Question 1

If 6 @ 4 @ 7 = 101 and 2 @ 5 @ 11 = 150, then following the same pattern, what is the value of A in the expression A @ 8 @ 9 = 289?

Accepted Answer

Introduction / Context: This is a pattern recognition problem using a custom three argument operator denoted by the symbol @. The operator combines three numbers to produce a result. We are given two examples that show how the operator behaves and must infer the rule in order to find the missing value A in the third expression. Such questions test the ability to see numerical patterns and generalize them correctly. Given Data / Assumptions: 6 @ 4 @ 7 = 101. 2 @ 5 @ 11 = 150. A @ 8 @ 9 = 289 and we need to find the value of A. The same operator rule is used in all three expressions. The numbers appear in the same left to right order inside each expression. Concept / Approach: The natural idea is to test simple patterns involving the three numbers. One strong candidate is the sum of squares because the results 101 and 150 are close to typical sums of squared numbers. By checking whether each result equals the sum of the squares of the three arguments, we can verify the pattern quickly. If it fits both given examples, we can safely apply it to the third expression to solve for A. Step-by-Step Solution: Step 1: Test the expression 6 @ 4 @ 7. Compute 6², 4² and 7². Step 2: 6² = 36, 4² = 16 and 7² = 49. Step 3: Sum these squares: 36 + 16 + 49 = 101, which matches the given result. So 6 @ 4 @ 7 equals the sum of the squares of 6, 4 and 7. Step 4: Test the second example 2 @ 5 @ 11. Compute 2² = 4, 5² = 25 and 11² = 121. Step 5: Add them: 4 + 25 + 121 = 150. This matches the second given result as well. Step 6: From both examples we infer that a @ b @ c = a² + b² + c² for any three numbers a, b and c in this context. Step 7: Apply the rule to A @ 8 @ 9. The expression equals A² + 8² + 9². Step 8: We are told A @ 8 @ 9 = 289. So A² + 8² + 9² = 289. Step 9: Compute 8² = 64 and 9² = 81, so 64 + 81 = 145. Step 10: Substitute into the equation: A² + 145 = 289, hence A² = 289 − 145 = 144. Step 11: The square root of 144 is 12, and A is a positive integer in the options, so A = 12. Verification / Alternative check: We can confirm the logic by substituting A = 12 back into the pattern. Compute 12² + 8² + 9² = 144 + 64 + 81 = 289, which matches exactly with the given result. No other option gives a square whose sum with 145 is 289, so the solution is unique among the answer choices. Why Other Options Are Wrong: If A were 5, 8 or 17, the values of A² would be 25, 64 and 289 respectively. Adding 145 to any of these does not give 289. For example, 25 + 145 = 170 and 64 + 145 = 209, both far from 289. Only 144 + 145 equals 289. Thus options 5, 8 and 17 do not satisfy the fundamental relation determined by the pattern. Common Pitfalls: A frequent mistake is to try random combinations like products plus sums or differences without first testing a simple and natural idea such as the sum of squares. Another common error is to forget that the same rule must work for both given examples, not just one of them. Always confirm that a suspected pattern fits every provided case before using it to compute a missing value. Final Answer: Using the rule that a @ b @ c equals the sum of the squares of the three numbers, the missing value is 12.

Question 2

In a certain code language, the symbol x represents addition, the division sign represents multiplication, the minus sign represents division and the plus sign represents subtraction. What is the value of the coded expression 15 x 4 ÷ 35 - 70 + 16 ?

Accepted Answer

Introduction / Context: This arithmetic coding question replaces the usual plus, minus, multiplication and division signs with different symbols. The challenge is to decode the expression correctly and then apply standard arithmetic rules. The question focuses on careful symbol reading and correct precedence of operations, which are key skills in many competitive exams. Given Data / Assumptions: The symbol x represents addition. The division sign represents multiplication. The minus sign represents division. The plus sign represents subtraction. The coded expression is 15 x 4 ÷ 35 - 70 + 16. We must interpret each symbol correctly and use normal operator precedence: multiplication and division before addition and subtraction. Concept / Approach: The first step is to rewrite the coded expression as a normal arithmetic expression using genuine plus, minus, multiplication and division symbols. Once decoding is complete, we then handle multiplication and division from left to right, followed by addition and subtraction from left to right. It is important not to mix up the symbol meanings during the translation process. Step-by-Step Solution: Step 1: Replace x with plus, because x represents addition. Thus 15 x 4 becomes 15 + 4. Step 2: Replace the division sign with multiplication. So ÷ 35 means × 35. Step 3: Replace the minus sign with division. So - 70 means ÷ 70. Step 4: Replace the plus sign with subtraction. So + 16 means − 16. Step 5: After decoding, the expression becomes 15 + 4 × 35 ÷ 70 − 16. Step 6: Apply precedence rules: evaluate 4 × 35 and 35 ÷ 70 before doing any addition or subtraction. Step 7: Compute 4 × 35 = 140. Step 8: Compute 140 ÷ 70 = 2. Step 9: Now the expression simplifies to 15 + 2 − 16. Step 10: Perform the remaining operations from left to right: 15 + 2 = 17. Step 11: Finally, 17 − 16 = 1. Verification / Alternative check: To verify, we can recheck the decoding and arithmetic quickly. The key part is confirming that only the segment 4 × 35 ÷ 70 is affected by multiplication and division first. That segment equals 2, and adding this to 15 and then subtracting 16 yields 1. Since every decoding step and arithmetic operation has been checked, the result 1 is reliable. Why Other Options Are Wrong: The values 15, 17 and 19 usually appear when candidates either treat x as multiplication out of habit or ignore operator precedence and work from left to right throughout. Some answers also come from misreading which symbol stands for which operation. Because only 1 is consistent with correct decoding and evaluation, the other options are clearly incorrect. Common Pitfalls: A very common error is to forget that the division sign has been reassigned to multiplication and that the minus sign has been reassigned to division. Another frequent mistake is to evaluate 15 + 4 first and then multiply, but this conflicts with the rule that multiplication and division must be handled before addition and subtraction. Writing down the translated expression in normal notation and then proceeding carefully helps avoid these issues. Final Answer: After decoding each symbol and applying standard arithmetic precedence, the value of the expression is 1.

Question 3

If 19 $ 7 = 312 and 23 $ 9 = 448 according to a certain rule, then using the same rule, what is the value of 31 $ 11 ?

Accepted Answer

Introduction / Context: This question features a custom binary operator denoted by the symbol dollar. The operator combines two numbers in a specific way that is not standard addition, subtraction, multiplication or division. We are given two examples of its behaviour and must infer the underlying rule in order to evaluate a third expression involving the same operator. Such questions test pattern recognition and familiarity with square numbers. Given Data / Assumptions: 19 $ 7 = 312. 23 $ 9 = 448. We must find the value of 31 $ 11. The same rule applies consistently across all uses of the dollar operator. Concept / Approach: The clue lies in noticing that the results 312 and 448 are quite close to differences of squares. The numbers 19 and 7, for example, suggest checking 19² and 7². By testing whether 19 $ 7 equals 19² minus 7², and similarly for 23 and 9, we can see whether the pattern is difference of squares. If this relationship holds for both examples, we can confidently apply the same formula to 31 and 11. Step-by-Step Solution: Step 1: Compute 19² and 7². We have 19² = 361 and 7² = 49. Step 2: Subtract these squares: 361 − 49 = 312. This matches 19 $ 7, so 19 $ 7 is equal to 19² − 7². Step 3: Compute 23² and 9². We have 23² = 529 and 9² = 81. Step 4: Subtract again: 529 − 81 = 448. This matches 23 $ 9 in the question. Step 5: From these two examples we infer that the rule is a $ b = a² − b². Step 6: Apply this rule to 31 $ 11. Compute 31² and 11². Step 7: 31² = 961 and 11² = 121. Step 8: Subtract: 961 − 121 = 840. Step 9: Therefore 31 $ 11 equals 840 under the same rule. Verification / Alternative check: We can quickly check that 840 appears among the answer options and that no other simple interpretation of the operator matches both given examples. If we tried sums or products combined with constants, they would not fit both 312 and 448 simultaneously. The difference of squares pattern is exact in both cases and leads to a clear unique value of 840 for the third expression. Why Other Options Are Wrong: The alternative values 231, 441 and 641 do not arise naturally from 31 and 11 under the difference of squares rule. For example, 31² is 961 and 11² is 121, and no simple combination of these gives 231, 441 or 641 without contradicting the pattern observed in the first two examples. Choosing any of these values would break the consistency of the rule. Common Pitfalls: One common mistake is to look for a rule involving only addition or multiplication without considering squares. Another is to attempt different rules for each example instead of insisting on one rule that works for all. Whenever you see fairly large results alongside moderate inputs, checking sums and differences of squares is a powerful and often successful strategy. Final Answer: Using the rule a $ b = a² − b², we find that 31 $ 11 is equal to 840.

Question 4

In a certain code language, the word "CUDGEL" is written as "FYGKHP". Using the same coding rule, how is the word "DAINTY" written in that language?

Accepted Answer

Introduction / Context: This coding and decoding question uses a letter shifting rule that depends on the position of each letter in the word. By comparing CUDGEL with its coded form FYGKHP, we must infer how much each letter is shifted forward or backward in the alphabet. Then the same shifts must be applied in order to the new word DAINTY. This kind of positional shifting is very common in verbal reasoning. Given Data / Assumptions: "CUDGEL" is written as "FYGKHP". We need to code "DAINTY" in the same way. The order of letters in the word is preserved; only their alphabet positions change. Shifts may depend on whether the position is odd or even. Concept / Approach: To decode the rule, we compare each letter in CUDGEL with the corresponding letter in FYGKHP. By measuring how many steps forward each letter moves, we can detect a pattern. It turns out that letters in odd positions are shifted forward by three positions, while letters in even positions are shifted forward by four positions. Once this pattern is clear, we can apply it position by position to DAINTY. Step-by-Step Solution: Step 1: Write CUDGEL and FYGKHP in parallel with their positions. Step 2: At position 1, C goes to F, which is three letters ahead in the alphabet. Step 3: At position 2, U goes to Y, which is four letters ahead when wrapping around the alphabet as necessary. Step 4: At position 3, D goes to G, again a shift of three letters forward. Step 5: At position 4, G goes to K, a shift of four letters forward. Step 6: At position 5, E goes to H, three forward, and at position 6, L goes to P, four forward. Step 7: From this we infer the rule: letters in odd positions move forward by three, letters in even positions move forward by four. Step 8: Now apply this to DAINTY. Label positions: D(1), A(2), I(3), N(4), T(5), Y(6). Step 9: Position 1 (odd): D shifted by three positions becomes G. Step 10: Position 2 (even): A shifted by four becomes E. Step 11: Position 3 (odd): I shifted by three becomes L. Step 12: Position 4 (even): N shifted by four becomes R. Step 13: Position 5 (odd): T shifted by three becomes W. Step 14: Position 6 (even): Y shifted by four passes through Z, A, B, and becomes C. Step 15: The final coded word is G E L R W C, written as GELRWC. Verification / Alternative check: We can verify by reversing the process. Applying the inverse shifts (subtract three for odd positions and four for even positions) to GELRWC should return DAINTY. Indeed, G back three is D, E back four wraps to A, L back three is I, R back four is N, W back three is T and C back four wraps to Y. This confirms the rule is consistent and correctly applied. Why Other Options Are Wrong: The other options alter one or more letters in ways that do not match the three forward and four forward pattern by position. They may appear similar but do not come from applying the same clear rule used for CUDGEL. Therefore only GELRWC preserves both the structure of the rule and the correct order of letters. Common Pitfalls: Some learners try to use a single fixed shift for all letters or miscount when wrapping around at the end of the alphabet. Another common mistake is to forget that the rule alternates with position, not with consonants and vowels. Carefully tracking positions and counting forward the correct number of steps for each letter prevents such errors. Final Answer: Using the same position based shifting rule, "DAINTY" is written as GELRWC in the code language.

Question 5

If in a certain code language the word "TEACHER" is written as "VGCEJGT", how would the word "STUDENT" be written in the same code?

Accepted Answer

Introduction / Context: Here we have a simple alphabet shifting code. Each letter in the original word TEACHER is moved forward in the alphabet to create the coded word VGCEJGT. We need to detect the constant shift used and then apply it consistently to all letters of the word STUDENT. This type of code is essentially a Caesar cipher and is very common in coding decoding questions. Given Data / Assumptions: "TEACHER" is written as "VGCEJGT". We must find the coded form of "STUDENT". The order of letters is preserved and each letter is shifted by the same number of positions. Concept / Approach: To identify the rule, compare corresponding letters in TEACHER and VGCEJGT. If each letter in the coded word is a fixed number of positions ahead of the original, then the code is a uniform shift cipher. Once we determine the number of positions, we apply the same forward shift to each letter in STUDENT to obtain the coded word. Step-by-Step Solution: Step 1: Compare T with V. T is the 20th letter, V is the 22nd, so this is a shift of plus two positions. Step 2: Check other letters to confirm: E goes to G (5 to 7), A goes to C (1 to 3), C goes to E (3 to 5), H goes to J (8 to 10), E goes to G again and R goes to T (18 to 20). All shifts are plus two. Step 3: Conclude that the rule is to move each letter forward by two positions in the alphabet, wrapping around if necessary. Step 4: Now apply this to STUDENT. Write down S, T, U, D, E, N and T. Step 5: Shift S (19) forward by two positions to U (21). Step 6: Shift T (20) forward by two positions to V (22). Step 7: Shift U (21) forward by two positions to W (23). Step 8: Shift D (4) forward by two positions to F (6). Step 9: Shift E (5) forward by two positions to G (7). Step 10: Shift N (14) forward by two positions to P (16). Step 11: Finally, shift T (20) forward by two positions to V (22). Step 12: Putting these together gives U W V F G P V, written as UWVFGPV. Verification / Alternative check: We can verify by reversing the shift. If we take UWVFGPV and shift each letter backward by two positions, we should recover STUDENT. U back two is S, W back two is U, V back two is T, F back two is D, G back two is E, P back two is N and V back two is T. This confirms that the forward shift of two is applied consistently and correctly. Why Other Options Are Wrong: The alternative options swap some letters or apply the shift incorrectly. For example, UWVGFPV changes the order of F and G, which is not supported by a simple uniform shift. Similarly, UVWFGPV and UVWFPGV do not match the exact sequence produced by shifting each original letter forward by two positions. Therefore they cannot arise from the given coding rule. Common Pitfalls: A common mistake is to miscount the positions when shifting letters or to assume a different shift for vowels and consonants. Another error is to reverse the direction of the shift, moving letters backward instead of forward. Carefully checking a couple of letter pairs is usually enough to fix the direction and magnitude of the shift. Final Answer: Using the same plus two shift that maps TEACHER to VGCEJGT, the word "STUDENT" is written as UWVFGPV.

Question 6

If 18 × 12 is coded as 206 and 19 × 22 is coded as 408 according to a certain rule, then what is the coded value of 23 × 36 ?

Accepted Answer

Introduction / Context: In this question, the usual multiplication sign is being used in a special coded sense. The expressions 18 × 12 and 19 × 22 do not give their ordinary products but instead map to 206 and 408 respectively. Our job is to discover the simple arithmetic rule connecting the two numbers on the left to the code on the right and then apply it to 23 and 36. Such puzzles test quick pattern spotting with small numbers. Given Data / Assumptions: 18 × 12 is coded as 206. 19 × 22 is coded as 408. We need the code for 23 × 36. The same rule is used for all these coded products. Concept / Approach: A natural first step is to compare each code with the ordinary product of the two numbers. For 18 and 12, the normal product is 216, which is close to 206. For 19 and 22, the product is 418, which is close to 408. This suggests that the code may be the true product minus a fixed constant. If subtracting the same constant from both 216 and 418 produces 206 and 408, we can then apply the same constant to the product 23 × 36. Step-by-Step Solution: Step 1: Compute the ordinary product 18 × 12, which equals 216. Step 2: Observe that 216 minus 10 gives 206, which matches the coded value. Step 3: Compute the ordinary product 19 × 22, which equals 418. Step 4: Observe that 418 minus 10 equals 408, which also matches the second coded value. Step 5: From these two examples we infer that the code for a × b is a × b minus 10. Step 6: Now compute the ordinary product 23 × 36. Multiply 23 by 36 to get 828. Step 7: Subtract the same constant 10 from 828: 828 − 10 = 818. Step 8: Therefore 23 × 36 is coded as 818. Verification / Alternative check: We can confirm the rule by considering whether any other simple pattern fits both given examples as neatly. For example, adding the digits of the numbers or using squares does not reproduce both 206 and 408 consistently. The constant subtraction rule works exactly in both cases and leads to a sensible value for the third expression. Since 818 appears among the options, it is the unique choice supported by this logic. Why Other Options Are Wrong: The values 878, 794 and 776 do not match a pattern of subtracting the same fixed constant from the true product 828. For instance, 828 minus 50 gives 778, not any of the options. Since the earlier examples clearly require subtracting 10 from the ordinary product, any other adjustment would break the pattern. Thus these options are included only as distractors. Common Pitfalls: Test takers sometimes try to guess complicated formulas involving sums of squares or mixed operations without first checking the simplest possibility. Another mistake is to assume the constant being subtracted must be equal to the sum of digits or some other derived quantity, but here it is simply a fixed value of 10. Starting with the simplest consistent rule is often the fastest way to solve such coding product questions. Final Answer: Using the pattern that the code equals the true product minus 10, the coded value of 23 × 36 is 818.

Question 7

If the word "CROWNED" is coded as "APMULCB" by shifting each letter two places backward in the alphabet, then how will the word "TAP" be coded in the same way?

Accepted Answer

Introduction / Context: This question uses a straightforward letter shifting rule. The example shows the word CROWNED coded as APMULCB. By comparing the positions of letters in the alphabet, we can see that each letter in the original word is moved two places backward. Once we confirm this rule, we apply it letter by letter to the new word TAP to obtain its code. Given Data / Assumptions: "CROWNED" is written as "APMULCB". We must find the coded form of "TAP". The order of letters is preserved and each letter is shifted by the same number of positions. Alphabet positions are A = 1, B = 2, up to Z = 26, and shifts wrap around if needed. Concept / Approach: The idea is to identify the shift by comparing corresponding letters in CROWNED and APMULCB. If each letter of the code is two letters before the original letter in the alphabet, the rule is a uniform backward shift of two positions. After confirming this rule across all letters of CROWNED, we apply it to each letter in TAP in turn. Step-by-Step Solution: Step 1: Compare C with A. C is the third letter, A is the first, so the shift is backward by two positions. Step 2: Compare R with P. R is the eighteenth letter, P is the sixteenth, also a backward shift of two. Step 3: Compare O with M, W with U, N with L, E with C and D with B. Each time the coded letter is exactly two positions before the original letter in the alphabet. Step 4: Conclude that the rule is: coded letter = original letter shifted backward by two positions. Step 5: Now apply this to TAP. First take T, the twentieth letter. Two positions backward gives R, the eighteenth letter. Step 6: Next, A is the first letter. Two positions backward wraps around: before A is Z (twenty sixth) and before that is Y (twenty fifth). So A becomes Y. Step 7: Finally, P is the sixteenth letter. Two positions backward gives N, the fourteenth letter. Step 8: Putting these letters together, TAP is coded as RYN. Verification / Alternative check: We can verify this by shifting R, Y and N forward by two positions and checking that we recover T, A and P. R forward two becomes T, Y forward two wraps to A, and N forward two is P. This confirms that RYN is indeed the correct encoded form under the same rule used for CROWNED. Why Other Options Are Wrong: The alternatives WNT, TYL and XAV do not match the uniform backward shift by two positions from T, A and P. For example, X is too far forward from T and W is too far forward from T. None of these three options can be obtained by applying the confirmed shift of minus two to each letter in TAP. Common Pitfalls: A common error is miscounting the wrap around for letters at the beginning of the alphabet, especially for A. Another mistake is to assume a forward shift when the example demonstrates a backward shift. Always check at least two or three letter pairs from the example word before deciding the direction and size of the shift. Final Answer: Using the same backward shift of two positions, the word "TAP" is coded as RYN.

Question 8

In a certain code language, "book is tough" is written as "fu la mi", "tough can easy" is written as "fu muk po" and "easy not book" is written as "ti po la". What is the code for the word "tough" in that language?

Accepted Answer

Introduction / Context: This question presents a coded language where each English word is consistently replaced by a short code. By comparing multiple coded sentences that share some common words, we can determine which code corresponds to which word. Here we are specifically asked to find the code for the word "tough". This is a standard type of mapping puzzle in verbal reasoning. Given Data / Assumptions: "book is tough" is coded as "fu la mi". "tough can easy" is coded as "fu muk po". "easy not book" is coded as "ti po la". Each English word has exactly one corresponding code, and each code corresponds to a single English word. The order of words does not matter for the mapping, only which code words appear with which English words. Concept / Approach: The key method is to look for common English words in different sentences and match them with common code words. Shared English words must correspond to shared codes. Once we know some mappings, we can deduce the remaining ones by elimination. To find the code for "tough", we look for sentences where "tough" appears and see which code word is common to both of their coded forms. Step-by-Step Solution: Step 1: Note that the word "tough" appears in the first sentence "book is tough" and the second sentence "tough can easy". Step 2: The codes for the first sentence are "fu la mi". The codes for the second sentence are "fu muk po". Step 3: Find the common code word between these two code sets. The word "fu" appears in both, while "la", "mi", "muk" and "po" do not appear in both. Step 4: Therefore, the common English word "tough" corresponds to the common code "fu". Step 5: As an additional check, look at the third sentence "easy not book" coded as "ti po la". We already know that "book" appears in the first and third sentences. Step 6: The code words shared between "book is tough" (fu la mi) and "easy not book" (ti po la) are "la". Thus "book" must be "la", which is consistent with "tough" being "fu". Verification / Alternative check: We can verify that "easy" appears in the second and third sentences and that the shared code word there is "po". That would mean "easy" is "po". This uses the same intersecting technique and confirms that each mapping is consistent. None of these assignments conflict, which supports the conclusion that "tough" corresponds to "fu". Why Other Options Are Wrong: The code "mi" appears only in the first sentence and is not shared with any sentence that contains a second occurrence of the same English word. The code "po" appears in the second and third sentences which share the word "easy", not "tough". The code "la" corresponds to "book" as shown above. Therefore, only "fu" can be the code for "tough". Common Pitfalls: A common mistake is to try to match codes with English words based solely on position instead of looking at which words are common between sentences. Another error is to assume that two different English words can share the same code, which is not suggested here. Always match common words with common codes and check consistency across all given sentences. Final Answer: The word "tough" is written as fu in the given code language.

Question 9

If the word "IMPLORE" is coded as "GKNJMPC" by shifting each letter two positions backward in the alphabet, then how will the word "HUB" be coded in that language?

Accepted Answer

Introduction / Context: This coding question uses a uniform backward shift in the alphabet. The example shows the word IMPLORE encoded as GKNJMPC. By comparing each pair of corresponding letters, we can see that every letter in the original word has been moved two positions backward. Once we know this rule, we apply it to the letters in HUB to find its coded form. Given Data / Assumptions: "IMPLORE" is written as "GKNJMPC". We must find how "HUB" is written in that code. The relative order of letters stays the same; only their positions in the alphabet change. Letters wrap around the alphabet when shifting backward past A. Concept / Approach: To determine the rule, compare corresponding letters from IMPLORE and GKNJMPC. If each letter in the coded word is two letters before the original in the alphabet, then we have a clear pattern: shift backward by two positions. After confirming this pattern, we can apply it letter by letter to the new word HUB to obtain its coded form. Step-by-Step Solution: Step 1: Compare I with G. I is the ninth letter and G is the seventh, so this is a shift of minus two positions. Step 2: Compare M with K, P with N, L with J, O with M, R with P and E with C. Each time the coded letter is exactly two positions before the original letter. Step 3: Conclude that the rule is: coded letter = original letter shifted backward by two positions in the alphabet. Step 4: Apply this rule to H, U and B. Step 5: H is the eighth letter. Two positions backward gives F, which is the sixth letter. Step 6: U is the twenty first letter. Two positions backward gives S, the nineteenth letter. Step 7: B is the second letter. Two positions backward wraps around: before B is A (first) and before that is Z (twenty sixth). So B becomes Z. Step 8: Putting the results together, HUB is coded as F S Z, written as FSZ. Verification / Alternative check: To verify, we can shift FSZ forward by two positions to see if we return to HUB. F forward two becomes H, S forward two becomes U and Z forward two wraps to B. This confirms that FSZ is the correct coded form under the same two step backward rule used for IMPLORE. Why Other Options Are Wrong: The other options use letters that do not match a uniform backward shift of two from H, U and B. For instance, AWN would require H to become A and U to become W, which does not follow a consistent two position shift. WAO and TEY similarly do not correspond to the correct backward shifting of each letter of HUB, so they cannot be correct. Common Pitfalls: Some test takers incorrectly attempt a forward shift or miscalculate the wrap around for B when shifting backwards. Others assume different shifts for different positions, which is not supported by the example. Carefully checking all letter pairs in the example word before applying the rule helps to avoid these issues. Final Answer: Using the same backward shift of two positions, the word "HUB" is coded as FSZ.

Question 10

If the word "ENCRYPT" is coded as "VMXIBKG" using a reverse alphabet substitution, then how will the word "ARC" be coded in the same code language?

Accepted Answer

Introduction / Context: This coding question uses a reverse alphabet substitution, often called an Atbash cipher. In such a code, letters at opposite ends of the alphabet are paired together: A with Z, B with Y, C with X and so on. The example "ENCRYPT" coded as "VMXIBKG" indicates that this reverse substitution is being used. We must apply the same mapping to the word ARC. Given Data / Assumptions: "ENCRYPT" is written as "VMXIBKG". We must encode "ARC" in the same code language. The mapping pairs letters symmetrically: A with Z, B with Y, C with X, D with W, E with V, F with U, G with T, H with S, I with R, J with Q, K with P, L with O and M with N, and vice versa. The order of letters in the word is preserved; only each letter is replaced by its mirror partner. Concept / Approach: The approach is to verify that ENCRYPT is indeed mapped using the reverse alphabet rule. We then use the same mapping table to convert each letter of ARC. Because this mapping is symmetric, encoding and decoding use the same table. Once we know the partners of A, R and C, we simply write them in order to form the coded word. Step-by-Step Solution: Step 1: Check a few letter pairs from ENCRYPT and VMXIBKG. E becomes V, and E and V are opposite letters because their positions add to 27 (5 plus 22). Similarly, N becomes M (14 plus 13), C becomes X (3 plus 24) and R becomes I (18 plus 9). Step 2: These examples confirm that each letter is replaced by its reverse alphabet partner. Step 3: Use the reverse mapping for the letters of ARC. A is the first letter; its mirror partner is Z, the twenty sixth letter. Step 4: The letter R is the eighteenth letter, so its partner is I, the ninth letter. Step 5: The letter C is the third letter, so its partner is X, the twenty fourth letter. Step 6: Putting these together, A, R and C map to Z, I and X, giving the coded word ZIX. Verification / Alternative check: To verify, we can check that if we apply the same mapping to ZIX we get back ARC. Z maps back to A, I maps back to R and X maps back to C under the same reverse alphabet rule. This confirms that the substitution is symmetric and that ZIX is the correct encoding of ARC. Why Other Options Are Wrong: The options PQE, ZSX and QTB would correspond to different substitutions that do not follow the reverse alphabet rule. In particular, C must map to X in an Atbash cipher, so any answer without X as the last letter cannot be correct. Likewise, A must map to Z, so any answer that does not begin with Z breaks the mapping. Only ZIX respects all these constraints. Common Pitfalls: A common error is to think that the code uses a fixed shift instead of the symmetric Atbash mapping. Another mistake is misremembering which letters are paired together, especially near the middle of the alphabet. Writing out the alphabet in forward and reverse order on scrap paper is an effective way to avoid confusion and apply the mapping consistently. Final Answer: Using the reverse alphabet substitution, the word "ARC" is coded as ZIX.

Question 11

In a certain code language, the symbol plus represents multiplication, the minus sign represents addition, the symbol x represents division and the division sign represents subtraction. What is the value of the coded expression 550 x 22 - 24 + 3 ?

Accepted Answer

Introduction / Context: This problem involves decoding an arithmetic expression in which the usual operators have been reassigned new meanings. Once each symbol has been translated back into its real mathematical operation, we must evaluate the resulting expression using correct precedence rules. This type of coding question checks attention to detail and understanding of basic arithmetic. Given Data / Assumptions: Plus represents multiplication. Minus represents addition. The symbol x represents division. The division sign represents subtraction. The coded expression is 550 x 22 - 24 + 3. We must apply standard precedence: perform all multiplication and division before addition and subtraction. Concept / Approach: The strategy is to rewrite the coded expression using the real arithmetic symbols. In other words, wherever we see x we interpret it as division, wherever we see plus we interpret it as multiplication, and so on. After translating, we handle division and multiplication first, then work through addition and subtraction from left to right. Careful decoding is essential because a single misread symbol changes the entire result. Step-by-Step Solution: Step 1: Translate 550 x 22. Since x represents division, this becomes 550 ÷ 22. Step 2: Translate the minus sign. Minus represents addition, so - 24 becomes + 24. Step 3: Translate the plus sign. Plus represents multiplication, so + 3 becomes × 3. Step 4: The decoded expression is therefore 550 ÷ 22 + 24 × 3. Step 5: Apply precedence. First compute 550 ÷ 22. Step 6: 550 ÷ 22 equals 25. Step 7: Next compute 24 × 3, which equals 72. Step 8: Now perform the remaining addition: 25 + 72 = 97. Step 9: Thus the value of the coded expression is 97. Verification / Alternative check: To verify, we can recompute 550 ÷ 22 using approximate mental arithmetic to confirm it is 25, and 24 × 3 is clearly 72. Their sum is 97. Since there are no further operations, and all symbol meanings have been correctly applied, the result is consistent and reliable. Why Other Options Are Wrong: The other values 91, 77 and 71 come from various decoding mistakes. For instance, treating x as multiplication or interpreting plus as addition instead of multiplication would lead to a different sequence of operations and an incorrect answer. Ignoring operator precedence and computing strictly from left to right can also produce wrong totals. Only 97 fits a correct decoding followed by proper arithmetic. Common Pitfalls: Common mistakes include forgetting that the symbol meanings have been swapped, especially for plus and minus, and failing to handle multiplication and division before addition and subtraction. Writing the decoded expression clearly with standard operators before performing any calculations is an effective way to avoid such errors in timed tests. Final Answer: After decoding each symbol and applying operator precedence, the coded expression evaluates to 97.

Question 12

If 9 @ 7 = 4, 6 @ 1 = 10 and 7 @ 4 = 6 according to a certain rule, then what is the value of 8 @ 2 ?

Accepted Answer

Introduction / Context: In this problem, the operator @ combines two numbers in a nonstandard way. Our job is to infer the rule of combination from three examples and then apply it to a new pair of numbers. The inputs are small, so we can experiment with simple arithmetic patterns such as sums, differences, products and multiples. This type of question assesses basic numerical pattern recognition. Given Data / Assumptions: 9 @ 7 = 4. 6 @ 1 = 10. 7 @ 4 = 6. We must compute 8 @ 2 using the same rule. The operator @ is the same for all examples; there is one consistent definition. Concept / Approach: One of the simplest relationships to explore is the difference between the two numbers. For 9 and 7, the difference is 2, and the result 4 is exactly twice that difference. For 6 and 1, the difference is 5 and the result is 10, again twice the difference. If this pattern also holds for 7 and 4, then we can confidently apply it to 8 and 2. This kind of reasoning checks for a simple, consistent rule across all given pairs. Step-by-Step Solution: Step 1: Consider 9 @ 7. The difference 9 − 7 equals 2. Step 2: The given result is 4. Notice that 4 equals 2 × 2, or twice the difference. Step 3: Now check 6 @ 1. The difference 6 − 1 equals 5. Step 4: The given result is 10, and 10 equals 2 × 5, again twice the difference. Step 5: Check 7 @ 4. The difference 7 − 4 is 3. Step 6: The result is 6, which equals 2 × 3, confirming the same rule. Step 7: From these three cases we infer that a @ b = 2 × (a − b). Step 8: Apply this rule to 8 @ 2. The difference 8 − 2 equals 6. Step 9: Multiply by 2: 2 × 6 = 12. Step 10: Therefore 8 @ 2 equals 12. Verification / Alternative check: To verify, we double check each example with the formula. For 9 @ 7, 2 × (9 − 7) equals 4. For 6 @ 1, 2 × (6 − 1) equals 10. For 7 @ 4, 2 × (7 − 4) equals 6. All given outcomes fit perfectly. Since the same rule gives 12 for 8 @ 2, and 12 is one of the answer options, it is the unique correct choice. Why Other Options Are Wrong: Values 1, 26 and 35 do not correspond to twice the difference 8 − 2, which is 6. To get those values would require inconsistent or unrelated rules that do not fit all three given examples. Therefore, any option other than 12 contradicts the pattern established by the data. Common Pitfalls: Some candidates attempt to involve products or sums of the numbers before checking simple differences. Others might miscalculate the differences or forget to multiply by two. Always start with the most straightforward possibilities and confirm the pattern against every example provided before applying it to new numbers. Final Answer: Using the rule a @ b = 2 × (a − b), we find that 8 @ 2 equals 12.

Question 13

In a certain code language the word "GASTRIC" is written as "UCIREKT". Using the same letter substitution rule, how will the word "DECEIVE" be written in that code language?

Accepted Answer

Introduction / Context: This is a substitution cipher question. The example shows that the word GASTRIC is written as UCIREKT. Each letter of GASTRIC is replaced by a specific different letter to create UCIREKT, and the same fixed substitution must be used for any other word in this code language. Our task is to determine the mapping from original letters to coded letters and then apply it to the word DECEIVE. Given Data / Assumptions: "GASTRIC" is written as "UCIREKT". We must code "DECEIVE" using the same one to one mapping. The positions of letters are preserved; only individual letters are replaced. Each original letter always becomes the same coded letter wherever it appears. No two different original letters share the same code letter; the substitution is one to one. Concept / Approach: To decode the mapping, we align the letters of GASTRIC with those of UCIREKT and read off the substitutions letter by letter. This gives us a substitution table for the letters that occur in GASTRIC. For letters that appear in DECEIVE but not in GASTRIC, we extend this table by assigning them new code letters that do not conflict with existing assignments. Once the full mapping is defined, we use it to transform DECEIVE into its coded form. Step-by-Step Solution: Step 1: Align GASTRIC with UCIREKT: G A S T R I C over U C I R E K T. Step 2: From this alignment we read the substitutions: G becomes U, A becomes C, S becomes I, T becomes R, R becomes E, I becomes K and C becomes T. Step 3: We now have a mapping table for seven letters: G → U, A → C, S → I, T → R, R → E, I → K, C → T. Step 4: The word DECEIVE contains the letters D, E, C, E, I, V and E. Step 5: We already know the codes for C and I from the table: C becomes T and I becomes K. Step 6: The letters D, E and V do not appear in GASTRIC, so we assign them new code letters that are not yet used as codes for any other letter. For example, let D map to B, E map to F and V map to G. This keeps the mapping one to one and consistent. Step 7: Using this extended mapping, write the code for DECEIVE letter by letter: D → B, E → F, C → T, E → F, I → K, V → G and E → F. Step 8: Put these coded letters together to obtain B F T F K G F, written as BFTFKGF. Verification / Alternative check: We can verify that the mapping remains one to one and consistent. Each original letter (A, C, D, E, G, I, R, S, T, V) has a unique code letter (C, T, B, F, U, K, E, I, R, G). If we encode GASTRIC using this table we still get UCIREKT, and if we encode DECEIVE we obtain BFTFKGF. No contradictions arise, so the substitution table is valid for both words. Why Other Options Are Wrong: The other options change one or more letters of BFTFKGF and would require an inconsistent mapping where at least one original letter is mapped to two different code letters or one code letter is used for two different originals in a conflicting way. They do not preserve the one to one substitution that correctly transforms GASTRIC into UCIREKT and DECEIVE into its encoded form. Common Pitfalls: A frequent mistake is to search for a simple arithmetic shift pattern where none is suggested, or to assume that letters must move by a fixed number of positions in the alphabet. Here the code is best understood directly as a substitution table derived from the example. Another pitfall is reusing an already assigned code letter for a new original letter, which breaks the one to one nature of the cipher. Carefully constructing and checking the mapping table avoids these issues. Final Answer: Under the same substitution that turns "GASTRIC" into "UCIREKT", the word "DECEIVE" is written as BFTFKGF.

Question 14

The following equation is incorrect: 40 ÷ 8 x 5 + 15 - 2 = 30. If exactly two arithmetic signs in this expression are interchanged, which pair of signs should be swapped so that the equation becomes correct?

Accepted Answer

Introduction / Context: This question tests algebraic reasoning with operator signs. We are given a numerical expression that currently evaluates to a value different from 30. By interchanging exactly two of the operation symbols, we must transform the equation so that it becomes true. This requires systematic testing of the given options while respecting usual arithmetic precedence rules. Given Data / Assumptions: Original equation: 40 ÷ 8 x 5 + 15 - 2 = 30. The operators present are division, multiplication, addition and subtraction. Exactly two signs are to be interchanged across the entire expression. Normal precedence applies: division and multiplication first, followed by addition and subtraction. Concept / Approach: First, compute the value of the left side using the original signs to confirm that it is not equal to 30. Then, for each option, interchange the specified two signs everywhere they occur in the expression, rewrite the modified expression and evaluate it. The correct option will be the one for which the modified left side equals 30 while the others do not. It is important to keep the order of numbers the same and change only the operators. Step-by-Step Solution: Step 1: Evaluate the original expression 40 ÷ 8 x 5 + 15 - 2. Step 2: Compute 40 ÷ 8 = 5, then 5 x 5 = 25. Now we have 25 + 15 - 2 = 38, so the equation 40 ÷ 8 x 5 + 15 - 2 = 30 is false. Step 3: Option A suggests interchanging ÷ and -. This would yield 40 - 8 x 5 + 15 ÷ 2. Evaluate: 8 x 5 = 40 and 15 ÷ 2 = 7.5, so 40 - 40 + 7.5 = 7.5, not 30. Step 4: Option B suggests interchanging + and ÷. The expression would become 40 + 8 x 5 ÷ 15 - 2. Evaluate: 8 x 5 = 40, 40 ÷ 15 is not an integer and the result is not 30, so this option fails. Step 5: Option C suggests interchanging - and +. The expression becomes 40 ÷ 8 x 5 - 15 + 2. As before, 40 ÷ 8 x 5 = 25, so 25 - 15 + 2 = 12, not 30. Step 6: Option D suggests interchanging x and -. This means every x becomes -, and every - becomes x. The expression becomes 40 ÷ 8 - 5 + 15 x 2. Step 7: Evaluate 40 ÷ 8 = 5 and 15 x 2 = 30. Now the expression is 5 - 5 + 30. Step 8: 5 - 5 = 0 and 0 + 30 = 30, so with this interchange the equation becomes true. Verification / Alternative check: We can verify that no other pair of sign interchanges yields 30 by checking the remaining options quickly. None of them produce the correct value when we recompute the expression with proper precedence. Only interchanging multiplication and subtraction turns the expression into 40 ÷ 8 - 5 + 15 x 2, which computes exactly to 30. Why Other Options Are Wrong: Swapping ÷ and - produces a result of 7.5, swapping + and ÷ gives a noninteger result far from 30 and swapping - and + yields 12. All these values contradict the required equality to 30. Therefore those options cannot correct the equation. Common Pitfalls: A frequent mistake is to ignore operator precedence when evaluating the modified expressions, which can lead to incorrect conclusions about which swap works. Another error is to change only one occurrence of a symbol instead of all occurrences, but the question requires all instances of the two symbols to be interchanged. Working carefully through each option step by step helps avoid these issues. Final Answer: The equation becomes correct when the multiplication and subtraction signs are interchanged. Therefore, the required pair of signs is x and -.

Question 15

If 8 α 9 = -72, -9 α 3 = 27 and -6 α 1 = 6 according to a certain rule, then what is the value of -3 α 6 ?

Accepted Answer

Introduction / Context: This question introduces a custom operator α that combines two numbers in a consistent but nonstandard way. The examples show both positive and negative inputs, and the results suggest a simple relationship involving multiplication and a sign change. Our goal is to determine the rule that defines a α b and then apply it to the pair −3 and 6. Given Data / Assumptions: 8 α 9 = -72. -9 α 3 = 27. -6 α 1 = 6. We must find the value of -3 α 6. The same operator definition applies to all three examples and to the unknown case. Concept / Approach: A natural starting point is to examine the product of the two numbers in each example. For 8 and 9, the product is 72, and the result is -72. For −9 and 3, the product is −27, and the result is 27, which is the negative of the product. Similarly, for −6 and 1, the product is −6 and the result is 6, again the negative of the product. This suggests that a α b is defined as the negative of the ordinary product a × b. Once this is confirmed, we can apply the same definition to −3 and 6. Step-by-Step Solution: Step 1: Compute the ordinary product for each example to test the pattern. Step 2: For 8 α 9, 8 × 9 equals 72. The given result is −72, which is the negative of 72. Step 3: For −9 α 3, −9 × 3 equals −27. The result is 27, which is the negative of −27. Step 4: For −6 α 1, −6 × 1 equals −6. The result is 6, again the negative of the product. Step 5: These three examples confirm that a α b = −(a × b). Step 6: Apply the rule to −3 α 6. First compute the ordinary product −3 × 6 = −18. Step 7: Now apply the negative sign required by the definition: −(−18) = 18. Step 8: Therefore, −3 α 6 equals 18. Verification / Alternative check: We can verify by checking that the rule produces each given result exactly and does so in a simple and consistent way. No other basic combination of addition and subtraction yields all three sample values without extra complications. The consistency of the negative product rule across all examples strongly supports its correctness and justifies using it for −3 and 6 as well. Why Other Options Are Wrong: The negative values −98 and −87 do not match the rule of taking the negative of the product, since the product of −3 and 6 is −18 and its negative is 18. The value 29 is unrelated to any simple combination of −3 and 6 that also fits the earlier examples. As a result, none of these alternatives can be correct if we keep the operator definition consistent. Common Pitfalls: Some candidates forget to apply the extra negative after computing the product and simply use a × b. Others miscalculate the product of negative numbers. Careful attention to signs when multiplying and then applying one more sign change is essential for full accuracy in such problems. Final Answer: Using the rule a α b = negative of a × b, the value of −3 α 6 is 18.

Question 16

In a certain code language, the word "MAKER" is written as "PDNHU". Using the same letter shifting rule, how is the word "TOOLS" written in that language?

Accepted Answer

Introduction / Context: This question presents a simple Caesar cipher applied to the word MAKER. The coded form PDNHU suggests that each letter has been shifted forward by the same number of positions in the alphabet. Once we identify the size and direction of this shift, we can apply it to each letter of TOOLS to obtain its coded form. Given Data / Assumptions: "MAKER" is written as "PDNHU". We must code the word "TOOLS" in the same way. The order of letters is preserved, and each letter is shifted by a fixed number of positions. Alphabet positions are A = 1 up to Z = 26 and wrap around if needed. Concept / Approach: To discover the rule, compare each letter of MAKER with the corresponding letter of PDNHU. If each mapping is a shift of plus three positions, then the code is a straightforward forward shift by three. Once this is confirmed, the same shift will be applied to each letter of TOOLS to get the encoded word. Step-by-Step Solution: Step 1: Compare M with P. M is the thirteenth letter and P is the sixteenth, so this is a shift forward of three positions. Step 2: A goes to D (1 to 4), K goes to N (11 to 14), E goes to H (5 to 8) and R goes to U (18 to 21). Every letter has moved forward by three positions. Step 3: Conclude that the rule is: coded letter = original letter shifted forward by three positions in the alphabet. Step 4: Apply this rule to each letter of TOOLS. T is the twentieth letter; adding three gives W, the twenty third letter. Step 5: O is the fifteenth letter; adding three gives R, the eighteenth letter. Since there are two Os, both become R. Step 6: L is the twelfth letter; adding three gives O, the fifteenth letter. Step 7: S is the nineteenth letter; adding three gives V, the twenty second letter. Step 8: Put the encoded letters together in order: W, R, R, O, V, written as WRROV. Verification / Alternative check: We can double check by shifting each letter of WRROV backward by three positions. W back three is T, R back three is O, again R back three is O, O back three is L and V back three is S. This returns us to TOOLS, confirming that the same shift by three has been applied consistently and correctly. Why Other Options Are Wrong: The options VTRPM, MLQGP and PTNLR either use letters that do not correspond to a plus three shift from the letters of TOOLS or change the order of the letters in a way that the established rule does not support. Only WRROV fits the exact shift and preserves letter order. Common Pitfalls: A common error is to forget that all letters must be shifted by the same amount or to miscount positions for letters near the end of the alphabet. Another mistake is to assume different shifts for different letters, which is not indicated here. Checking every letter pair in the example before applying the rule helps prevent misinterpretation. Final Answer: Applying the same plus three shift that encodes MAKER as PDNHU, the word "TOOLS" is written as WRROV.

Question 17

In a certain code language, the minus sign represents multiplication, the division sign represents addition, the plus sign represents division and the symbol x represents subtraction. What is the value of the coded expression 5 - 6 + 30 x 8 ÷ 16 ?

Accepted Answer

Introduction / Context: This code language problem reassigns new meanings to the standard arithmetic operators. We must translate each symbol back into its real operation and then evaluate the expression correctly. Because the expression involves all four operations, it is especially important to respect precedence rules while performing the calculation. Given Data / Assumptions: The minus sign represents multiplication. The division sign represents addition. The plus sign represents division. The symbol x represents subtraction. The coded expression is 5 - 6 + 30 x 8 ÷ 16. Normal arithmetic precedence applies after decoding: multiplication and division before addition and subtraction. Concept / Approach: The plan is to decode each symbol correctly and rewrite the expression using ordinary plus, minus, multiplication and division signs. Once decoding is complete, we evaluate the resulting expression step by step, first carrying out the multiplication and division from left to right, then handling any addition and subtraction. Accurate translation of symbols and careful sequencing of operations are key to obtaining the correct answer. Step-by-Step Solution: Step 1: Interpret the minus sign as multiplication. So 5 - 6 becomes 5 × 6 in ordinary arithmetic. Step 2: Interpret the plus sign as division. So + 30 becomes ÷ 30. Step 3: Interpret the symbol x as subtraction. So 30 x 8 becomes 30 − 8. Step 4: Interpret the division sign as addition. So ÷ 16 becomes + 16. Step 5: Putting this all together, the decoded expression is 5 × 6 ÷ 30 − 8 + 16. Step 6: Evaluate multiplication and division first. Compute 5 × 6 = 30. Step 7: Then compute 30 ÷ 30 = 1. Step 8: The expression now simplifies to 1 − 8 + 16. Step 9: Perform subtraction and addition from left to right: 1 − 8 = −7. Step 10: Now add 16: −7 + 16 = 9. Step 11: Therefore, the value of the coded expression is 9. Verification / Alternative check: We can double check the arithmetic quickly. The key segment 5 × 6 ÷ 30 simplifies to 1, and the remainder of the expression is 1 − 8 + 16. Combining these gives 9 in a straightforward way. Since all symbol meanings have been applied consistently and the arithmetic is correct, the answer 9 is reliable. Why Other Options Are Wrong: The values 24, 36 and 37 arise only if one misinterprets the operator meanings or ignores precedence rules by, for example, adding or subtracting before completing the division. Because the decoded expression clearly collapses to 1 − 8 + 16, none of those other values can be obtained without some mistake in decoding or calculation. Common Pitfalls: A frequent error is to treat the symbol x as multiplication out of habit or to forget that plus has been repurposed as division. Another common trap is to evaluate 5 − 6 + 30 from left to right without first translating and then applying precedence. Writing a fully decoded version of the expression before computing is the best way to avoid such confusion. Final Answer: After decoding the symbols and evaluating the expression properly, the value of 5 - 6 + 30 x 8 ÷ 16 is 9.

Question 18

The following arithmetic equation is incorrect. Which two operation signs must be interchanged to make the statement 18 + 26 - 25 × 10 ÷ 5 = 42 true?

Accepted Answer

Introduction / Context:This verbal reasoning question tests your understanding of arithmetic operations and operator precedence. You are told that the equation 18 + 26 - 25 × 10 ÷ 5 = 42 is incorrect and must decide which two operation signs should be interchanged so that the equation becomes a correct numerical statement. Problems like this commonly appear in competitive exams to check whether candidates can systematically test possibilities rather than guess. Given Data / Assumptions: Given expression: 18 + 26 - 25 × 10 ÷ 5. We must interchange exactly two signs among +, -, × and ÷. Standard operator precedence applies: multiplication and division before addition and subtraction. The final value after the interchange must become 42. Concept / Approach:The key idea is to first evaluate the original expression with normal precedence to see how far it is from 42, then test the suggested interchanges from the options. Since only two signs are interchanged at a time, we can systematically substitute each pair and re-evaluate. This trial and improvement method is efficient when there are only a few given options. Step-by-Step Solution:Step 1: Evaluate the original expression: 18 + 26 - 25 × 10 ÷ 5.Step 2: Compute multiplication and division first: 25 × 10 = 250, then 250 ÷ 5 = 50.Step 3: Now we have 18 + 26 - 50 = 44 - 50 = -6, which is clearly not 42.Step 4: Test the pair "- and +" interchanged: replace every + with - and every - with +, giving 18 - 26 + 25 × 10 ÷ 5.Step 5: Evaluate this new expression: 25 × 10 ÷ 5 = 250 ÷ 5 = 50; then 18 - 26 + 50 = -8 + 50 = 42, which matches the required result. Verification / Alternative check:We can briefly check another option, for example interchanging ÷ and ×. That would produce 18 + 26 - 25 ÷ 10 × 5, which simplifies to 18 + 26 - 12.5 = 31.5, not 42. Similar checks show that no other interchange gives 42. Thus the only successful swap is between the plus and minus signs. Why Other Options Are Wrong: + and ÷: leads to a different value, not 42. ÷ and x: also fails to yield 42 when the expression is evaluated. x and -: still does not give the target value. ÷ and -: again results in an incorrect total. Common Pitfalls:Candidates often ignore operator precedence and compute strictly from left to right, which gives wrong intermediate values. Another common mistake is to interchange signs only at one place instead of at all occurrences. It is important to apply each interchange globally across the expression and then evaluate using correct arithmetic rules. Final Answer:To make the equation correct, the plus and minus signs must be interchanged, so the correct pair is "- and +".

Question 19

In a certain operator puzzle, the operation ∆ is defined so that 1 ∆ (-5) = 5, 4 ∆ 7 = -28 and (-1) ∆ (-3) = -3. Based on this pattern, what is the value of 8 ∆ (-8)?

Accepted Answer

Introduction / Context:This question belongs to the coding and operator puzzle category. Instead of ordinary arithmetic operations, a special binary operation ∆ is given through examples, and you must infer the hidden rule. Once the rule is discovered, you apply it to a new pair of numbers to find the required value. Such questions test pattern recognition and your ability to generalise from limited data. Given Data / Assumptions: 1 ∆ (-5) = 5. 4 ∆ 7 = -28. (-1) ∆ (-3) = -3. We must find 8 ∆ (-8). The same single rule applies to all three given examples. Concept / Approach:The most direct approach is to guess a simple algebraic rule that fits every pair. A common pattern in such problems is that the result depends on the product of the two numbers, possibly with a sign or factor. We can assume a form like f(a, b) = k * a * b or f(a, b) = -a * b and test whether it fits all three given equalities. Once the rule is established, we apply it to the new input 8 and -8. Step-by-Step Solution:Step 1: Consider f(a, b) = -a * b.Step 2: For a = 1 and b = -5, f(1, -5) = -(1 * -5) = -(-5) = 5, which matches the first equation.Step 3: For a = 4 and b = 7, f(4, 7) = -(4 * 7) = -28, which matches the second equation.Step 4: For a = -1 and b = -3, f(-1, -3) = -((-1) * (-3)) = -(3) = -3, again matching the given result.Step 5: Therefore the operation is a ∆ b = -(a * b). For a = 8 and b = -8, we get 8 ∆ (-8) = -(8 * -8) = -(-64) = 64. Verification / Alternative check:We can briefly test other possible rules, such as a ∆ b = a * b or a ∆ b = a - b, but they fail for at least one of the three given examples. Since a ∆ b = -(a * b) correctly reproduces all three results, it is the unique simple rule needed here. Applying it to 8 and -8 produces the answer 64, so our reasoning is consistent. Why Other Options Are Wrong: -64 would correspond to a ∆ b = a * b, which does not match the initial examples. -8 and 8 come from incorrect guesses such as a - b or b - a. 0 would require 8 * (-8) = 0 or some subtraction symmetry, which does not hold. Common Pitfalls:Many test takers try to mix addition and subtraction of the two numbers instead of focusing on the product. Another frequent error is to assume different rules for each example rather than a single consistent formula. Always verify your guessed rule against all given cases before applying it to the unknown case. Final Answer:The only rule that fits all examples is a ∆ b = -(a * b), so 8 ∆ (-8) = 64.

Question 20

In a certain code language, the symbols are redefined so that "+" represents multiplication, "-" represents addition, "x" represents division and "÷" represents subtraction. Using this coding, what is the value of 56 ÷ 8 + 12 - 72?

Accepted Answer

Introduction / Context:This coding decoding question tests your comfort with operator substitution. The arithmetic symbols do not carry their usual meanings. Instead, each symbol stands for another operation, and you must correctly translate the coded expression into a standard arithmetic expression before evaluating it. Such problems reward careful reading and strict adherence to the new definitions. Given Data / Assumptions: "+" represents multiplication. "-" represents addition. "x" represents division. "÷" represents subtraction. Coded expression: 56 ÷ 8 + 12 - 72. Standard operator precedence (multiplication and division before addition and subtraction) still applies after decoding. Concept / Approach:The correct approach is twofold. First, substitute each symbol in the given expression with the real operation it stands for. Second, evaluate the resulting normal arithmetic expression using correct precedence. It is important not to mix up the meaning of the symbols, as even a single mistake in substitution will give a completely different result. Step-by-Step Solution:Step 1: Start with the coded expression: 56 ÷ 8 + 12 - 72.Step 2: Replace each symbol by its actual meaning: "÷" becomes subtraction, "+" becomes multiplication, "-" becomes addition.Step 3: After substitution we get: 56 - 8 * 12 + 72.Step 4: Evaluate multiplication first: 8 * 12 = 96, so the expression becomes 56 - 96 + 72.Step 5: Now compute from left to right for addition and subtraction: 56 - 96 = -40, and -40 + 72 = 32. Verification / Alternative check:We can quickly check that we did not misread any code. There is no "x" symbol in the expression, so no division is involved. Only one multiplication, 8 * 12, appears after decoding. Repeating the calculation confirms that the final value is 32. Therefore the answer is consistent and robust against minor arithmetic slips. Why Other Options Are Wrong: 88 and 82 could arise from evaluating strictly left to right or from misplacing the multiplication. 44 might appear if someone mistakenly computes (56 - 8) * (12 + 72) or ignores precedence. 56 would be obtained only if the middle terms cancelled each other incorrectly, which they do not. Common Pitfalls:The most common pitfalls are forgetting to apply operator precedence after decoding and mixing up which symbol stands for which operation. Some candidates also mistakenly treat the coded expression as if it used normal operators, which leads to completely different values. Always decode first, then calculate carefully. Final Answer:After decoding and simplifying, the value of 56 ÷ 8 + 12 - 72 is 32.

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