1 → D (4). Step 5: Combine all six coded letters to obtain P S B M F D. Verification / Alternative check: To verify, we can again check the original MOTHER example: the first three letters are all shifted up by one and the last three down by one. This pattern is consistent and simple. Applying it to ORANGE gives PSBMFD, which is present as option B. Recalculating the shifts for each letter confirms that no mistakes were made in counting alphabet positions, and none of the other options follow this exact split pattern of +1 then -1. Why Other Options Are Wrong: Option A (PSBMDF) swaps the positions of the last two letters compared to the correct PSBMFD. Option C and option D change one or more letters in ways that break the +1 then -1 pattern. Option E introduces additional changes that do not match the transformation shown by MOTHER → NPUGDQ. Since the coding rule is clearly defined by that original pair, only PSBMFD respects the same position wise shifts for ORANGE. Common Pitfalls: Common mistakes include assuming all letters are shifted in the same direction or miscounting forward and backward moves in the alphabet. Another pitfall is failing to notice that the first half and second half of the word follow different shift directions. Writing down the alphabet and marking the movement for each letter step by step prevents these errors. Always verify your rule on the given example before applying it elsewhere. Final Answer: Using the same rule that converts MOTHER to NPUGDQ, the word ORANGE is written as PSBMFD in the code language.

Question 15

In a code language, each letter of a word is replaced by a number to form a single continuous code. The word FATE is written as 61205 using this rule. Using the same coding pattern, what is the code for the word LEAN?

Accepted Answer

Introduction / Context: This question is about converting letters to numbers using a positional code. The word FATE is written as 61205 in the code language. The challenge is to figure out how each letter's position in the alphabet is reflected in the code, and then to use the same system to encode the word LEAN. This tests familiarity with alphabetical positions and careful grouping of digits. Given Data / Assumptions: FATE is written as 61205. Target word: LEAN. Letters are encoded based on their positions in the English alphabet. The final code is a single string formed by concatenating the numeric codes of each letter without spaces. Concept / Approach: To decode the pattern, compare FATE with 61205. The letters F, A, T and E are the 6th, 1st, 20th and 5th letters of the alphabet respectively. If we take these positions and write them in sequence as 6, 1, 20, 5 we get the string 61205, which matches the given code exactly. Therefore, the rule is simply to replace each letter by its alphabetical position and then write the numbers consecutively. We can apply this straightforward rule to LEAN. Step-by-Step Solution: Step 1: Confirm the pattern for FATE. Positions are F = 6, A = 1, T = 20, E = 5. Step 2: Concatenate these values: 6, 1, 20, 5 → 61205, matching the given code. Step 3: Now take the word LEAN. Identify each letter's position in the alphabet: L is the 12th letter → 12. E is the 5th letter → 5. A is the 1st letter → 1. N is the 14th letter → 14. Step 4: Write these numeric codes in order: 12, 5, 1, 14. Step 5: Concatenate them to form a single code string: 125114. Verification / Alternative check: To verify, we can reverse the process for FATE: splitting 61205 as 6, 1, 20, 5 leads back to F, A, T and E based on positions, confirming that the encoding rule is correct. For LEAN, splitting 125114 as 12, 5, 1, 14 gives back L, E, A and N. Checking the answer options shows that 125114 is present as option A and agrees perfectly with this reasoning. Why Other Options Are Wrong: Options B, C, D and E change one or more digits in the sequence, which would correspond to different alphabet positions and therefore different letters. For example, 125115 would map to L, E, A, O, not LEAN. Since the code must strictly follow the alphabetical positions of L, E, A and N, any deviation from 12, 5, 1 and 14 is incorrect. Only 125114 reproduces the intended word. Common Pitfalls: A typical mistake is to miscount the position of a letter, particularly for letters in the second half of the alphabet. Another common error is misgrouping the digits when reading or writing codes, especially for letters with two digit positions like 12 or 14. To avoid this, it helps to memorise or quickly reconstruct the alphabet with position numbers and to carefully check each letter's index before writing the final code string. Final Answer: Using alphabetical positions as the code, the word LEAN is written as 125114.

Question 16

In a coded arithmetic system, letters stand for operations as follows: A denotes division (/), B denotes multiplication (x), C denotes addition (+) and D denotes subtraction (-). Which one of the following coded statements is mathematically true whe…

Accepted Answer

Introduction / Context: This question mixes coding of operators with verification of equations. Each letter stands for a basic arithmetic operation, and we are given several candidate equations written in this coded form. Our task is to decode each equation into normal arithmetic, evaluate the left hand side and see which equation is actually true. This combines understanding of operator mapping with ordinary numerical calculation and the use of precedence rules. Given Data / Assumptions: A denotes division (/). B denotes multiplication (x). C denotes addition (+). D denotes subtraction (-). We have four main candidates: 1) 11B34A17D8A3 = 38/3 2) 6B18D26A13C7 = 173/13 3) 32C8A16D4 = -3/4 4) 9C9A9D9B9 = -71 We must find which equality holds true after decoding and correct evaluation. Concept / Approach: The method is straightforward but requires care. For each option, we replace A, B, C and D with /, ×, + and - respectively and write the resulting arithmetic expression. Then we evaluate the left hand side using standard precedence rules, and compare it with the right hand side value. The option where both sides match exactly is the correct answer. It is often efficient to start with the simplest looking expression to check for correctness quickly. Step-by-Step Solution: Step 1: Decode option D, which is shorter: 9C9A9D9B9 = -71. Step 2: Substitute operators: C → +, A → /, D → -, B → ×. Step 3: The left side becomes 9 + 9 / 9 - 9 × 9. Step 4: Apply precedence. First 9 / 9 = 1 and 9 × 9 = 81. Step 5: Substitute these results: 9 + 1 - 81. Step 6: Now evaluate from left to right: 9 + 1 = 10. Step 7: Then 10 - 81 = -71, which matches the right hand side exactly. Step 8: Therefore, option D is a true equation after decoding. Step 9: Briefly check that other options fail. For example, option A decodes to 11 × 34 / 17 - 8 / 3 which simplifies to 22 - 8/3 = 58/3, not 38/3. Step 10: Option B decodes to 6 × 18 - 26 / 13 + 7, which equals 108 - 2 + 7 = 113, not 173/13. Step 11: Option C decodes to 32 + 8 / 16 - 4, which equals 32.5 - 4 = 28.5, not -3/4. Verification / Alternative check: Recomputing the arithmetic for option D confirms that 9 / 9 is 1 and 9 × 9 is 81, and that 9 + 1 - 81 is indeed -71. The right hand side of the equation is -71, so the equality holds exactly. Because all other options give left hand sides that do not match their right hand side values, option D is uniquely correct. There is no alternative interpretation of the operator mapping that would change this conclusion. Why Other Options Are Wrong: In option A, the difference between 22 and 8/3 is 58/3, which does not equal 38/3. In option B, the left side equals 113, while the right side is a fraction around 13.3. Option C's left side is 28.5, not -0.75. Option E is an extra distractor that would decode to a different false equation. Since these values do not match their claimed right hand sides, they cannot be correct in this coded system. Common Pitfalls: A common error is to misremember the mapping of letters to operations, for instance swapping A and B or confusing C with D. Another is to ignore operator precedence and evaluate from left to right, which will change the result. To avoid this, it is helpful to rewrite each coded expression slowly with the proper symbols and type it mentally as an ordinary arithmetic expression before evaluating it step by step. Final Answer: The only coded equation that is true after decoding and evaluation is 9C9A9D9B9 = -71.

Question 17

In a certain coding scheme, each word is assigned a value based on the letters it contains. If A = 26 and the word X-RAY has the coded value 40, then what is the coded value of the word WHAT?

Accepted Answer

Introduction / Context: This question defines a letter based numerical code with two examples. We are told that A has a code value of 26 and that X-RAY has a total code of 40. Using this information, we must deduce the coding rule and then find the code for WHAT. This problem tests understanding of alphabetical positions and a simple transformation involving reversing the order of the alphabet to assign numeric values to letters. Given Data / Assumptions: Single letter A has code 26. Word X-RAY has code 40. Hyphens in words like X-RAY are not letters and do not get codes. The same coding rule is used consistently for all letters. We must find the code value for the word WHAT. Concept / Approach: In the standard alphabet, A has position 1, B has 2 and Z has 26. Here, however, A is assigned the value 26. This suggests that letter values are counted from the end of the alphabet: A gets 26, B gets 25, C gets 24 and so on down to Z which would get 1. Thus, each letter's code value is 27 minus its normal alphabetical position. For X-RAY, we sum the reversed positions of X, R, A and Y and check whether this equals 40. If it does, we apply the same reversed position rule to W, H, A and T in WHAT. Step-by-Step Solution: Step 1: Confirm the reversed alphabet rule using A. In normal order, A is 1st, so reversed value is 27 - 1 = 26, matching the given A = 26. Step 2: Compute reversed values for X-RAY. Normal positions: X = 24, R = 18, A = 1, Y = 25. Step 3: Reversed values are: X → 27 - 24 = 3, R → 27 - 18 = 9, A → 27 - 1 = 26, Y → 27 - 25 = 2. Step 4: Sum these values: 3 + 9 + 26 + 2 = 40, which matches the given code for X-RAY. Step 5: Now apply the same rule to WHAT: letters W, H, A and T. Step 6: Normal positions: W = 23, H = 8, A = 1, T = 20. Step 7: Reversed values: W → 27 - 23 = 4, H → 27 - 8 = 19, A → 27 - 1 = 26, T → 27 - 20 = 7. Step 8: Add these: 4 + 19 + 26 + 7. Step 9: 4 + 19 = 23, then 23 + 26 = 49, and 49 + 7 = 56. Verification / Alternative check: We can verify by checking that the pattern works on both A and X-RAY without any inconsistencies. The reversed values for X-RAY add up exactly to 40, and A's reversed index is 26. No other simple coding scheme involving letter values would fit both pieces of information so neatly. Recomputing the sum for WHAT confirms that 56 is correct, and among the answer options, only 56 matches this computation. Why Other Options Are Wrong: Options 52, 54, 58 and 60 do not equal the sum of the reversed positions of W, H, A and T. To obtain any of these values, one would have to miscount one or more letter positions or apply an inconsistent rule. Since the code is clearly defined by A and X-RAY, and 56 is uniquely determined by that rule for WHAT, the other options cannot be correct. Common Pitfalls: A frequent mistake is to assume that the letters use their normal alphabetical positions and then to sum those. Another is to misinterpret A = 26 as a special case for A only, instead of seeing it as a clue that the entire alphabet is reversed. Some may also forget to include all letters in the word WHAT or miscalculate the arithmetic sum. Carefully computing 27 minus each normal position and adding the values step by step prevents these errors. Final Answer: Using the reversed alphabet coding rule, the value of WHAT is 56.

Question 18

In a coded arithmetic expression, the letters represent operations as follows: P denotes division (÷), Q denotes multiplication (x), R denotes addition (+) and S denotes subtraction (-). Using this code, what is the value of 18Q12P4R5S6?

Accepted Answer

Introduction / Context: This question presents an arithmetic expression where each letter stands for a basic operation. P, Q, R and S are used in place of division, multiplication, addition and subtraction. To find the value of the coded expression 18Q12P4R5S6, we must translate it into a normal numerical expression and then evaluate it using standard precedence rules. This tests careful symbol decoding and accurate arithmetic. Given Data / Assumptions: P denotes division (÷). Q denotes multiplication (x). R denotes addition (+). S denotes subtraction (-). Expression to evaluate: 18Q12P4R5S6. Standard arithmetic precedence applies after substitution. Concept / Approach: The problem is solved in two main stages. First, we replace P, Q, R and S with their corresponding standard operators to obtain an ordinary arithmetic expression. Second, we evaluate this expression by performing multiplication and division first, from left to right, and then performing addition and subtraction. Failure to respect the correct order of operations will give a wrong result even if the substitution is done correctly. Step-by-Step Solution: Step 1: Start from the coded expression: 18Q12P4R5S6. Step 2: Substitute operators: Q → ×, P → ÷, R → +, S → -. Step 3: The translated expression is 18 × 12 ÷ 4 + 5 - 6. Step 4: Apply precedence. First compute 18 × 12 = 216. Step 5: Next compute 216 ÷ 4 = 54. Step 6: Substitute these results back to get 54 + 5 - 6. Step 7: Now perform addition and subtraction from left to right: 54 + 5 = 59. Step 8: Then 59 - 6 = 53. Step 9: Therefore, the value of 18Q12P4R5S6 is 53. Verification / Alternative check: Rewriting the expression with brackets for emphasis, (18 × 12 ÷ 4) + 5 - 6 shows clearly that 18 and 12 are multiplied, the result is divided by 4, and only then are 5 and 6 combined. Re-evaluating 18 × 12 = 216 and 216 ÷ 4 = 54, and then performing the remaining operations confirms that 53 is the final result. No other interpretation of the operator mapping is consistent with the definitions provided, so the answer is robust. Why Other Options Are Wrong: Options 64, 81 and 24 typically arise from common miscalculations, such as treating 12 ÷ 4 as 3 and then multiplying 18 by 3 incorrectly, or ignoring the order of operations. Option 72 might occur if someone multiplies 18 and 4 instead of dividing 216 by 4. Since the correct computation following proper substitution and precedence gives 53, all other options are incorrect. Common Pitfalls: Frequent mistakes include mixing up which letter stands for which operation, for example treating P as multiplication rather than division, or Q as addition instead of multiplication. Another pitfall is to perform all operations sequentially from left to right without first carrying out the multiplication and division. To avoid these errors, always write the fully translated expression, underline the multiplication and division steps, and only then evaluate step by step. Final Answer: After decoding the operators and applying correct arithmetic, the value of 18Q12P4R5S6 is 53.

Question 19

Some equations are solved according to a certain coding pattern. For example, 4 x 5 x 8 is written as 584 and 7 x 3 x 9 is written as 397. Using the same pattern, what is the code for 9 x 7 x 3?

Accepted Answer

Introduction / Context: This question uses a simple rearrangement code applied to three numbers. The expressions 4 x 5 x 8 and 7 x 3 x 9 are not being evaluated arithmetically; instead, the numbers are being rearranged to form a three digit code. Our task is to identify this rearrangement pattern and apply it to the numbers 9, 7 and 3. The problem tests pattern recognition more than numerical calculation. Given Data / Assumptions: 4 x 5 x 8 is coded as 584. 7 x 3 x 9 is coded as 397. The symbol x here is just a separator, not actual multiplication. The same rearrangement rule applies to all three number expressions. We must find the code for 9 x 7 x 3. Concept / Approach: Instead of computing products, we treat the three numbers as digits a, b and c. We then compare the given expressions and their codes digit by digit. For 4 x 5 x 8, the numbers are a = 4, b = 5, c = 8, and the code is 584. For 7 x 3 x 9, a = 7, b = 3, c = 9, and the code is 397. Observing these pairs suggests that the code is formed by placing the middle number first, the last number second and the first number third: (b, c, a). Once this rearrangement is identified, we simply apply it to 9, 7 and 3. Step-by-Step Solution: Step 1: For 4 x 5 x 8, note that the numbers are 4, 5 and 8. Step 2: The code 584 has digits 5, 8 and 4. This is the sequence middle, last, first. Step 3: For 7 x 3 x 9, the numbers are 7, 3 and 9. Step 4: The code 397 has digits 3, 9 and 7. Again, this is middle, last, first. Step 5: Conclude that the rule is: if the three numbers are a, b and c, the code is b c a. Step 6: Apply this to 9 x 7 x 3, where a = 9, b = 7, c = 3. Step 7: Arrange them as middle, last, first: 7, 3, 9. Step 8: Combine the digits to form the code 739. Verification / Alternative check: Checking the two examples again confirms that the pattern b c a is consistently used. There is no arithmetic involved and no other obvious rearrangement that would fit both 584 and 397 so neatly. Applying this rule to 9 x 7 x 3 gives 739, and among the options, only 739 matches this pattern. Thus the answer is unique and consistent with the given coding system. Why Other Options Are Wrong: Option 397 is the code for 7 x 3 x 9 and does not correspond to 9 x 7 x 3. Option 793 would be the arrangement c a b, and 973 would be a c b, neither of which follows the b c a pattern demonstrated in the examples. Option 937 similarly rearranges the digits in an unsupported order. Only 739 preserves the middle, last, first order required by the code. Common Pitfalls: Some students mistakenly try to multiply the numbers, since the symbol x is often used for multiplication, and then attempt to extract digits from the product. This leads away from the actual pattern, which is purely positional. Others may guess a reversal (c b a) or keep the original order (a b c) without carefully checking against the examples. Always compare all given codes and look for the simplest consistent rearrangement before assuming any arithmetic is involved. Final Answer: Using the same rearrangement pattern as in the examples, the code for 9 x 7 x 3 is 739.

Question 20

In a certain code language, each letter of a word is replaced by its position in the English alphabet, written in sequence without spaces. For example, "HARMS" is written as "81181319" and "FLUTE" is written as "61221205". Using this code, how is "W…

Accepted Answer

Introduction / Context: This question uses a direct letter to number coding based on alphabetical positions. Each letter of a word is replaced by its position in the alphabet and these numbers are written in order to form a single code string. We are told that HARMS and FLUTE are encoded this way and asked to determine the code for WOUND. This type of question tests basic knowledge of letter positions and careful concatenation of the resulting numbers. Given Data / Assumptions: HARMS is written as 81181319. FLUTE is written as 61221205. Each letter is replaced by its alphabetical position: A = 1, B = 2, C = 3, ..., Z = 26. The numeric codes for the letters are simply concatenated to form a single string. We must find the code for WOUND. Concept / Approach: First, verify the coding rule using the given examples. For HARMS, the letters H, A, R, M, S have positions 8, 1, 18, 13 and 19. Concatenating these gives 8 1 18 13 19 → 81181319, which matches the example. For FLUTE, the letters F, L, U, T, E have positions 6, 12, 21, 20 and 5, which produce 6 12 21 20 5 → 61221205. Once we are confident that this is the rule, we apply it to the letters W, O, U, N and D in WOUND. Step-by-Step Solution: Step 1: Confirm the pattern for HARMS: H → 8, A → 1, R → 18, M → 13, S → 19. Step 2: Writing these consecutively gives 81181319, which matches the example. Step 3: Confirm the pattern for FLUTE: F → 6, L → 12, U → 21, T → 20, E → 5. Step 4: Writing these in order gives 61221205, matching the example again. Step 5: Apply this rule to WOUND. Write the letters: W, O, U, N, D. Step 6: Find their positions: W is 23rd, O is 15th, U is 21st, N is 14th, D is 4th. Step 7: Replace each letter by its number: 23, 15, 21, 14, 4. Step 8: Concatenate these values: 231521144. Verification / Alternative check: Rechecking the positions of each letter (W, O, U, N, D) confirms that 23, 15, 21, 14 and 4 are correct. Concatenating these numbers in order produces exactly 231521144. Comparing with the answer choices shows that 231521144 appears as option C. No other option follows the simple position based rule applied to WOUND, so the answer is unique and fully consistent with the coding system. Why Other Options Are Wrong: Other options such as 21144152 or 21441514 rearrange digits or change them in ways that correspond to different letter positions or orders. They do not map directly to W, O, U, N and D when decoded. Option 171921415 would represent a completely different sequence of letters. Because the rule is straightforward and based purely on alphabetical order, any deviation from the exact positions of WOUND is incorrect. Common Pitfalls: Common mistakes include miscounting the position of letters near the end of the alphabet, such as W or Y, or inadvertently swapping the order of digits for letters with two digit positions. Another pitfall is trying to add or manipulate the numbers instead of simply concatenating them. To avoid these issues, it helps to write down the alphabet with position numbers or to memorise key anchor points like A = 1, M = 13 and Z = 26 and count carefully from these when needed. Final Answer: Using alphabetical positions as the code, the word WOUND is written as 231521144.

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