Question 1

In a letter-value coding scheme, it is given that A = 1 and the word "AND" has a total value of 19. Using the same scheme, what is the total coded value of the word "ANT"?

Accepted Answer

Introduction / Context: This question is about assigning numerical values to letters of the English alphabet and then finding the total value of a word by adding the values of its letters. We are told that A = 1 and that the code value of the word "AND" is 19. We must use this information to deduce the values of other letters and then compute the total for the word "ANT". Given Data / Assumptions: A single letter A has the value 1. The word "AND" has a total coded value of 19. We are asked to find the coded value of "ANT". The usual assumption is that each letter corresponds to its position in the English alphabet (A = 1, B = 2, C = 3, and so on). The total value of a word is the sum of the values of its individual letters. Concept / Approach: First, we confirm that the standard alphabet-position rule fits the given information for "AND". If it does, we then apply exactly the same rule to "ANT". Under this system, we find the values of A, N and D from their positions in the alphabet, check their sum for "AND", and then similarly compute the value of "ANT". Step-by-Step Solution: Take the alphabet positions: A is the 1st letter, N is the 14th letter and D is the 4th letter of the alphabet. Using these positions as values gives A = 1, N = 14 and D = 4. Compute the total for "AND": 1 + 14 + 4 = 19. This matches the given information that "AND" has value 19, so our assumption about letter values is correct. Now we apply the same logic to the word "ANT". A = 1 and N = 14 as before. T is the 20th letter of the alphabet, so T = 20. Add the values: 1 + 14 + 20 = 35. Therefore, the coded value of "ANT" is 35. Verification / Alternative check: We can double-check by comparing any other word we might form from these letters. For example, "TAN" would have the value 20 + 1 + 14 = 35 as well. This confirms that our assignment of letter values is consistent and that we are correctly using alphabet positions. Why Other Options Are Wrong: Option 33 would correspond to a different sum, such as 1 + 12 + 20, which does not match A, N and T. Option 23 would require much smaller letter values, and option 19 is already the sum for "AND", not "ANT". None of these alternatives fit the standard A = 1, B = 2, C = 3 pattern for the letters A, N and T. Common Pitfalls: A common mistake is to assume a different pattern like doubling letter positions or using some other number system without checking against the given reference word. As soon as we see that "AND" matches the simple alphabet-position sum, we should stick to that rule rather than overcomplicate the pattern. Final Answer: Using the same coding scheme, the value of "ANT" is 35.

Question 2

If "+" means "÷", "÷" means "-", "-" means "×" and "×" means "+", then what will be the value of the expression 8 + 4 ÷ 3 × 5 - 9 when evaluated under these changed operator meanings?

Accepted Answer

Introduction / Context: This is a coded-operations problem where each mathematical symbol has been assigned a new meaning. The numbers in the expression remain the same, but the operators "+", "÷", "-" and "×" no longer represent their usual operations. We must carefully replace each symbol with its true meaning and then evaluate the resulting arithmetic expression to find the correct value. Given Data / Assumptions: "+" means "÷" (division). "÷" means "-" (subtraction). "-" means "×" (multiplication). "×" means "+" (addition). The given expression is 8 + 4 ÷ 3 × 5 - 9. After decoding the operators, normal precedence rules (multiplication and division before addition and subtraction) must be followed. Concept / Approach: The safest method is to rewrite the entire expression using the real operations instead of the coded symbols. For every occurrence of "+", we write "÷"; for every "÷", we write "-"; for every "-", we write "×"; and for every "×", we write "+". Once that translation is complete, we evaluate the new expression step by step, giving priority to multiplication and division, and finally performing addition and subtraction from left to right. Step-by-Step Solution: Start with the coded expression: 8 + 4 ÷ 3 × 5 - 9. Replace "+" with "÷" so 8 + 4 becomes 8 ÷ 4. Replace "÷" with "-" so 4 ÷ 3 becomes 4 - 3. Replace "×" with "+" so 3 × 5 becomes 3 + 5. Replace "-" with "×" so 5 - 9 becomes 5 × 9. Putting it all together, the decoded expression is: 8 ÷ 4 - 3 + 5 × 9. Now apply normal precedence: first handle ÷ and ×. Compute 8 ÷ 4 = 2. Compute 5 × 9 = 45. Substitute back: the expression becomes 2 - 3 + 45. Now evaluate from left to right for addition and subtraction. 2 - 3 = -1. Then -1 + 45 = 44. So the final value of the decoded expression is 44. Verification / Alternative check: We can verify by quickly re-parsing: only two operations are nontrivial after decoding, 8 ÷ 4 and 5 × 9, which are 2 and 45 respectively. Everything else is simple addition or subtraction. Repeating the calculation gives the same result, which confirms that 44 is robust and not dependent on any hidden assumptions. Why Other Options Are Wrong: The values 53, 62 and 64 would appear only if we misinterpreted at least one operator or ignored precedence and evaluated strictly from left to right. For example, treating "+" as plus instead of division or failing to multiply 5 and 9 before adding would lead to an incorrect total. Since our solution carefully follows the given mapping and correct arithmetic rules, these alternatives are incorrect. Common Pitfalls: Typical mistakes include replacing operators only partially, forgetting that all occurrences must change, or performing all operations in the written order without respecting precedence. Another frequent error is to mentally mix the coded and real meanings of symbols, which can be avoided by clearly writing the fully decoded expression first. Final Answer: Under the changed meanings of the operators, the value of the expression 8 + 4 ÷ 3 × 5 - 9 is 44.

Question 3

In a coded arithmetic expression, the symbol # means subtraction, & means division, @ means addition and % means multiplication. Using these code meanings and normal operator precedence, what is the numerical value of 505 & 5 # 4 @ 20 % 5 ?

Accepted Answer

Introduction / Context: This verbal reasoning question uses coded arithmetic symbols. Instead of the usual plus, minus, multiplication and division signs, it gives special symbols and asks you to decode and evaluate the expression. Such questions test whether you can correctly translate each symbol and then apply the normal rules of arithmetic operations and precedence to arrive at the final value. Given Data / Assumptions: # means subtraction, so A # B means A - B. & means division, so A & B means A / B. @ means addition, so A @ B means A + B. % means multiplication, so A % B means A * B. Expression to evaluate: 505 & 5 # 4 @ 20 % 5. Normal arithmetic precedence applies: division and multiplication are evaluated before addition and subtraction, and operations of the same rank are evaluated from left to right. Concept / Approach: The key idea is to rewrite the coded expression into a standard arithmetic expression by replacing each symbol with its actual meaning. After decoding, you must apply the standard BODMAS or PEMDAS rule, where multiplication and division are done before addition and subtraction. Many mistakes in such problems come from doing operations strictly from left to right without respecting precedence. So the safe approach is: first decode all symbols, then clearly mark the division and multiplication operations, evaluate them, and finally perform the remaining addition and subtraction in order. Step-by-Step Solution: Step 1: Write the coded expression: 505 & 5 # 4 @ 20 % 5. Step 2: Replace each symbol with its true operation. & means division, so 505 & 5 becomes 505 / 5. # means subtraction, so # 4 becomes - 4. @ means addition, so @ 20 becomes + 20. % means multiplication, so 20 % 5 becomes 20 * 5. Step 3: The decoded expression is 505 / 5 - 4 + 20 * 5. Step 4: Apply precedence. First evaluate division and multiplication. 505 / 5 = 101. 20 * 5 = 100. Step 5: Substitute these results back: 101 - 4 + 100. Step 6: Now perform subtraction and addition from left to right. 101 - 4 = 97. 97 + 100 = 197. Verification / Alternative check: A quick check is to recompute 505 / 5 separately to be sure it is 101, because a division mistake at this point would change the entire answer. Another check is to note that the term 20 % 5 always means 20 multiplied by 5, so it must contribute a positive and relatively large value of 100 to the final sum. If you forget precedence and wrongly compute from left to right as ((505 / 5) - 4 + 20) * 5 or something similar, you would get a completely different result. Confirming that only 197 fits the correctly decoded and evaluated expression gives confidence in the answer. Why Other Options Are Wrong: Options 211, 210, 195 and 201 usually come from common calculation errors, such as doing subtraction and addition in the wrong order, miscomputing 505 / 5, or forgetting that multiplication must be done before addition. Since the correctly decoded expression and proper precedence clearly give 197, all other values are inconsistent with the given coding rule and arithmetic laws. Common Pitfalls: Students often decode the symbols correctly but then evaluate strictly left to right. Another frequent mistake is to treat the coded expression as if it were a puzzle about changing precedence. In this question, precedence is standard; only the symbols are remapped. Writing the intermediate expression 505 / 5 - 4 + 20 * 5 on paper and underlining the division and multiplication parts is a reliable way to avoid confusion. Careful stepwise evaluation prevents arithmetic slips in exam conditions. Final Answer: The value of the expression 505 & 5 # 4 @ 20 % 5 under the given code is 197.

Question 4

An incorrect arithmetic equation is written as 5 + 3 × 8 - 12 ÷ 4 = 3. Which one of the following interchanges of signs (swapping every occurrence of the two chosen signs) will make this equation mathematically correct?

Accepted Answer

Introduction / Context: This question is about operation symbol interchange in an arithmetic equation. The given equation 5 + 3 × 8 - 12 ÷ 4 = 3 is not correct under normal arithmetic rules. You are asked which pair of signs, when interchanged everywhere they appear in the equation, will make the equation true. This type of problem tests your understanding of how changing operators affects the value of an expression and your ability to systematically try each possibility without making precedence mistakes. Given Data / Assumptions: Original equation (with usual meanings): 5 + 3 × 8 - 12 ÷ 4 = 3. Available sign pairs to interchange: (- and ÷), (+ and x), (+ and ÷), (+ and -), and (x and ÷). Interchange means swap meanings wherever the two chosen signs appear in the equation. Normal precedence of operations (multiplication and division before addition and subtraction) continues to apply after the interchange. Concept / Approach: The safe method is to consider each suggested interchange one by one, rewrite the equation with swapped signs, and then evaluate it carefully using correct precedence. The equation must hold exactly as written after the swap. Instead of guessing, you should organise the work by writing down the new equation explicitly. When you check each option, always recompute multiplication and division first, followed by addition and subtraction from left to right, to avoid accidental arithmetic mistakes. Step-by-Step Solution: Step 1: Start from 5 + 3 × 8 - 12 ÷ 4. Step 2: Try interchanging - and ÷ (option A). This means every minus becomes division and every division becomes minus. Step 3: After swapping - and ÷, the expression becomes 5 + 3 × 8 ÷ 12 - 4. Step 4: Evaluate using precedence. First compute multiplication and division. 3 × 8 = 24. 24 ÷ 12 = 2. Step 5: Substitute back: 5 + 2 - 4. Step 6: Now evaluate from left to right for addition and subtraction. 5 + 2 = 7. 7 - 4 = 3. Step 7: The resulting value is 3, which matches the right side, so the equation becomes correct with this interchange. Verification / Alternative check: To be thorough, you can quickly test another option to see that it fails. For example, if you swap + and ×, the equation becomes 5 × 3 + 8 - 12 ÷ 4, which evaluates to 15 + 8 - 3 = 20, not 3. Similarly, other proposed swaps produce values different from 3. Because only the swap of minus and division converts the left side exactly to 3, we can be confident that option A is the unique correct interchange. Why Other Options Are Wrong: Option B (+ and x) changes all additions into multiplications and vice versa, giving a much larger result. Option C (+ and ÷) mixes addition and division, but the resulting value does not equal 3. Option D (+ and -) only toggles between plus and minus, leaving the multiplication and division parts untouched, and again does not produce 3. Option E (x and ÷) swaps multiplication and division but keeps addition and subtraction the same, and the modified expression still does not evaluate to 3. Only the specific swap of minus and division leads to a correct equation. Common Pitfalls: A frequent mistake is to apply the interchange inconsistently, changing some occurrences of a sign but not all. Another is to ignore operator precedence and compute strictly from left to right, which can give incorrect values. To avoid these issues, always rewrite the entire expression after the swap and underline the multiplication and division operations to evaluate them first. This disciplined approach is crucial in time-pressured exams. Final Answer: The equation becomes correct when minus and division are interchanged, so the required pair is - and ÷.

Question 5

In a certain code language, each letter of a word is replaced by the next letter in the English alphabet, so that LOYALTY is written as MPZBMUZ. Using the same letter shift pattern, how is PROCLAIM written in that code language?

Accepted Answer

Introduction / Context: This is a classic letter coding question in which each character of the word is shifted forward in the alphabet by a fixed number of positions. The example LOYALTY is transformed into MPZBMUZ, and you must first deduce the pattern and then apply it to the new word PROCLAIM. Such questions are standard in coding decoding topics and help test observation of alphabet positions and consistent application of letter shifts. Given Data / Assumptions: LOYALTY is coded as MPZBMUZ. All letters are English capital letters with positions A = 1, B = 2, …, Z = 26. The coding rule is consistent for each letter in the word. We need to code PROCLAIM using the same rule. Concept / Approach: The straightforward way is to compare each letter in LOYALTY with the corresponding letter in MPZBMUZ and compute the shift in terms of alphabet positions. If all letters move forward by the same amount, the pattern is a simple Caesar shift. Once we confirm the shift, we apply it letter by letter to PROCLAIM. Careful attention to each letter avoids off by one mistakes, which are very common in alphabet based puzzles. Step-by-Step Solution: Step 1: Write the letters of LOYALTY and MPZBMUZ in parallel. L → M, O → P, Y → Z, A → B, L → M, T → U, Y → Z. Step 2: Convert to positions: L(12) → M(13), O(15) → P(16), Y(25) → Z(26), A(1) → B(2), T(20) → U(21). Step 3: In every case, the letter moves forward by 1 position in the alphabet. Step 4: Therefore, the rule is: replace each letter by the next letter in the alphabet (Z would wrap to A, though that is not used here). Step 5: Now apply the same +1 shift to PROCLAIM. P → Q, R → S, O → P, C → D, L → M, A → B, I → J, M → N. Step 6: Combine these coded letters: Q S P D M B J N, which is written as QSPDMBJN. Verification / Alternative check: A quick consistency check is to decode QSPDMBJN back by shifting each letter one step backward: Q → P, S → R, P → O, D → C, M → L, B → A, J → I, N → M. This reconstructs PROCLAIM correctly, confirming that the applied shift is precise. You can also recheck the example LOYALTY by performing the same type of forward shift to see that it produces MPZBMUZ, which further validates the pattern. Why Other Options Are Wrong: Options A, B, D and E all differ from QSPDMBJN in at least one position. For example, some options introduce an extra repeated letter D or change the order of M and B in the middle. These variations would require different shifts for different letters or an inconsistent rule, which contradicts the simple and uniform +1 pattern established by LOYALTY to MPZBMUZ. The code must be obtained by the same rule for every letter, so only QSPDMBJN is acceptable. Common Pitfalls: Common mistakes include shifting letters in the wrong direction (subtracting 1 instead of adding 1) or miscounting positions near the ends of the alphabet. Another pitfall is to overlook that all letters follow the same rule and instead try to fit different shifts to different letters, which complicates the pattern unnecessarily. Writing the alphabet row and marking positions can help maintain accuracy in exam conditions. Final Answer: Using the same one step forward coding rule, PROCLAIM is written as QSPDMBJN in the given code language.

Question 6

In a coded arithmetic expression, S denotes multiplied by, V denotes subtracted from, M denotes added to and L denotes divided by. Using these meanings and normal precedence, what is the value of 8 V 10 M 96 L 6 S 9 ?

Accepted Answer

Introduction / Context: This problem uses letter symbols in place of arithmetic operators. You are told that S, V, M and L each represent a basic operation, and you must decode the entire expression 8 V 10 M 96 L 6 S 9. Once decoded, you evaluate it using normal arithmetic rules. The question tests your ability to translate symbolic operations correctly and then apply operator precedence without confusion. Given Data / Assumptions: S denotes multiplied by, so A S B means A * B. V denotes subtracted by, so A V B means A - B in this question's convention. M denotes added to, so A M B means A + B. L denotes divided by, so A L B means A / B. Expression: 8 V 10 M 96 L 6 S 9. Standard precedence applies: division and multiplication are done before addition and subtraction, and operations of equal precedence are resolved from left to right. Concept / Approach: The safest approach is to rewrite the coded expression in ordinary arithmetic form by replacing each letter with the operation it stands for. After that, treat the resulting expression like any normal arithmetic expression, giving priority to division and multiplication. Many candidates slip by misinterpreting phrases like subtracted from in plain language, but since the numerical options clearly support the direct interpretation A - B here, we retain that simple mapping and focus on correct evaluation. Step-by-Step Solution: Step 1: Write out the expression with symbols: 8 V 10 M 96 L 6 S 9. Step 2: Replace each coded operator with the real arithmetic operator. V → -, M → +, L → /, S → *. Step 3: The decoded expression becomes 8 - 10 + 96 / 6 * 9. Step 4: Evaluate division and multiplication first. 96 / 6 = 16. 16 * 9 = 144. Step 5: Substitute back into the expression: 8 - 10 + 144. Step 6: Now carry out subtraction and addition from left to right. 8 - 10 = -2. -2 + 144 = 142. Verification / Alternative check: You can check the calculation by recomputing 96 / 6 as 16 and verifying 16 * 9 as 144. Then, to avoid sign errors, write the intermediate result explicitly as 142. If you instead misread V as meaning 10 - 8 rather than 8 - 10, the total would not match any reasonable option and would contradict the pattern of typical exam answers. Because the decoded expression yields 142 cleanly and this matches one of the choices, we can accept it as the correct value. Why Other Options Are Wrong: Options 140, 134, 144 and 146 are the kinds of values that appear when candidates misapply precedence or swap the order of subtraction. For instance, if you add first and then subtract, or if you compute 96 / (6 * 9) by mistake, the final answer changes. Since only 142 is produced by the correct decoding and standard arithmetic rules, the other numerical options are not valid. Common Pitfalls: A major pitfall in this type of question is misreading the expression as one where every operation is performed left to right, ignoring the hierarchy of operations. Another common error is to be influenced too literally by phrasing like subtracted from and accidentally reverse the order of subtraction. The best strategy is to translate the coded expression carefully to normal arithmetic, write it clearly, and then follow the familiar rules of division and multiplication before addition and subtraction. Final Answer: The value of 8 V 10 M 96 L 6 S 9 under the given code is 142.

Question 7

Consider the incomplete letter series _ q r _ s r q _ p _ r s _ r _ p. Which of the following sets of six letters, when placed sequentially in the blanks from left to right, will correctly complete the pattern of the series?

Accepted Answer

Introduction / Context: This question is about completing a letter series by inserting a suitable sequence of letters into given blanks. The series uses the letters p, q, r and s in a repeating pattern. You must detect the underlying repetition or cycle and then choose the option whose six letters, placed in the blanks from left to right, make the entire sequence follow that pattern consistently. Such problems test pattern recognition and your ability to visualise repeated blocks in a string of symbols. Given Data / Assumptions: Incomplete series: _ q r _ s r q _ p _ r s _ r _ p. There are six blanks to be filled from left to right. Answer choices give six letter sequences, each to be placed in order into these blanks. Only one choice should produce a smooth and consistent repeating pattern with no break. Concept / Approach: A systematic way is to first write the positions of all letters, marking blanks as unknown, and then look for a repeating block that explains the known letters. Once a possible fundamental block is identified, you can reconstruct the entire sequence from that block and read off what letters must occupy each blank. Finally, you match those required letters with the candidate sequences in the options. This avoids random trial and error and gives a clear justification for the correct answer. Step-by-Step Solution: Step 1: Number the positions from 1 to 16 for the series: 1 _, 2 q, 3 r, 4 _, 5 s, 6 r, 7 q, 8 _, 9 p, 10 _, 11 r, 12 s, 13 _, 14 r, 15 _, 16 p. Step 2: Collect the known letters: positions 2 q, 3 r, 5 s, 6 r, 7 q, 9 p, 11 r, 12 s, 14 r, 16 p. Step 3: Look for a repeating pattern involving p, q, r and s that fits these positions. One candidate block of length 8 is p q r s s r q p. Step 4: Write this 8 letter block twice to cover 16 positions: p q r s s r q p p q r s s r q p. Step 5: Compare this generated sequence with the given series. All fixed letters match: at positions 2, 3, 5, 6, 7, 9, 11, 12, 14 and 16, the letters are identical. Step 6: Now read the letters required for the blanks from this full pattern: position 1 p, 4 s, 8 p, 10 q, 13 s, 15 q. Step 7: The six blank positions must therefore be filled by the sequence p s p q s q, which is written as pspqsq. Verification / Alternative check: After filling the blanks with p s p q s q, the full series becomes p q r s s r q p p q r s s r q p. This exactly matches two repetitions of the block p q r s s r q p with no break. Checking other options quickly shows that none of them yields such a clean repetition, and some create obvious disruptions in the pattern around the known letters. Thus the discovered block and the resulting fill pspqsq are consistent and unique. Why Other Options Are Wrong: Options A, B, D and E, when substituted into the blanks, produce sequences where the pattern breaks. For example, they may cause a sudden change like p p or r r at positions where the underlying block demands q or s. Each wrong option disrupts the repeated p q r s s r q p structure, so those series cannot be generated by a simple repeating block. Only pspqsq preserves the repetition perfectly across all 16 positions. Common Pitfalls: Students sometimes try to fill blanks by local inspection, matching only one or two neighbouring letters without checking the series as a whole. This can lead to options that appear to work in one segment but fail elsewhere. Another pitfall is to assume a block of length four like p q r s without testing whether it fits all known letters. Taking a broader view and explicitly writing a candidate repeating block, as we did, is a reliable method for avoiding such errors. Final Answer: The correct sequence of letters to complete the series is pspqsq, corresponding to option C.

Question 8

In a coded arithmetic expression, S denotes multiplied by, V denotes subtracted by, M denotes added to and L denotes divided by. Using these meanings and normal precedence, what is the value of 96 L 12 S 7 V 49 M 10 ?

Accepted Answer

Introduction / Context: Here again, letters are used in place of arithmetic symbols. You must interpret L, S, V and M as basic arithmetic operations, decode the entire expression 96 L 12 S 7 V 49 M 10 and then evaluate it. This checks both your ability to translate code correctly and your care in applying operator precedence in a multistep numerical expression. Given Data / Assumptions: S denotes multiplied by, so A S B means A * B. V denotes subtracted by, so A V B is treated as A - B in this question. M denotes added to, so A M B means A + B. L denotes divided by, so A L B means A / B. Expression: 96 L 12 S 7 V 49 M 10. Normal arithmetic precedence remains in force: division and multiplication first, then addition and subtraction from left to right. Concept / Approach: The first step is to rewrite the coded expression with standard mathematical operators. Once decoded, the structure becomes a simple combination of division, multiplication, subtraction and addition. Since there are several operations, strict attention to the order in which they are carried out is crucial. Evaluating division and multiplication before handling subtraction and addition ensures that the result is correct. Step-by-Step Solution: Step 1: Start with 96 L 12 S 7 V 49 M 10. Step 2: Replace L with /, S with *, V with - and M with +. Step 3: The decoded expression is 96 / 12 * 7 - 49 + 10. Step 4: Evaluate division and multiplication from left to right. 96 / 12 = 8. 8 * 7 = 56. Step 5: Now the expression reduces to 56 - 49 + 10. Step 6: Perform subtraction and addition from left to right. 56 - 49 = 7. 7 + 10 = 17. Verification / Alternative check: To verify, you can recompute 96 / 12 as 8 and confirm that 8 * 7 is indeed 56. Then check that 56 - 49 simplifies to 7 and finally that adding 10 gives 17. If you mistakenly change the order and compute 12 * 7 first or combine 49 + 10 before performing the earlier operations, you will obtain a different result. Because the correctly decoded and ordered calculation gives 17, and this matches exactly one of the options, the answer is reliable. Why Other Options Are Wrong: Options 18, 20, 21 and 23 can arise from common missteps, such as treating V as 49 - 56 instead of 56 - 49, or adding 10 at the wrong stage. Some candidates also incorrectly evaluate 96 / (12 * 7) or modify precedence rules, which again leads to incorrect totals. Since only 17 satisfies the expression under the specified coding and standard rules, the other values cannot be accepted. Common Pitfalls: A typical pitfall is forgetting that division and multiplication have equal precedence and must be done from left to right. Another pitfall is mentally rewriting the expression in a different order that looks simpler but no longer corresponds to the given structure. Writing each intermediate step explicitly on paper helps reduce arithmetic errors, especially when both negative and positive operations are involved. Final Answer: Using the given coded operations, the value of 96 L 12 S 7 V 49 M 10 is 17.

Question 9

In a coded arithmetic expression, # means subtraction, & means division, @ means addition and % means multiplication. Using these meanings and standard precedence, what is the value of 315 & 3 # 9 @ 4 % 6 ?

Accepted Answer

Introduction / Context: This question continues the pattern of coded arithmetic expressions, where ordinary operators are replaced with special symbols. You must translate the expression 315 & 3 # 9 @ 4 % 6 into standard arithmetic and then evaluate it. These problems assess precision in decoding and in applying precedence rules during multi step calculations. Given Data / Assumptions: & means division, so A & B represents A / B. # means subtraction, so A # B represents A - B. @ means addition, so A @ B represents A + B. % means multiplication, so A % B represents A * B. Expression to evaluate: 315 & 3 # 9 @ 4 % 6. Standard precedence: division and multiplication first, then addition and subtraction from left to right. Concept / Approach: As in previous coded arithmetic questions, you first decode each symbol into its true arithmetic meaning. Once the expression is written with ordinary operators, you handle division and multiplication before any addition or subtraction. Writing each intermediate result clearly is important to avoid mixing the operations and accidentally changing the intended computation. Step-by-Step Solution: Step 1: Start with 315 & 3 # 9 @ 4 % 6. Step 2: Replace & with /, # with -, @ with + and % with *. Step 3: The decoded expression becomes 315 / 3 - 9 + 4 * 6. Step 4: Evaluate division and multiplication first. 315 / 3 = 105. 4 * 6 = 24. Step 5: Substitute these into the expression: 105 - 9 + 24. Step 6: Now perform subtraction and addition from left to right. 105 - 9 = 96. 96 + 24 = 120. Verification / Alternative check: You can verify quickly by checking the extreme operations. The division 315 / 3 must give a round value of 105, otherwise you know an arithmetic slip has occurred. Since 4 * 6 is a simple multiplication equal to 24, combining 105, 9 and 24 is straightforward. If you mistakenly ignore precedence and compute 315 / (3 - 9 + 4) * 6 or any other rearrangement, you will not arrive at 120 and will conflict with the defined coding rule. Thus 120 is the only value consistent with both the decoding and standard arithmetic evaluation. Why Other Options Are Wrong: Options 190, 221 and 420 are typical of errors where candidates either add before multiplying or confuse which numbers are involved in division. Option 315 is the original number and might attract those who misread the problem as a trick. None of these values follow directly from the expression 315 / 3 - 9 + 4 * 6 with correct precedence. Since only 120 matches the exact evaluation, the other options must be rejected. Common Pitfalls: Students often rush through coded arithmetic and treat the symbols as if they were ordinary operators in their usual positions, without rewriting the expression. This leads to mixing old and new meanings. Another common pitfall is evaluating from left to right and ignoring the rule that division and multiplication come before addition and subtraction. To avoid these mistakes, always decode first, rewrite the expression, and then carefully apply the operation hierarchy. Final Answer: So, according to the given symbol meanings, the value of 315 & 3 # 9 @ 4 % 6 is 120.

Question 10

In a certain code language, each letter in a word is replaced by the letter two places ahead in the English alphabet, so that OUTLINE is written as QWVNKPG. Using this same pattern, how is MAN written in that code language?

Accepted Answer

Introduction / Context: This question deals with a simple alphabet shifting code. You are told that OUTLINE is coded as QWVNKPG and asked to find the code for MAN. It is a straightforward Caesar shift style pattern where each letter moves a fixed number of positions ahead in the alphabet. Recognising this shift and applying it consistently is the key to solving the problem. Given Data / Assumptions: OUTLINE is written as QWVNKPG in the code language. All letters are uppercase English letters A to Z. The coding rule is consistent for each letter and each position. We must determine the code for MAN. Concept / Approach: We compare each letter in OUTLINE with the corresponding letter in QWVNKPG and calculate how many positions forward it has shifted in the alphabet. If all shifts are identical, we have a uniform code. Once the shift is known, we apply the same shift to each letter of MAN. This method avoids guesswork and ensures that the entire mapping is consistent with the example provided. Step-by-Step Solution: Step 1: Pair the letters of OUTLINE with QWVNKPG: O → Q, U → W, T → V, L → N, I → K, N → P, E → G. Step 2: Convert each letter to its alphabetical position: O(15) → Q(17), U(21) → W(23), T(20) → V(22), L(12) → N(14), I(9) → K(11), N(14) → P(16), E(5) → G(7). Step 3: Each letter has moved forward by 2 positions (15 to 17, 21 to 23, and so on). Step 4: Therefore, the coding rule is: shift every letter forward by 2 in the alphabet, wrapping around after Z if necessary. Step 5: Now apply this rule to MAN: M(13) → O(15), A(1) → C(3), N(14) → P(16). Step 6: Combine the coded letters: O C P, written as OCP. Verification / Alternative check: To verify, you can decode OCP back by shifting each letter backward by 2 positions. O moves to M, C moves to A and P moves to N, recovering the original word MAN. Re checking the OUTLINE example by shifting each letter forward by 2 also returns QWVNKPG, so the pattern is confirmed. Since this uniform shift accounts for all letters and yields the given example accurately, we can trust it when applied to MAN. Why Other Options Are Wrong: Options FVQ, RNE, RKX and NCQ would correspond to different and inconsistent shifts. For example, FVQ would require M to map to F, which is a backward shift of many positions, not the same plus two rule used in OUTLINE. Any option that does not match a uniform +2 shift for each letter of MAN must be discarded. OCP is the only option that matches the correctly deduced pattern. Common Pitfalls: Typical errors include shifting by the wrong amount, miscounting positions in the alphabet or changing the direction of the shift. It is helpful to write a quick row of letters from A to Z and count steps carefully. Another pitfall is assuming that the shift might vary from letter to letter; however, the example clearly shows the same +2 movement throughout, so consistency is the guiding principle here. Final Answer: Thus, when each letter is shifted forward by two positions, MAN is written as OCP in the code language.

Question 11

In a coded arithmetic system, the operators are redefined as follows: "+" means "×", "-" means "÷", "x" means "-" and "÷" means "+". Under these meanings, what is the value of the expression 16 ÷ 4 x 10 - 5 + 8 ?

Accepted Answer

Introduction / Context: This coded arithmetic question reassigns new meanings to the familiar operators plus, minus, multiplication and division. The expression 16 ÷ 4 x 10 - 5 + 8 must be evaluated after replacing each symbol by the operation it now represents. The question checks whether you can correctly interpret each symbol and then apply conventional precedence rules to compute the final value. Given Data / Assumptions: "+" means "×" (multiplication). "-" means "÷" (division). "x" means "-" (subtraction). "÷" means "+" (addition). Expression to interpret: 16 ÷ 4 x 10 - 5 + 8. Standard arithmetic precedence applies after decoding: division and multiplication are evaluated before addition and subtraction, working left to right within each level. Concept / Approach: The main idea is to translate the coded expression into a normal one where each symbol has its usual meaning. This is done by writing a new expression in terms of actual plus, minus, multiplication and division based on the given mapping. Once you have the decoded expression, ignore the original symbols and evaluate the new expression using standard precedence. It is essential not to mix the old and new meanings during calculations. Step-by-Step Solution: Step 1: Start from the coded expression: 16 ÷ 4 x 10 - 5 + 8. Step 2: Replace each symbol by the operation it represents. "÷" means "+", so 16 ÷ 4 becomes 16 + 4. "x" means "-", so 4 x 10 becomes 4 - 10. "-" means "÷", so 10 - 5 becomes 10 ÷ 5. "+" means "×", so + 8 becomes × 8. Step 3: Writing in standard form, the entire decoded expression is 16 + 4 - 10 ÷ 5 × 8. Step 4: Apply the usual precedence, handling division and multiplication first. 10 ÷ 5 = 2. 2 × 8 = 16. Step 5: Substitute this back: 16 + 4 - 16. Step 6: Evaluate from left to right: 16 + 4 = 20, then 20 - 16 = 4. Verification / Alternative check: To confirm, you can redo the translation more slowly to be sure each operator has been replaced correctly and that no code meaning has been confused. Checking the critical part 10 ÷ 5 × 8 after decoding shows that it equals 16, which balances out the 16 + 4 at the start to give 4 overall. Any misinterpretation, such as swapping the meanings of plus and minus, will result in a different answer that does not match the consistent rule described in the question statement. Why Other Options Are Wrong: Options 12, 8, 2 and 16 correspond to typical mistakes like ignoring operator precedence, computing from left to right, or misassigning one of the code meanings. For example, if you treat x as multiplication instead of subtraction, you will obtain a larger number. Because the decoded and correctly evaluated expression yields exactly 4, none of the other options can be considered correct. Common Pitfalls: Students sometimes attempt to compute directly on the original expression while mentally remembering the new meanings, which increases the chance of mixing semantics mid calculation. Another pitfall is not rewriting the full decoded expression before starting to compute, leading to an error in one of the replacements. The safest strategy is always to translate the entire expression into standard operators first, then apply normal arithmetic rules step by step. Final Answer: Under the given operator definitions, the value of 16 ÷ 4 x 10 - 5 + 8 is 4.

Question 12

In a certain code language, the operators are redefined as follows: "+" represents "×", "-" represents "+", "x" represents "÷" and "÷" represents "-". Using these definitions, what is the value of the expression 60 x 5 + 3 ÷ 24 - 6 ?

Accepted Answer

Introduction / Context: This is another operator coding question. Here, the symbols plus, minus, multiplication and division are all given new meanings. You need to evaluate the expression 60 x 5 + 3 ÷ 24 - 6 using these new meanings, not the usual ones. The problem ensures that you can systematically translate each symbol and still follow standard precedence rules when actually doing the calculations. Given Data / Assumptions: "+" represents "×" (multiplication). "-" represents "+" (addition). "x" represents "÷" (division). "÷" represents "-" (subtraction). Expression to interpret: 60 x 5 + 3 ÷ 24 - 6. Once decoded, ordinary precedence of operations (division and multiplication before addition and subtraction) applies. Concept / Approach: The strategy is to rewrite the entire expression in terms of actual operations. After replacing each symbol with its true meaning, you ignore the original coding and evaluate the resulting standard expression. This avoids confusion and ensures that you do not accidentally use mixed meanings. Correctly dealing with precedence is important because division and multiplication segments must be evaluated before using the addition and subtraction represented by other symbols. Step-by-Step Solution: Step 1: Take the coded expression: 60 x 5 + 3 ÷ 24 - 6. Step 2: Replace each operator with the operation it represents. "x" represents "÷", so 60 x 5 becomes 60 ÷ 5. "+" represents "×", so + 3 becomes × 3. "÷" represents "-", so 3 ÷ 24 becomes 3 - 24. "-" represents "+", so - 6 becomes + 6. Step 3: Writing the whole decoded expression, we get 60 ÷ 5 × 3 - 24 + 6. Step 4: Now apply normal precedence, starting with division and multiplication. 60 ÷ 5 = 12. 12 × 3 = 36. Step 5: Substitute back into the expression: 36 - 24 + 6. Step 6: Evaluate from left to right for addition and subtraction. 36 - 24 = 12. 12 + 6 = 18. Verification / Alternative check: Rechecking the mapping is a good sanity check: x is division, plus is multiplication, division is subtraction and minus is addition. With that mapping, any alternative decoding that does not give 60 ÷ 5 × 3 - 24 + 6 violates the problem statement. Computing 60 ÷ 5 × 3 as 36 and then combining it with -24 and +6 gives 18 consistently. This result matches one of the options and fits the logic of the coding pattern, so it can be accepted with confidence. Why Other Options Are Wrong: Option B, 94, is far larger than what the decoded expression can produce and would require incorrect multiplication in place of division. Option C, 9, and option D, 57, typically arise if candidates perform some operations in the wrong order or misapply the code meanings for one of the symbols. Option E, 21, could be produced by adding before doing multiplication. Since none of these values arise from the correctly decoded and properly evaluated expression, they are incorrect. Common Pitfalls: A common error is to treat x as the usual multiplication instead of division and to treat ÷ as division instead of subtraction. Another pitfall is to start manipulating the coded expression directly without writing the fully decoded version. This leads to mixed semantics and inconsistent steps. To avoid such issues, translate carefully first, then work step by step with normal arithmetic rules on the translated expression only. Final Answer: With the given redefined operators, the value of 60 x 5 + 3 ÷ 24 - 6 is 18.

Question 13

In this coding puzzle, three digit numbers are transformed according to a fixed pattern: 462 is written as 551 and 398 is written as 487. On the same basis, what should 856 be written as?

Accepted Answer

Introduction / Context: This is a simple number coding question where each three digit number is converted into another three digit number using a consistent rule. You are given two examples, 462 = 551 and 398 = 487, and must apply the same rule to the number 856. Questions like this test your ability to identify an underlying arithmetic relationship between input and output values and to generalise that pattern to a new case. Given Data / Assumptions: 462 maps to 551 under the coding rule. 398 maps to 487 under the same rule. The rule is uniform for all numbers in this question. We must find the coded form of 856. The transformation is numerical, not a rearrangement of digits. Concept / Approach: The easiest approach is to consider whether the output is obtained by adding or subtracting a fixed number from the input. You can compute the difference between the given pairs and see whether it is constant. If the difference is the same for both examples, then the code is simply adding that constant. This is a very common pattern in such three digit number puzzles and is quick to apply once detected. Step-by-Step Solution: Step 1: Compute the difference between 462 and 551. 551 - 462 = 89. Step 2: Compute the difference between 398 and 487. 487 - 398 = 89. Step 3: The difference is the same in both cases, suggesting that the coding rule is "add 89 to the original number". Step 4: Now apply this rule to 856. 856 + 89 = 945. Verification / Alternative check: To verify, you can check whether any plausible alternative pattern, such as manipulating digits separately, would produce the given outputs and also match 856. For instance, reversing digits or adding digit sums does not produce 551 from 462 or 487 from 398. The clean and consistent explanation is that each number is simply increased by 89. Because 462 + 89 = 551 and 398 + 89 = 487 both hold, and 856 + 89 gives 945, the pattern appears stable and unambiguous. Why Other Options Are Wrong: Options 745, 773, 743 and 851 do not equal 856 + 89 and therefore cannot match the discovered rule. If you attempted to fit a different rule to produce one of those values, it would not simultaneously work for both 462 and 398 without becoming contrived. Since the question explicitly tells you that the same basis must be used for each transformation, only the output that preserves the constant difference of 89 is acceptable, which is 945. Common Pitfalls: A common mistake is to immediately manipulate digits separately (for example, adding or subtracting tens and units) without first checking the simple possibility of adding a fixed number. Another pitfall is to find a rule that works for the first example but not verify it with the second, leading to a mismatched pattern. Always test any hypothesised transformation on all given examples before applying it to the unknown case. Final Answer: On the same basis as the given examples, 856 should be written as 945.

Question 14

In this coding puzzle, three digit expressions are rewritten by rearranging their digits. It is given that 6 x 2 x 9 is written as 269 and 8 x 7 x 1 is written as 781. On the same basis, how should 4 x 1 x 3 be written?

Accepted Answer

Introduction / Context: This question involves a simple digit rearrangement code. The numbers 6, 2 and 9 are written with multiplication signs between them as 6 x 2 x 9, but the answer 269 is just a three digit number whose digits are drawn from 6, 2 and 9. The second example, 8 x 7 x 1 = 781, follows the same pattern. You need to identify how the digits are being rearranged and then apply that to 4 x 1 x 3. Given Data / Assumptions: 6 x 2 x 9 is coded as 269. 8 x 7 x 1 is coded as 781. The multiplication signs are not used for actual multiplication; they simply separate the three input digits. The code rearranges the three digits into a new three digit number. We must determine the rearrangement for 4 x 1 x 3. Concept / Approach: The first step is to examine the relation between the ordered triple (6, 2, 9) and the digits of 269. Observing that the coded form starts with the middle number, followed by the first number and then the last number, suggests a (second, first, third) pattern. Confirming this with the second example (8, 7, 1) and 781 will show whether the pattern is consistent. Once it is confirmed, we can apply the same rearrangement to the new triple (4, 1, 3). Step-by-Step Solution: Step 1: For 6 x 2 x 9 → 269, note the original digits in order: first 6, second 2, third 9. Step 2: The coded number 269 has digits 2, 6 and 9 in that order. Step 3: So the pattern is: take the second digit first, then the first digit, then the third digit, yielding (second, first, third). Step 4: Check this rule with the second example, 8 x 7 x 1 → 781. Original digits: first 8, second 7, third 1. Applying (second, first, third): 7, 8, 1 → 781, which matches the coded result. Step 5: Now apply the same rearrangement to 4 x 1 x 3. Original digits: first 4, second 1, third 3. Rearranged as (second, first, third): 1, 4, 3 → 143. Verification / Alternative check: Since the same (second, first, third) rule explains both given examples perfectly, there is no need to search for a more complicated pattern. Any other rearrangement, such as (first, third, second) or (third, second, first), would fail to match at least one of the examples. Thus, applying the (second, first, third) pattern to every triple is consistent and simple. For 4, 1, 3, this gives 143, which matches option D exactly. Why Other Options Are Wrong: Options 431, 413, 341 and 314 correspond to different permutations of the digits 4, 1 and 3, but none of these follows the consistent (second, first, third) mapping that the given examples confirm. Choosing any of these would require a different rearrangement rule that would not hold for both 6, 2, 9 and 8, 7, 1. Therefore, they cannot represent the intended coding system. Common Pitfalls: A common mistake is to try to interpret the x symbols as actual multiplication, which leads to numerical products instead of digit rearrangements and does not fit the outputs. Another error is to choose a permutation that works for only one example but not both. Always use all given examples to identify a single consistent rule, then apply that rule to the new case. Final Answer: Using the same rearrangement pattern, 4 x 1 x 3 is written as 143.

Question 15

In a certain code language, the operators are redefined as follows: "-" represents "×", "÷" represents "+", "+" represents "÷" and "x" represents "-". Using these meanings, what is the value of 160 + 8 x 30 ÷ 4 - 6 ?

Accepted Answer

Introduction / Context: This is a coded arithmetic expression where each familiar operator symbol has been assigned a different arithmetic meaning. The expression 160 + 8 x 30 ÷ 4 - 6 must be evaluated under these new definitions. The question assesses whether you can accurately decode every occurrence of each operator and then evaluate the resulting standard expression using correct precedence rules. Given Data / Assumptions: "-" represents "×" (multiplication). "÷" represents "+" (addition). "+" represents "÷" (division). "x" represents "-" (subtraction). Expression: 160 + 8 x 30 ÷ 4 - 6. After decoding, usual arithmetic precedence applies: division and multiplication before addition and subtraction. Concept / Approach: We should systematically replace each symbol in the given expression with the operation it now stands for. This creates a new expression written entirely in terms of the ordinary plus, minus, multiplication and division operators. Then we evaluate this new expression as usual, respecting precedence. It is important not to mix the original and redefined meanings during calculation, so writing down the decoded expression clearly is essential. Step-by-Step Solution: Step 1: Begin with the coded expression: 160 + 8 x 30 ÷ 4 - 6. Step 2: Apply the mapping of each operator. "+" represents "÷", so 160 + 8 becomes 160 ÷ 8. "x" represents "-", so 8 x 30 becomes 8 - 30. "÷" represents "+", so 30 ÷ 4 becomes 30 + 4. "-" represents "×", so - 6 becomes × 6. Step 3: Writing the full decoded expression, we have 160 ÷ 8 - 30 + 4 × 6. Step 4: Evaluate division and multiplication first. 160 ÷ 8 = 20. 4 × 6 = 24. Step 5: Substitute these results: 20 - 30 + 24. Step 6: Now carry out subtraction and addition from left to right. 20 - 30 = -10. -10 + 24 = 14. Verification / Alternative check: You can verify by re doing the decoding more slowly to confirm that each symbol has been replaced correctly: plus as division, x as subtraction, division as addition and minus as multiplication. Evaluating 160 ÷ 8 gives 20 and 4 × 6 gives 24, so the reduced expression must be 20 - 30 + 24. Because this simplifies neatly to 14 and this value appears among the answer choices, it is a strong indication that the decoding is correct. Alternative decodings that misassign any symbol do not match both the given mapping and the final value simultaneously. Why Other Options Are Wrong: Options 42, 5, 17 and 10 generally arise from computing with the wrong operation precedence or forgetting to recode one of the operators. For instance, treating x as multiplication or ÷ as division changes the structure dramatically and leads to a different answer. Because the properly decoded expression yields 14 and the mapping is clearly specified, all other numerical values must be rejected. Common Pitfalls: A typical pitfall is performing operations directly on the original expression without writing the decoded version. This invites confusion between old and new meanings. Another common error is to do addition and subtraction before resolving the multiplications and divisions in the decoded form. Carefully writing the full translated expression and underlining the division and multiplication operations before evaluating them is a good habit to avoid such mistakes. Final Answer: Using the given code for the operators, the value of 160 + 8 x 30 ÷ 4 - 6 is 14.

Question 16

In a coded arithmetic operation denoted by the symbol α, it is given that 18 α 1 = 34, 19 α 2 = 34 and 18 α 3 = 30. Using the same rule, what is the value of 18 α 8 ?

Accepted Answer

Introduction / Context: This question defines a custom binary operation represented by the symbol α. You are given three input output pairs involving 18, 19 and small integers, and you must find the result when 18 operates with 8 under the same rule. This type of reasoning problem tests your ability to infer a formula linking two input numbers to an output and then apply it to a new pair. Given Data / Assumptions: 18 α 1 = 34. 19 α 2 = 34. 18 α 3 = 30. The same functional rule applies to all instances of α. We need to compute 18 α 8. Concept / Approach: Because the results are close to twice the first operand, it is natural to test expressions involving multiples of the first number and the second number. Observing that 18 α 1 and 19 α 2 both equal 34 suggests that the operation combines the first number and the second number in a symmetric way. One efficient approach is to check a simple linear expression like 2 * first minus k times second and determine whether a single consistent k fits all the examples. Once the rule is discovered, it can be directly applied to 18 α 8. Step-by-Step Solution: Step 1: Look at 18 α 1 = 34. Twice the first operand is 2 * 18 = 36. Step 2: To get 34 from 36, we subtract 2, which is 2 * 1. This suggests a pattern 2 * first - 2 * second. Step 3: Test this tentative rule with 19 α 2 = 34. 2 * 19 = 38 and 2 * 2 = 4, so 38 - 4 = 34, which matches the given value. Step 4: Test again with 18 α 3 = 30. 2 * 18 = 36 and 2 * 3 = 6, so 36 - 6 = 30, again matching the given result. Step 5: This confirms that the operation is defined by the formula: first α second = 2 * first - 2 * second. Step 6: Now compute 18 α 8 using this rule. 2 * 18 = 36. 2 * 8 = 16. 36 - 16 = 20. Verification / Alternative check: The rule can be written compactly as α(a, b) = 2a - 2b. Applying this to each given pair reproduces the outputs exactly, which makes it much more convincing than a rule that works only for some examples. For 18 and 8, no alternative simple formula is needed, because 2 * 18 - 2 * 8 gives 20 cleanly. Since 20 appears among the answer choices, and the rule is fully consistent, the answer is reliable. Why Other Options Are Wrong: Options 6, 16, 8 and 18 would require a different formula, such as 2 * first - second or first - 2 * second, but these do not match all the given examples simultaneously. For instance, 2 * 19 - 2 does not give 34, and 2 * 18 - 3 does not give 30. Because the question assumes one fixed definition of α, only the value that agrees with all examples can be correct. That value is 20. Common Pitfalls: A common mistake is to try complex patterns involving squares or digit sums when a simple linear relation is sufficient. Another is to stop once a formula fits one example without checking the others. Always verify a proposed rule against all given pairs; if it fails even once, it must be rejected or modified. Testing a small set of reasonable formulas, such as weighted sums or differences of the operands, usually reveals the correct one quickly. Final Answer: Using the rule first α second = 2 * first - 2 * second, the value of 18 α 8 is 20.

Question 17

In this coding puzzle, a custom set of operations is defined: A*B means A multiplied by B, A@B means A divided by B, A?B means B added to A and A=B means B subtracted from A. Using these rules and normal precedence, what is the value of 10*10 = 5*10…

Accepted Answer

Introduction / Context: This question introduces four non standard operator symbols *, @, ? and =, each with a specific arithmetic meaning. The expression 10*10 = 5*10 ? 50@10 must be evaluated using these coded meanings, not the usual ones. The problem checks whether you can translate the entire expression carefully and then apply normal precedence of multiplication and division before performing addition and subtraction. Given Data / Assumptions: A*B means A multiplied by B, so A*B = A * B. A@B means A divided by B, so A@B = A / B. A?B means add B to A, so A?B = A + B. A=B means subtract B from A, so A=B = A - B. Expression: 10*10 = 5*10 ? 50@10. Normal precedence applies after decoding: multiplication and division first, then addition and subtraction from left to right. Concept / Approach: The method is to replace each occurrence of *, @, ? and = with their true operations. Once this is done, the expression becomes an ordinary arithmetic expression with standard operators. You then carry out multiplication and division first, followed by addition and subtraction. Care must be taken to interpret = as subtraction and not as equality, as its meaning has been overloaded here in a special way. Step-by-Step Solution: Step 1: Start from the coded expression: 10*10 = 5*10 ? 50@10. Step 2: Decode each operator using the given meanings. 10*10 means 10 * 10. Between 10 and 5, the symbol = means subtract B from A, so 10 = 5 means 10 - 5. 5*10 means 5 * 10. ? represents addition, so ? 50 means + 50. @ represents division, so 50@10 means 50 / 10. Step 3: Combine these pieces into one standard expression. 10 * 10 - 5 * 10 + 50 / 10. Step 4: Evaluate all multiplication and division first. 10 * 10 = 100. 5 * 10 = 50. 50 / 10 = 5. Step 5: Substitute back: 100 - 50 + 5. Step 6: Evaluate subtraction and addition from left to right. 100 - 50 = 50. 50 + 5 = 55. Verification / Alternative check: A quick check is to recompute each of the three basic operations to confirm no multiplication or division error has occurred. Since 10 * 10 = 100, 5 * 10 = 50 and 50 / 10 = 5 are all straightforward, combining them as 100 - 50 + 5 gives 55 without ambiguity. Any alternative interpretation of = as equality or ? as something other than addition would conflict with the definitions given in the question. Because the decoded expression is unique and evaluates to 55, this answer is secure. Why Other Options Are Wrong: Options 100, 45 and 65 often come from errors such as adding instead of subtracting at the = position, or failing to perform the division by 10. Option 1000 misinterprets the expression as 10 * 10 * 5 * 10 with no subtraction or division at all. These values do not follow from the explicit definitions of the coded operators and so cannot be valid. Only 55 is consistent with both the decoding and correct arithmetic evaluation. Common Pitfalls: A major pitfall is to treat = as an equality sign and try to form an equation to be solved, which is not what the question intends. Another is to work directly on the coded expression without writing the decoded version, which can lead to mixing up the meanings of @ and ?, especially under time pressure. Always rewrite the expression clearly in terms of normal operators and then apply precedence rules methodically. Final Answer: Under the given definitions of *, @, ? and =, the value of 10*10 = 5*10 ? 50@10 is 55.

Question 18

In a word to number coding system, the value of a word is obtained by adding the alphabetical positions of its letters (A = 1, B = 2, …, Z = 26). If REDUCE has value 56 and RECYCLE has value 71 under this system, what is the value of REUSE?

Accepted Answer

Introduction / Context: This is a straightforward coding question where words are converted into numbers by summing the positions of their letters in the English alphabet. You are told that REDUCE equals 56 and RECYCLE equals 71 in this system, and you must find the corresponding value for REUSE. This tests your familiarity with letter positions and basic addition. Given Data / Assumptions: Alphabet positions: A = 1, B = 2, …, Z = 26. REDUCE has value 56. RECYCLE has value 71. The code value of a word is the sum of the positions of its letters. We need to find the value of REUSE. Concept / Approach: To confirm the coding rule, we compute the sum of letter positions for REDUCE and RECYCLE and check that they match the given values. Once verified, we apply exactly the same procedure to REUSE. The main tasks are to recall or determine the numeric position of each letter and to add these values accurately. Step-by-Step Solution: Step 1: Check the rule with REDUCE. Letters: R, E, D, U, C, E. Positions: R(18), E(5), D(4), U(21), C(3), E(5). Sum: 18 + 5 = 23, 23 + 4 = 27, 27 + 21 = 48, 48 + 3 = 51, 51 + 5 = 56. This matches the given value 56. Step 2: Check the rule with RECYCLE. Letters: R, E, C, Y, C, L, E. Positions: R(18), E(5), C(3), Y(25), C(3), L(12), E(5). Sum: 18 + 5 = 23, 23 + 3 = 26, 26 + 25 = 51, 51 + 3 = 54, 54 + 12 = 66, 66 + 5 = 71. This matches the given value 71. Step 3: Now apply the same rule to REUSE. Letters: R, E, U, S, E. Positions: R(18), E(5), U(21), S(19), E(5). Step 4: Add these values. 18 + 5 = 23. 23 + 21 = 44. 44 + 19 = 63. 63 + 5 = 68. Verification / Alternative check: Since the sums for REDUCE and RECYCLE match the given values exactly, the coding rule of summing letter positions is fully confirmed. The computation for REUSE uses the same positions and simple additions, so there is no ambiguity in obtaining 68. You can quickly re add 18 + 5 + 21 + 19 + 5 on a calculator or by grouping as (18 + 21) + (5 + 19 + 5) to double check that the result is 39 + 29 = 68. Why Other Options Are Wrong: Options 65, 69, 70 and 72 are all close to the correct value but do not match the exact sum of the letter positions for REUSE. Choosing any of these would reflect either a misremembered letter position or a small arithmetic mistake, such as adding 19 as 20 or miscounting the total. Because the rule is precise and the calculation is straightforward, only 68 fits the definition exactly. Common Pitfalls: A common pitfall is misassigning positions to letters near the middle or end of the alphabet, for example treating S as 20 instead of 19. Another is rushing through the addition and making a small mistake. Writing down each intermediate partial sum as shown and grouping numbers cleverly can help prevent simple arithmetic errors, especially under time pressure. Final Answer: Therefore, under this letter position coding system, the value of REUSE is 68.

Question 19

In a special operation denoted by "x", the result is defined by the difference between the sum of the digits of the first number and the sum of the digits of the second number. It is given that 56 x 11 = 9, 37 x 13 = 6 and 42 x 12 = 3. Using the sam…

Accepted Answer

Introduction / Context: This question defines a custom operation "x" between two numbers based on their digit sums. You are told the outcomes for three pairs and asked to compute the outcome for a fourth pair. These problems test pattern recognition with a focus on simple properties of digits rather than full numeric operations like multiplication. Given Data / Assumptions: 56 x 11 = 9. 37 x 13 = 6. 42 x 12 = 3. The same rule is used to compute 87 x 77. The definition suggested in the question description is that the result equals the sum of the digits of the first number minus the sum of the digits of the second number. Concept / Approach: We first verify that the supposed rule, digit sum difference, fits all three given examples. To do this, we compute the sum of digits for each of the two numbers in an example pair, subtract the second sum from the first sum and check if the result matches the given output. If the rule holds for all examples, we can safely apply it to 87 and 77. This approach is simpler and more consistent than trying to interpret x as normal multiplication. Step-by-Step Solution: Step 1: For 56 x 11, find the digit sums. Sum of digits of 56: 5 + 6 = 11. Sum of digits of 11: 1 + 1 = 2. Difference: 11 - 2 = 9, which matches the given value. Step 2: For 37 x 13, compute the digit sums. Sum of digits of 37: 3 + 7 = 10. Sum of digits of 13: 1 + 3 = 4. Difference: 10 - 4 = 6, matching the given value. Step 3: For 42 x 12, compute the digit sums. Sum of digits of 42: 4 + 2 = 6. Sum of digits of 12: 1 + 2 = 3. Difference: 6 - 3 = 3, which again matches the given value. Step 4: The rule is confirmed: a x b = (sum of digits of a) - (sum of digits of b). Step 5: Now apply this rule to 87 x 77. Sum of digits of 87: 8 + 7 = 15. Sum of digits of 77: 7 + 7 = 14. Difference: 15 - 14 = 1. Verification / Alternative check: Because the same simple digit sum difference rule explains all three given examples exactly, there is no need to look for a more complicated pattern. Any other rule would need to be at least as consistent and would also have to give 1 for 87 and 77, which is unlikely without becoming contrived. Recomputing 8 + 7 and 7 + 7 confirms that the difference really is 1, so the result is secure. Why Other Options Are Wrong: Options 2, 3, 4 and 5 would require different differences of digit sums or alternative definitions, but these would fail when tested on 56, 11 and the other given pairs. Because the question insists that all equations are solved on the basis of one system, only the value that arises from that system can be accepted. Thus, a difference of exactly 1 is the only correct outcome for 87 x 77. Common Pitfalls: A common pitfall is to read x as normal multiplication and attempt to relate the product to the output, which does not yield a consistent pattern here. Another is to test a rule on just one example and not verify it across all three. Always ensure that any pattern you conjecture matches every given instance before applying it to the unknown case. Final Answer: Using the digit sum difference rule, the value of 87 x 77 is 1.

Question 20

In a certain code language, each letter is replaced by its alphabetical mirror so that the sum of the positions of a letter and its code partner is always 27 (A ↔ Z, B ↔ Y, …, M ↔ N). Under this Atbash style code, UNCOVER is written as FMXLEVI. Usin…

Accepted Answer

Introduction / Context: This coding question uses an Atbash style substitution, where each letter is replaced by its mirror in the alphabet. In such a code, the first letter A pairs with Z, B pairs with Y, C with X and so on, so that the sum of the positions of a letter and its coded partner is 27. You are told that UNCOVER is coded as FMXLEVI and must determine how TIP is written under the same system. Given Data / Assumptions: Word: UNCOVER, Code: FMXLEVI. The mapping is letter by letter and one to one. For Atbash, if a letter is at position n in the alphabet, its code partner is at position 27 - n. We need to code TIP using this mapping. Concept / Approach: The approach is to verify that UNCOVER → FMXLEVI follows the Atbash rule. If it does, then the same mapping can be applied to any new word, including TIP. To verify, we compute the alphabetical positions of each letter in UNCOVER and FMXLEVI and check that each pair sums to 27. Once confirmed, we use the rule code(letter) = letter at position 27 - original position for T, I and P. Step-by-Step Solution: Step 1: Write the letter pairs from UNCOVER to FMXLEVI. U → F, N → M, C → X, O → L, V → E, E → V, R → I. Step 2: Convert each letter to its alphabetical position. U(21) and F(6): 21 + 6 = 27. N(14) and M(13): 14 + 13 = 27. C(3) and X(24): 3 + 24 = 27. O(15) and L(12): 15 + 12 = 27. V(22) and E(5): 22 + 5 = 27. E(5) and V(22): 5 + 22 = 27. R(18) and I(9): 18 + 9 = 27. Step 3: Every pair sums to 27, confirming that this is an Atbash mapping. Step 4: Now code the letters in TIP. T is the 20th letter, so its partner has position 27 - 20 = 7, which is G. I is the 9th letter, so its partner has position 27 - 9 = 18, which is R. P is the 16th letter, so its partner has position 27 - 16 = 11, which is K. Step 5: Combine the coded letters: G R K → GRK. Verification / Alternative check: To verify, decode GRK back to TIP by reversing the Atbash mapping. G is 7, so its mirror is 27 - 7 = 20, which is T. R is 18, so its mirror is 27 - 18 = 9, which is I. K is 11, so its mirror is 27 - 11 = 16, which is P. This reconstructs TIP exactly, confirming that GRK is the correct code. Because the mapping works both ways and matches all examples from UNCOVER, it is a robust and consistent rule. Why Other Options Are Wrong: Options LTV, NQV, KHQ and LHR involve different letters that do not satisfy the Atbash mirror rule for T, I and P. For example, if T were mapped to L, the positions would be 20 and 12, which sum to 32, not 27. Each wrong option would require abandoning the consistent mirror property established by UNCOVER. Since the code must be the same across all words, only GRK is compatible with the confirmed mapping. Common Pitfalls: A common mistake is to look for a fixed forward or backward shift instead of noticing the mirror property. Another is to miscount positions when letters are near the end of the alphabet. Writing down the full alphabet with position numbers and explicitly checking that each letter pair sums to 27 is an effective way to avoid errors. Once the Atbash pattern is recognised, encoding any other word is straightforward. Final Answer: Under the Atbash style coding where partnered positions sum to 27, TIP is written as GRK.

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