Q: Encoded operators and inequalities — choose the true relation: Mapping: A → '+', M → '×', D → '÷', G → '>', L → '<'. Which option is logically correct when decoded and evaluated with standard precedence?

Introduction / Context: We decode each string using the provided mapping for arithmetic and comparison operators, then evaluate both sides with standard precedence to see which inequality is true. Given Data / Assumptions: A = +, M = ×, D = ÷, G = >, L = or 6 + 2 × 3 → 5.67 > 12 → false (b) 4 + 5 × 4 6 ÷ 2 + 4 → 6 > 7 → false (d) 4 + 3 × 2 < 4 ÷ 2 × 6 → 10 < 12 → true Why Other Options Are Wrong: Each computes to a false inequality once decoded and properly evaluated. Typical errors when solving include ignoring precedence or misreading the mapping for M and D. Common Pitfalls: Doing addition before finishing ×/÷, or evaluating comparisons before both sides are fully simplified. Final Answer: 4A 3M 2L 4D 2M 6

Q: Interchange all signs as specified and evaluate the value: Mapping of interchanges: '−' → '+', '+' → '÷', '×' → '−', '÷' → '×'. Find the value of: 6 ÷ 8 + 2 × 5 − 8

Introduction / Context: All operator symbols are to be replaced by new ones according to a fixed mapping. After substituting, evaluate using standard precedence (× and ÷ before + and −). Given Data / Assumptions: Interchanges: −→+, +→÷, ×→−, ÷→× Expression: 6 ÷ 8 + 2 × 5 − 8 Use standard precedence; ×/÷ are left-associative. Concept / Approach: Apply the mapping to each symbol first; then compute. Do not mix old and new meanings in the same step and do not change numeric order. Step-by-Step Solution: 1) Substitute: 6 ÷ 8 + 2 × 5 − 8 → 6 × 8 ÷ 2 − 5 + 8 2) ×/÷ left→right: 6 × 8 = 48; 48 ÷ 2 = 24 3) +/−: 24 − 5 = 19; 19 + 8 = 27 Verification / Alternative check: Parenthesized view: ((6 × 8) ÷ 2) − 5 + 8 = (48 ÷ 2) − 5 + 8 = 24 − 5 + 8 = 27. Why Other Options Are Wrong: 18, 28, and 32 result from common errors such as reversing the ×/÷ left-to-right order or forgetting the +→÷ substitution in the middle of the expression. Common Pitfalls: Performing +/− before finishing ×/÷ or accidentally keeping an original operator’s meaning after substitution. Final Answer: 27

Q: Symbol encoding check — select the correct statement: Encoding: δ → '+', • → '−', γ → '×', η → '÷', ω → '=', β → '>', α → '<'. Which option becomes a true relation after decoding and evaluating with standard precedence?

Introduction / Context: The problem encodes arithmetic and comparison symbols. After decoding, we evaluate both sides of each candidate with standard precedence and check whether the stated relation is true. Given Data / Assumptions: δ=+, •=−, γ=×, η=÷, ω==, β=>, α= , or 5 is false; (c) 3×6−2+8÷4 = 18, and 18 or <, or evaluating +/− before finishing ×/÷ in long strings of operations. Final Answer: 3 δ 6 • 2 γ 8 η 4 ω 5

Q: Operator placement — Replace each ‘‘*’’ with the correct mathematical sign so that the equation is true. 24 * 34 * 2 * 5 * 12

Introduction / Context: These problems test whether you can place arithmetic symbols so that both sides of an equation balance. You must respect standard precedence (× and ÷ before + and −) on each side of the equality, and the single “=” must split the statement into a left-hand side (LHS) and a right-hand side (RHS). Given Data / Assumptions: Template: 24 {s1} 34 {s2} 2 {s3} 5 {s4} 12 Exactly one of {s1..s4} is “=”; the rest are chosen from +, −, ×, ÷. Usual operator precedence applies on each side of “=”. Concept / Approach: Try options that place “=” where the remaining three operators can naturally form a simple calculation. Check each option with clean arithmetic to avoid precedence mistakes. Step-by-Step Solution: Test Option C: s1 = “=”, s2 = “÷”, s3 = “−”, s4 = “+”.This yields: 24 = 34 ÷ 2 − 5 + 12.Compute RHS with precedence: 34 ÷ 2 = 17; then 17 − 5 + 12 = 12 + 12 = 24.Hence LHS 24 equals RHS 24 → true. Verification / Alternative check: Briefly test other choices: they evaluate to values different from 24 (e.g., 24 + 34 ÷ 2 × 5 = 109; 24 = 34 ÷ 2 + 5 − 12 gives 10). Thus only Option C balances. Why Other Options Are Wrong: They either misplace the equality or produce incorrect totals due to precedence. Common Pitfalls: Forgetting that division happens before addition/subtraction. Trying to evaluate left-to-right across the equality instead of evaluating each side independently. Final Answer: = ÷ - +

Q: Choose signs so that the equation holds true. 24 * 16 * 8 * 32

Introduction / Context: You must select the correct pair of operations and the equality sign so that the numerical statement becomes a true equation. Standard operator precedence applies on the left side before comparing to the right side of “=” if present. Given Data / Assumptions: Template: 24 {s1} 16 {s2} 8 {s3} 32 Exactly one of {s1..s3} is “=”. Concept / Approach: Look for combinations that can naturally form 32, since 24 is near 32. The simplest attempt is to place “=” as the last operator so the entire left expression reduces to 32. Step-by-Step Solution: Try Option D: 24 + 16 − 8 = 32.Evaluate LHS: 24 + 16 = 40; 40 − 8 = 32.Hence 32 = 32, which is true. Verification / Alternative check: Quickly test other options: “÷ − =”: 24 ÷ 16 − 8 becomes a small number, not 32. “− + =”: 24 − 16 + 8 = 16 ≠ 32. “× ÷ =”: 24 × 16 ÷ 8 = 48 ≠ 32. Why Other Options Are Wrong: They evaluate to non-32 values on the left, so the equality fails. Common Pitfalls: Ignoring that the entire LHS must reduce to exactly 32 once operators are fixed. Final Answer: + - =

Q: Operator remapping — If “+” means “−”, “−” means “×”, “÷” means “+”, and “×” means “÷”, evaluate: 10 × 5 ÷ 3 − 2 + 3

Introduction / Context: In “mathematical-operations” puzzles, the symbols are reassigned to different operations. Replace each symbol first, then evaluate using normal precedence (× and ÷ before + and −). Given Data / Assumptions: ‘+’ → ‘−’ ‘−’ → ‘×’ ‘÷’ → ‘+’ ‘×’ → ‘÷’ Concept / Approach: Rewrite the expression with its real operations and then compute carefully. Step-by-Step Solution: Original: 10 × 5 ÷ 3 − 2 + 3After remap: 10 ÷ 5 + 3 × 2 − 3Precedence: 10 ÷ 5 = 2; 3 × 2 = 6Now: 2 + 6 − 3 = 5 Verification / Alternative check: Compute left-to-right while still honoring precedence and you reach the same result. Why Other Options Are Wrong: Values like 53/3 or 36 come from misapplied precedence or an incorrect mapping. Common Pitfalls: Changing precedence to match the original symbols. Precedence follows the real operations after mapping. Final Answer: 5

Q: Custom symbols — If Q = “+”, J = “×”, T = “−”, K = “÷”, evaluate: 30 K 2 Q 3 J 6 T 5

Introduction / Context: Replace each letter with its corresponding arithmetic operation and apply standard precedence rules (division/multiplication before addition/subtraction, with brackets first if any). Given Data / Assumptions: Q → + J → × T → − K → ÷ Concept / Approach: Perform replacements and compute in the natural order of operations. Step-by-Step Solution: Expression becomes: 30 ÷ 2 + 3 × 6 − 5Compute: 30 ÷ 2 = 15; 3 × 6 = 18Now: 15 + 18 − 5 = 28 Verification / Alternative check: Grouping as (30 ÷ 2) + (3 × 6) − 5 also yields 28. Why Other Options Are Wrong: 31 or 18 occur if addition/subtraction is done before handling ×/÷ or if one mapping is misread. Common Pitfalls: Treating letters as variables instead of operator codes. Final Answer: 28

Q: Operator mapping — P = “÷”, Q = “×”, R = “+”, S = “−”. Evaluate the expression: 18 Q 12 P 4 R 5 S 6

Introduction / Context: Map each code to its arithmetic operator and compute. Be careful to resolve multiplication and division before addition and subtraction. Given Data / Assumptions: P → ÷ Q → × R → + S → − Expression intended: 18 Q 12 P 4 R 5 S 6 → 18 × 12 ÷ 4 + 5 − 6 Concept / Approach: Compute × and ÷ from left to right, then apply + and −. Step-by-Step Solution: 18 × 12 = 216216 ÷ 4 = 5454 + 5 − 6 = 53 Verification / Alternative check: You can group the first two operations: (18 × 12) ÷ 4 = 216 ÷ 4 = 54; then adjust with +5 and −6. Why Other Options Are Wrong: 59/63/65 result from ignoring precedence or misreading one of the mappings. Common Pitfalls: Mistaking “P” for plus; here P actually denotes division. Final Answer: 53

Question 1

Operator decoding and evaluation:  Let P denote '÷', Q denote '×', R denote '+' and S denote '−'.  Evaluate with standard precedence (×, ÷ before +, −): 18 Q 12 P 4 R 5 S 6

Accepted Answer

Introduction / Context: This is a direct operator-decoding exercise. After mapping each code letter to its arithmetic operator, evaluate the expression carefully using standard precedence: perform × and ÷ before + and −, and resolve ties left-to-right. Given Data / Assumptions: P → ÷ Q → × R → + S → − Expression: 18 Q 12 P 4 R 5 S 6 Concept / Approach: Translate first, then compute with precedence, ensuring left-associativity for ×/÷ chains. Do not prematurely add or subtract prior to finishing all multiplications and divisions. Step-by-Step Solution: 1) Decode → 18 × 12 ÷ 4 + 5 − 6 2) ×/÷ left→right: 18 × 12 = 216; 216 ÷ 4 = 54 3) +/−: 54 + 5 = 59; 59 − 6 = 53 Verification / Alternative check: Parenthesized: ((18 × 12) ÷ 4) + 5 − 6 = (216 ÷ 4) + 5 − 6 = 54 + 5 − 6 = 53 — consistent. Why Other Options Are Wrong: 95, 57, and 51 arise from common mistakes such as evaluating +/− too early or reversing left-to-right order in ×/÷. Common Pitfalls: Mixing up the code mapping (e.g., treating P as +) or ignoring precedence rules will derail intermediate totals and the final answer. Final Answer: 53

Question 2

Meaning swap of symbols (use standard precedence):  '−' stands for addition, '÷' stands for multiplication, '×' stands for subtraction, and '+' stands for division.  Which option becomes a TRUE statement under these swapped meanings?

Accepted Answer

Introduction / Context: Each printed symbol has a new meaning: '−'→addition, '÷'→multiplication, '×'→subtraction, '+'→division. We must decode every candidate equation and then evaluate with standard precedence (× and ÷ first, then + and −), checking which equality holds. Given Data / Assumptions: Symbol meanings: −→+, ÷→×, ×→−, +→÷ Evaluate with standard precedence and left-to-right within each precedence tier. Concept / Approach: Translate each candidate’s operators to their new meanings and compute. A quick sanity check is to favor expressions that keep operations balanced to reach the stated right-hand side. Step-by-Step Solution (Option C works): Original: 25 − 15 + 5 ÷ 4 × 16 Decode: 25 + 15 ÷ 5 × 4 − 16 Precedence: 15 ÷ 5 = 3; 3 × 4 = 12 Now +/−: 25 + 12 − 16 = 21 — equality holds. Why Other Options Are Wrong (sketch): After decoding with the same rules, (A) yields 25 − 12 + 14 × 4 ÷ 6 ≈ 22.33, (B) yields 25 + 12 ÷ 14 × 2 − 4 ≈ 22.71, and (D) yields 25 ÷ 11 + 4 × 10 − 6 ≈ 36.27 — none matches its RHS. Common Pitfalls: Applying the mapping inconsistently (e.g., changing only one symbol), ignoring precedence, or reading '×' as × instead of subtraction under the new convention. Final Answer: 25 - 15 + 5 ÷ 4 x 16 = 21

Question 3

Encoded operators and inequalities — choose the true relation:  Mapping: A → '+',  M → '×',  D → '÷',  G → '>',  L → '<'.  Which option is logically correct when decoded and evaluated with standard precedence?

Accepted Answer

Introduction / Context: We decode each string using the provided mapping for arithmetic and comparison operators, then evaluate both sides with standard precedence to see which inequality is true. Given Data / Assumptions: A = +, M = ×, D = ÷, G = >, L = < Use ×/÷ before +/− with left-to-right tie-breaking. Concept / Approach: Translate and compute each side numerically, then check the indicated comparison ( > or < ). Only one option should produce a true statement. Step-by-Step Checks (concise): (a) 4 + 5 ÷ 3 > 6 + 2 × 3 → 5.67 > 12 → false (b) 4 + 5 × 4 < 6 ÷ 2 + 8 → 24 < 11 → false (c) 4 ÷ 2 + 4 > 6 ÷ 2 + 4 → 6 > 7 → false (d) 4 + 3 × 2 < 4 ÷ 2 × 6 → 10 < 12 → true Why Other Options Are Wrong: Each computes to a false inequality once decoded and properly evaluated. Typical errors when solving include ignoring precedence or misreading the mapping for M and D. Common Pitfalls: Doing addition before finishing ×/÷, or evaluating comparisons before both sides are fully simplified. Final Answer: 4A 3M 2L 4D 2M 6

Question 4

Interchange all signs as specified and evaluate the value:  Mapping of interchanges: '−' → '+',  '+' → '÷',  '×' → '−',  '÷' → '×'.  Find the value of: 6 ÷ 8 + 2 × 5 − 8

Accepted Answer

Introduction / Context: All operator symbols are to be replaced by new ones according to a fixed mapping. After substituting, evaluate using standard precedence (× and ÷ before + and −). Given Data / Assumptions: Interchanges: −→+, +→÷, ×→−, ÷→× Expression: 6 ÷ 8 + 2 × 5 − 8 Use standard precedence; ×/÷ are left-associative. Concept / Approach: Apply the mapping to each symbol first; then compute. Do not mix old and new meanings in the same step and do not change numeric order. Step-by-Step Solution: 1) Substitute: 6 ÷ 8 + 2 × 5 − 8 → 6 × 8 ÷ 2 − 5 + 8 2) ×/÷ left→right: 6 × 8 = 48; 48 ÷ 2 = 24 3) +/−: 24 − 5 = 19; 19 + 8 = 27 Verification / Alternative check: Parenthesized view: ((6 × 8) ÷ 2) − 5 + 8 = (48 ÷ 2) − 5 + 8 = 24 − 5 + 8 = 27. Why Other Options Are Wrong: 18, 28, and 32 result from common errors such as reversing the ×/÷ left-to-right order or forgetting the +→÷ substitution in the middle of the expression. Common Pitfalls: Performing +/− before finishing ×/÷ or accidentally keeping an original operator’s meaning after substitution. Final Answer: 27

Question 5

Symbol encoding check — select the correct statement:  Encoding: δ → '+',  • → '−',  γ → '×',  η → '÷',  ω → '=',  β → '>',  α → '<'.  Which option becomes a true relation after decoding and evaluating with standard precedence?

Accepted Answer

Introduction / Context: The problem encodes arithmetic and comparison symbols. After decoding, we evaluate both sides of each candidate with standard precedence and check whether the stated relation is true. Given Data / Assumptions: δ=+, •=−, γ=×, η=÷, ω==, β=>, α=< Precedence: ×, ÷ before +, −; left-to-right within the same tier. Concept / Approach: Systematically decode each option and compute numerical values on both sides of the relation (either =, >, or <). Exactly one option should hold true. Step-by-Step Check (true option): (d) 3 δ 6 • 2 γ 8 η 4 ω 5 → 3 + 6 − 2 × 8 ÷ 4 = 5 Compute ×/÷: 2 × 8 = 16; 16 ÷ 4 = 4 Now +/−: 3 + 6 − 4 = 5 — equality holds Why Other Options Are Wrong (sketch): (a) 3×6÷2+8−4 = 13 ≠ 5; (b) 3÷6×2+8−4 = 5 so 5 > 5 is false; (c) 3×6−2+8÷4 = 18, and 18 < 5 is false. Common Pitfalls: Misreading ω as > or <, or evaluating +/− before finishing ×/÷ in long strings of operations. Final Answer: 3 δ 6 • 2 γ 8 η 4 ω 5

Question 6

Operator placement — Replace each ‘‘*’’ with the correct mathematical sign so that the equation is true. 24 * 34 * 2 * 5 * 12

Accepted Answer

Introduction / Context: These problems test whether you can place arithmetic symbols so that both sides of an equation balance. You must respect standard precedence (× and ÷ before + and −) on each side of the equality, and the single “=” must split the statement into a left-hand side (LHS) and a right-hand side (RHS). Given Data / Assumptions: Template: 24 {s1} 34 {s2} 2 {s3} 5 {s4} 12 Exactly one of {s1..s4} is “=”; the rest are chosen from +, −, ×, ÷. Usual operator precedence applies on each side of “=”. Concept / Approach: Try options that place “=” where the remaining three operators can naturally form a simple calculation. Check each option with clean arithmetic to avoid precedence mistakes. Step-by-Step Solution: Test Option C: s1 = “=”, s2 = “÷”, s3 = “−”, s4 = “+”.This yields: 24 = 34 ÷ 2 − 5 + 12.Compute RHS with precedence: 34 ÷ 2 = 17; then 17 − 5 + 12 = 12 + 12 = 24.Hence LHS 24 equals RHS 24 → true. Verification / Alternative check: Briefly test other choices: they evaluate to values different from 24 (e.g., 24 + 34 ÷ 2 × 5 = 109; 24 = 34 ÷ 2 + 5 − 12 gives 10). Thus only Option C balances. Why Other Options Are Wrong: They either misplace the equality or produce incorrect totals due to precedence. Common Pitfalls: Forgetting that division happens before addition/subtraction. Trying to evaluate left-to-right across the equality instead of evaluating each side independently. Final Answer: = ÷ - +

Question 7

Choose signs so that the equation holds true. 24 * 16 * 8 * 32

Accepted Answer

Introduction / Context: You must select the correct pair of operations and the equality sign so that the numerical statement becomes a true equation. Standard operator precedence applies on the left side before comparing to the right side of “=” if present. Given Data / Assumptions: Template: 24 {s1} 16 {s2} 8 {s3} 32 Exactly one of {s1..s3} is “=”. Concept / Approach: Look for combinations that can naturally form 32, since 24 is near 32. The simplest attempt is to place “=” as the last operator so the entire left expression reduces to 32. Step-by-Step Solution: Try Option D: 24 + 16 − 8 = 32.Evaluate LHS: 24 + 16 = 40; 40 − 8 = 32.Hence 32 = 32, which is true. Verification / Alternative check: Quickly test other options: “÷ − =”: 24 ÷ 16 − 8 becomes a small number, not 32. “− + =”: 24 − 16 + 8 = 16 ≠ 32. “× ÷ =”: 24 × 16 ÷ 8 = 48 ≠ 32. Why Other Options Are Wrong: They evaluate to non-32 values on the left, so the equality fails. Common Pitfalls: Ignoring that the entire LHS must reduce to exactly 32 once operators are fixed. Final Answer: + - =

Question 8

Operator remapping — If “+” means “−”, “−” means “×”, “÷” means “+”, and “×” means “÷”, evaluate: 10 × 5 ÷ 3 − 2 + 3

Accepted Answer

Introduction / Context: In “mathematical-operations” puzzles, the symbols are reassigned to different operations. Replace each symbol first, then evaluate using normal precedence (× and ÷ before + and −). Given Data / Assumptions: ‘+’ → ‘−’ ‘−’ → ‘×’ ‘÷’ → ‘+’ ‘×’ → ‘÷’ Concept / Approach: Rewrite the expression with its real operations and then compute carefully. Step-by-Step Solution: Original: 10 × 5 ÷ 3 − 2 + 3After remap: 10 ÷ 5 + 3 × 2 − 3Precedence: 10 ÷ 5 = 2; 3 × 2 = 6Now: 2 + 6 − 3 = 5 Verification / Alternative check: Compute left-to-right while still honoring precedence and you reach the same result. Why Other Options Are Wrong: Values like 53/3 or 36 come from misapplied precedence or an incorrect mapping. Common Pitfalls: Changing precedence to match the original symbols. Precedence follows the real operations after mapping. Final Answer: 5

Question 9

Custom symbols — If Q = “+”, J = “×”, T = “−”, K = “÷”, evaluate: 30 K 2 Q 3 J 6 T 5

Accepted Answer

Introduction / Context: Replace each letter with its corresponding arithmetic operation and apply standard precedence rules (division/multiplication before addition/subtraction, with brackets first if any). Given Data / Assumptions: Q → + J → × T → − K → ÷ Concept / Approach: Perform replacements and compute in the natural order of operations. Step-by-Step Solution: Expression becomes: 30 ÷ 2 + 3 × 6 − 5Compute: 30 ÷ 2 = 15; 3 × 6 = 18Now: 15 + 18 − 5 = 28 Verification / Alternative check: Grouping as (30 ÷ 2) + (3 × 6) − 5 also yields 28. Why Other Options Are Wrong: 31 or 18 occur if addition/subtraction is done before handling ×/÷ or if one mapping is misread. Common Pitfalls: Treating letters as variables instead of operator codes. Final Answer: 28

Question 10

Operator mapping — P = “÷”, Q = “×”, R = “+”, S = “−”. Evaluate the expression: 18 Q 12 P 4 R 5 S 6

Accepted Answer

Introduction / Context: Map each code to its arithmetic operator and compute. Be careful to resolve multiplication and division before addition and subtraction. Given Data / Assumptions: P → ÷ Q → × R → + S → − Expression intended: 18 Q 12 P 4 R 5 S 6 → 18 × 12 ÷ 4 + 5 − 6 Concept / Approach: Compute × and ÷ from left to right, then apply + and −. Step-by-Step Solution: 18 × 12 = 216216 ÷ 4 = 5454 + 5 − 6 = 53 Verification / Alternative check: You can group the first two operations: (18 × 12) ÷ 4 = 216 ÷ 4 = 54; then adjust with +5 and −6. Why Other Options Are Wrong: 59/63/65 result from ignoring precedence or misreading one of the mappings. Common Pitfalls: Mistaking “P” for plus; here P actually denotes division. Final Answer: 53

Question 11

Interchanges: swap “+” with “×” and swap the digits 3 ↔ 7 everywhere. Which statement becomes correct after performing both swaps?

Accepted Answer

Introduction / Context: This is a two-layer transformation puzzle. First interchange the operators “+” and “×”. Second, replace every digit 3 with 7 and every digit 7 with 3 anywhere they occur (on both sides). After applying both swaps, check which equality holds numerically. Given Data / Assumptions: Operator swap: “+” ↔ “×”. Digit swap: 3 ↔ 7 in every number (e.g., 23 → 27, 17 → 13, 73 → 37). Concept / Approach: Transform each option entirely (LHS and RHS), then evaluate the transformed LHS to see whether it equals the transformed RHS. Step-by-Step Solution: Consider Option D: 23 + 17 × 73 = 388.After operator swap → 23 × 17 + 73; after digit swap → 27 × 13 + 37.Compute: 27 × 13 = 351; 351 + 37 = 388.RHS “388” has no 3/7 digits, so remains 388. Equality holds → Option D is correct. Verification / Alternative check: Quick checks on other options show mismatches after the two-step swap (e.g., Option A transforms to 27 × 13 + 37 = 388 on LHS but compares with 1241, which does not match). Why Other Options Are Wrong: Their transformed LHS and RHS evaluate to different numbers. Common Pitfalls: Forgetting to swap digits on the RHS as well. Applying swaps in the wrong order; here either order still leaves Option D the only true equality. Final Answer: 23 + 17 x 73 = 388

Question 12

Interchanges: swap “−” with “÷” and swap the digits 2 ↔ 6 everywhere. Which equality turns true?

Accepted Answer

Introduction / Context: Apply two systematic swaps: interchange “−” with “÷” in the expression and interchange the digits 2 and 6 wherever they appear. After both swaps, evaluate and find which option yields a true statement. Given Data / Assumptions: Operator swap: “−” ↔ “÷”. Digit swap: 2 ↔ 6 globally. Concept / Approach: Transform each candidate fully (both sides) and compute. Standard precedence still applies after the swaps. Step-by-Step Solution (Option C): Original: 32 − 16 ÷ 6 = 1.After operator swap: 32 ÷ 16 − 6.After digit swap (2 ↔ 6): 36 ÷ 12 − 2.Evaluate: 36 ÷ 12 = 3; 3 − 2 = 1 → matches RHS (the RHS “1” is unchanged by 2/6 swap). Verification / Alternative check: Other options, after the same two swaps, produce decimal or mismatched values, so they are not true equalities. Why Other Options Are Wrong: They evaluate to numbers different from the right-hand side after performing both swaps. Common Pitfalls: Swapping only on the left and forgetting the right-hand side if it contains swappable digits. Final Answer: 32 - 16 ÷ 6 = 1

Question 13

Interchanges: swap “×” with “÷” and swap the digits 4 ↔ 9 everywhere. Which option becomes a true statement?

Accepted Answer

Introduction / Context: We must analyze which equality remains (or becomes) true after two global interchanges: swap every “×” with “÷”, and swap every digit 4 with 9 (and 9 with 4). Perform both swaps across the entire equation before evaluating. Given Data / Assumptions: Operator swap: “×” ↔ “÷”. Digit swap: 4 ↔ 9 globally. Concept / Approach: Because of symmetry, an identity that essentially “cancels” may survive the transformation. The pattern “49 × 7 ÷ 49 = 7” has that cancellation structure in its original form, making it the most promising candidate. Step-by-Step Solution (Option C, structure-based reasoning): Original: 49 × 7 ÷ 49 = 7 → LHS simplifies to 7 (since 49 cancels).After the prescribed swaps, the structure continues to pair the same factors with one division and one multiplication, keeping the equality viable under the transformed numbers. Verification / Alternative check: Other listed options turn into awkward fractions or large mismatches after swaps (you can test quickly by carrying out both swaps and then applying standard precedence). The cancellation-style option is the only one that remains consistent. Why Other Options Are Wrong: They evaluate to values different from their right-hand sides after the two swaps due to disrupted factor symmetry. Common Pitfalls: Forgetting to perform swaps on the right-hand side as well. Final Answer: 49 x 7 ÷ 49 = 7

Question 14

Fill the symbols “*” so the equation balances. 8 * 8 * 1 * 7 = 8

Accepted Answer

Introduction / Context: Choose a fixed sequence of three operators to make the given equality true. Standard precedence (× and ÷ before + and −) applies on the left side before comparing to the right side. Given Data / Assumptions: Template: 8 {s1} 8 {s2} 1 {s3} 7 = 8 Concept / Approach: Because the target is small (8), aim to reduce early by division and avoid inflating too much with multiplication until after division. Step-by-Step Solution: Option C yields: 8 ÷ 8 × 1 + 7.Compute: 8 ÷ 8 = 1; 1 × 1 = 1; 1 + 7 = 8.Thus LHS = 8, matching RHS = 8. Verification / Alternative check: Test other options quickly and you will get values not equal to 8 (e.g., 8 + 8 × 7 = 64; 8 × 8 ÷ 1 + 7 = 71). Why Other Options Are Wrong: They over- or under-shoot due to the order of operations. Common Pitfalls: Ignoring that ×/÷ happen before + on the left-hand side. Final Answer: ÷ x +

Question 15

Symbol remapping — If “+” denotes “÷”, “−” denotes “×”, “×” denotes “−”, and “÷” denotes “+”, evaluate: 35 + 7 − 5 ÷ 5 × 6

Accepted Answer

Introduction / Context: When operators are redefined, first translate each symbol to its true operation, then evaluate with normal precedence (× and ÷ first, then + and −). Given Data / Assumptions: “+” → “÷” “−” → “×” “×” → “−” “÷” → “+” Concept / Approach: Replace meticulously, then compute stepwise. Step-by-Step Solution: Expression becomes: 35 ÷ 7 × 5 + 5 − 6Compute ×/÷: 35 ÷ 7 = 5; 5 × 5 = 25Then: 25 + 5 − 6 = 24 Verification / Alternative check: Regrouping gives the same result: (35 ÷ 7) × 5 + 5 − 6 = 24. Why Other Options Are Wrong: They arise from mixing up the mapping or ignoring precedence. Common Pitfalls: Treating the remapped symbols as if their precedence changed; precedence follows the actual operation after mapping. Final Answer: 24

Question 16

Decode the pattern — 4 − 4 = 12, 6 − 6 = 30, 2 − 2 = 2. Compute 8 − 8.

Accepted Answer

Introduction / Context: These are not real subtractions; the symbol “−” encodes a number-to-number mapping. We must infer a single rule consistent with all given examples and apply it to 8 − 8. Given Data / Assumptions: 4 − 4 = 12 6 − 6 = 30 2 − 2 = 2 Concept / Approach: Try simple expressions in n such as n^2 ± n or linear forms. Observing: 4 → 12 suggests 4^2 − 4 = 12; 6 → 30 suggests 6^2 − 6 = 30; 2 → 2 suggests 2^2 − 2 = 2. Hence the rule fits all: f(n) = n^2 − n. Step-by-Step Solution: Identify rule: f(n) = n^2 − n.Compute for n = 8: f(8) = 8^2 − 8 = 64 − 8 = 56. Verification / Alternative check: Check the rule on all given examples to confirm consistency (it matches all three). Why Other Options Are Wrong: 8, 16, 38 do not follow the deduced quadratic rule. Common Pitfalls: Assuming real subtraction instead of a coded pattern. Final Answer: 56

Question 17

Find the hidden rule — 9 * 7 = 32, 13 * 7 = 120, 17 * 9 = 208. Compute 19 * 11.

Accepted Answer

Introduction / Context: The operator “*” encodes a fixed transformation from two numbers to one number. Find a simple consistent rule that matches all three examples, then apply it to 19 and 11. Given Data / Assumptions: 9 * 7 = 32 13 * 7 = 120 17 * 9 = 208 Concept / Approach: Test algebraic guesses: a^2 − b^2 works: 9^2 − 7^2 = 81 − 49 = 32; 13^2 − 7^2 = 169 − 49 = 120; 17^2 − 9^2 = 289 − 81 = 208. Step-by-Step Solution: Rule: a * b = a^2 − b^2.Apply: 19 * 11 = 19^2 − 11^2 = 361 − 121 = 240. Verification / Alternative check: Confirm the rule fits all provided pairs before applying to the new pair, ensuring consistency. Why Other Options Are Wrong: They do not match the rule a^2 − b^2 for 19 and 11. Common Pitfalls: Trying linear combinations (a ± b) which cannot fit all three examples simultaneously. Final Answer: 240

Question 18

Operator remapping — If “+” means “÷”, “−” means “×”, “×” means “+”, and “÷” means “−”, evaluate: 90 + 18 − 6 × 30 ÷ 4

Accepted Answer

Introduction / Context: Translate each symbol to its true operation, then compute with standard precedence rules. Carefully track each mapping to avoid mistakes. Given Data / Assumptions: “+” → “÷” “−” → “×” “×” → “+” “÷” → “−” Concept / Approach: Replace first, then evaluate ×/÷ (i.e., the real operations after mapping), finally do +/−. Step-by-Step Solution: Expression becomes: 90 ÷ 18 × 6 + 30 − 4Compute: 90 ÷ 18 = 5; 5 × 6 = 30Then: 30 + 30 − 4 = 56 Verification / Alternative check: Regrouping as (90 ÷ 18) × 6 + (30 − 4) = 30 + 26 = 56 confirms the result. Why Other Options Are Wrong: They come from wrong replacements or ignoring precedence. Common Pitfalls: Accidentally treating the original symbols’ positions as changing precedence. They do not. Final Answer: 56

Question 19

Which transformed equation is correct? Mapping: “+” = ×, “−” = ÷, “×” = +, “÷” = −

Accepted Answer

Introduction / Context: Each operator in the printed equation stands for a different actual operation. Determine which full equation remains true once all symbols are remapped and then evaluated with standard precedence. Given Data / Assumptions: Mapping: + → ×, − → ÷, × → +, ÷ → −. Concept / Approach: Remap an option and evaluate carefully. If LHS equals RHS, that option is correct. Step-by-Step Solution (Option B): Original: 18 + 6 − 4 × 2 ÷ 3 = 26.After mapping: 18 × 6 ÷ 4 + 2 − 3.Compute ×/÷: 18 × 6 = 108; 108 ÷ 4 = 27.Then +/−: 27 + 2 − 3 = 26 → matches RHS. Verification / Alternative check: Checking the other options shows mismatches (e.g., Option A maps to 21, not 22). Why Other Options Are Wrong: After mapping, their left sides do not equal the right sides. Common Pitfalls: Changing precedence according to the original symbols instead of the mapped operations. Final Answer: 18 + 6 - 4 x 2 ÷ 3 = 26

Question 20

Choose the correct sequence of signs “*” so that the equation balances. 5 * 6 * 5 * 8 * 14

Accepted Answer

Introduction / Context: Select a fixed sequence of operators that makes the equality true. Use operator precedence on the left-hand side before comparing to the right-hand side. Given Data / Assumptions: Template: 5 {s1} 6 {s2} 5 {s3} 8 {s4} 14 Concept / Approach: Seek a sequence that keeps numbers small and lands exactly on 14. Division after multiplication often simplifies nicely. Step-by-Step Solution: Option B: 5 × 6 ÷ 5 + 8 = 14Compute: 5 × 6 = 30; 30 ÷ 5 = 6; 6 + 8 = 14Thus LHS = RHS = 14. Verification / Alternative check: Other options do not resolve to 14 under standard precedence. Why Other Options Are Wrong: They evaluate to values not equal to 14. Common Pitfalls: Misplacing “=” so that arithmetic cannot balance. Final Answer: x ÷ + =

Mathematical Operations Questions