Question 1

Operator substitution — If “+” means minus, “−” means multiply, “÷” means plus, and “×” means divide, evaluate: 10 × 5 ÷ 3 − 2 + 3

Accepted Answer

Introduction / Context: This is a symbol-redefinition puzzle. Each symbol is reassigned to a different arithmetic operation. After replacing symbols, evaluate respecting normal precedence (× and ÷ before + and −) unless brackets dictate otherwise. Given Data / Assumptions: “+” → minus (−) “−” → multiply (×) “÷” → plus (+) “×” → divide (÷) Concept / Approach: Rewrite the expression with the true operations, then compute with standard precedence. Step-by-Step Solution: Original: 10 × 5 ÷ 3 − 2 + 3Swap: 10 ÷ 5 + 3 × 2 − 3Compute ×/÷ first: 10 ÷ 5 = 2; 3 × 2 = 6.Then +/−: 2 + 6 − 3 = 5. Verification / Alternative check: Evaluate left-to-right but still honoring precedence: same result 5. Why Other Options Are Wrong: 53/3, 21, 36 arise from misapplying precedence or incorrect replacements. Common Pitfalls: Treating the redefined symbols with altered precedence; precedence stays the same as the actual operation. Final Answer: 5

Question 2

If "T" means (x), 'U' means (-) , 'X' means (÷) and W means (+), then what will be the value of the following expression? (50 X 2) W (28 T 4 )

Accepted Answer

Given expression, (50 x 2) W (28 T 4) After interchanging the letters with symbols, we get (50 ÷ 2) + (28 x 4) = 25 + 112 = 137

Question 3

Custom symbols — If T = ×, U = −, X = ÷, and W = +, evaluate (50 X 2) W (28 T 4)

Accepted Answer

Introduction / Context: Symbol mapping questions require careful replacement followed by normal arithmetic precedence. Brackets must be evaluated first. Given Data / Assumptions: T → multiply (×) U → minus (−) X → divide (÷) W → plus (+) Expression: (50 X 2) W (28 T 4) Concept / Approach: Replace symbols, compute bracketed parts, then add. Step-by-Step Solution: 1) Replace: (50 ÷ 2) + (28 × 4).2) Inside brackets: 50 ÷ 2 = 25; 28 × 4 = 112.3) Sum: 25 + 112 = 137. Verification / Alternative check: Compute again independently: both parts 25 and 112 are straightforward, summing to 137. Why Other Options Are Wrong: Values like 142/158/163 stem from swapping one of the mappings or adding before multiplying. Common Pitfalls: Forgetting that X means division here. Final Answer: 137

Question 4

Fill in operators — Choose the correct sequence of signs to make the statement true: 40 * 2 * 4 * 3 * 8

Accepted Answer

Introduction / Context: We must pick a sequence of operations so that the expression becomes a true equation. The four symbols fill in order and one of them must be “=” to split left and right sides. Given Data / Assumptions: Template: 40 {s1} 2 {s2} 4 {s3} 3 {s4} 8 Standard precedence applies on each side of the “=”. Concept / Approach: Try choices that place “=” as the third symbol so the last symbol lands on the right-hand side. Option B yields a clean check. Step-by-Step Solution: Option B: s1=÷, s2=+, s3==, s4=× → 40 ÷ 2 + 4 = 3 × 8.Left: 40 ÷ 2 + 4 = 20 + 4 = 24.Right: 3 × 8 = 24.Equality holds → Option B is correct. Verification / Alternative check: Test other options quickly; they fail to balance to a single equality. Why Other Options Are Wrong: They either misplace “=” or produce unequal numeric results. Common Pitfalls: Ignoring precedence and evaluating strictly left-to-right across the equality. Final Answer: ÷ + = x

Question 5

Pattern decoding — Given 4 − 4 = 17, 6 − 6 = 37, 2 − 2 = 5, find the value of 5 − 5

Accepted Answer

Introduction / Context: These items encode a number transformation rather than ordinary subtraction. We must infer the hidden rule consistent with the examples and apply it to 5 − 5. Given Data / Assumptions: 4 − 4 = 17, 6 − 6 = 37, 2 − 2 = 5. Concept / Approach: Observe that n − n maps to n^2 + 1: 4^2+1=17, 6^2+1=37, 2^2+1=5. The same rule applied to n=5 yields 5^2+1=26. Step-by-Step Solution: 1) Hypothesize mapping f(n) = n^2 + 1.2) Validate on given examples.3) Compute f(5) = 25 + 1 = 26. Verification / Alternative check: Try alternate simple forms (e.g., 2n+9); they do not fit all three examples, reinforcing n^2+1. Why Other Options Are Wrong: 27/24/20 do not match the consistent quadratic pattern across the exemplars. Common Pitfalls: Assuming standard subtraction instead of a coded rule. Final Answer: 26

Question 6

Custom operators — If S = “−”, Q = “×”, R = “÷”, and P = “+”, evaluate 1 P 45 R 2 Q 2 S 4

Accepted Answer

Introduction / Context: Again, symbols are remapped to standard operations. Compute with normal precedence after replacement. Given Data / Assumptions: P → +, R → ÷, Q → ×, S → −. Expression: 1 + 45 ÷ 2 × 2 − 4. Concept / Approach: Evaluate division and multiplication first, then addition and subtraction. Step-by-Step Solution: 1) 45 ÷ 2 = 22.5.2) 22.5 × 2 = 45.3) 1 + 45 − 4 = 42. Verification / Alternative check: Re-group as (1 + (45 ÷ 2 × 2)) − 4; same result 42. Why Other Options Are Wrong: 40/36/46 arise from left-to-right evaluation ignoring precedence or swapping a mapping. Common Pitfalls: Changing the precedence based on the original letters; precedence follows the real operation after mapping. Final Answer: 42

Question 7

Insert two operators — Make 6 ▢ 5 ▢ 4 equal 34 by choosing the correct pair of signs

Accepted Answer

Introduction / Context: We must place two operators to obtain a specific target. Try multiplication first since 34 is near 6×5=30. Given Data / Assumptions: Expression: 6 ▢ 5 ▢ 4 = 34. Standard precedence applies. Concept / Approach: Check combinations that can produce a low-30s number. Step-by-Step Solution: 1) Try “× and +”: 6 × 5 + 4 = 30 + 4 = 34 → works.2) Other pairs quickly fail: 6 − 5 + 4 = 5; ÷ with × gives non-integer or large deviations. Verification / Alternative check: Ensure no other pair yields 34 under standard precedence; they do not. Why Other Options Are Wrong: They do not equal 34. Common Pitfalls: Forgetting that × precedes +; but even left-to-right, “× and +” still yields 34 here. Final Answer: x and +

Question 8

Insert two operators — Make 65 ▢ 40 ▢ 11 equal 36 by choosing the correct pair of signs

Accepted Answer

Introduction / Context: We choose two operators to achieve 36. Consider subtraction then addition to pull 65 down towards 36. Given Data / Assumptions: Expression: 65 ▢ 40 ▢ 11 = 36. Concept / Approach: Try 65 − 40 + 11 = 36. Step-by-Step Solution: Compute: 65 − 40 + 11 = 25 + 11 = 36. Verification / Alternative check: Other operator pairs overshoot or undershoot the target. Why Other Options Are Wrong: They fail to evaluate to 36 under standard precedence. Common Pitfalls: Trying multiplication/division first, which pushes the value far from 36. Final Answer: - and +

Question 9

Choose symbol sequence — Replace * to balance: 7 * 7 * 2 * 1 = 12

Accepted Answer

Introduction / Context: We must select a fixed sequence of three operators to make the equation true. Standard precedence applies. Given Data / Assumptions: Template: 7 {s1} 7 {s2} 2 {s3} 1 = 12. Concept / Approach: Test reasonable combinations; subtraction and multiplication often help reach small targets like 12. Step-by-Step Solution: Option B (+ - ×): 7 + 7 - 2 × 1 = 14 - 2 = 12 → true.Other options do not yield 12 under precedence. Verification / Alternative check: Explicitly compute each alternative; only Option B balances. Why Other Options Are Wrong: They produce values like 47, 12 by chance only if precedence is ignored, or other mismatches. Common Pitfalls: Ignoring that × takes precedence over +/−. Final Answer: + - x

Question 10

Decode the rule — If 73 + 82 = 14 and 91 + 21 = 11, then compute 86 + 24

Accepted Answer

Introduction / Context: These are not ordinary additions; a hidden rule maps two two-digit numbers to a small result. We must infer and apply the rule. Given Data / Assumptions: 73 + 82 = 14 91 + 21 = 11 Concept / Approach: A compact rule fitting both examples is: (sum of tens digits) − (difference of units digits) = result. Formally, (a + c) − |b − d| for numbers ab and cd. Step-by-Step Solution: 1) Check 73 & 82: (7 + 8) − |3 − 2| = 15 − 1 = 14 → fits.2) Check 91 & 21: (9 + 2) − |1 − 1| = 11 − 0 = 11 → fits.3) Apply to 86 & 24: (8 + 2) − |6 − 4| = 10 − 2 = 8. Verification / Alternative check: Other simple hypotheses (sum of all digits, etc.) fail on both examples, confirming this rule. Why Other Options Are Wrong: 9/6/62 do not follow the established mapping. Common Pitfalls: Assuming ordinary addition. Final Answer: 8

Question 11

Fill in signs — Choose the sequence that makes a true equation: 24 * 4 * 5 * 4

Accepted Answer

Introduction / Context: We must choose three symbols that, when substituted in order, yield a correct equality. The placement of “=” divides left and right sides. Given Data / Assumptions: Template: 24 {s1} 4 {s2} 5 {s3} 4 Try options where “=” yields a sensible split. Concept / Approach: Testing Option B produces an immediately true equality. Step-by-Step Solution: Option B: s1 = “=”, s2 = “×”, s3 = “+”. Statement becomes 24 = 4 × 5 + 4.Right side: 4 × 5 + 4 = 20 + 4 = 24.Hence balanced. Verification / Alternative check: Other placements of “=” do not simplify to an identity with remaining symbols. Why Other Options Are Wrong: They evaluate to values not equal to 24. Common Pitfalls: Forgetting that multiplication precedes addition on the right side. Final Answer: = x +

Question 12

Operator placement to balance equality — Replace each asterisk with the correct mathematical sign so that the equation is true.  16 * 6 * 4 * 24

Accepted Answer

Introduction / Context: These 'mathematical operations' questions ask you to replace placeholders with the correct arithmetic signs so that the expression forms a valid equality. The expression has three gaps (between four numbers), so we must choose a triple of signs such that, when evaluated with standard precedence (× and ÷ before + and −), both sides of the equality are consistent. Given Data / Assumptions: Target string: 16 * 6 * 4 * 24 Each * must be replaced by one of {+, −, ×, ÷, =} according to the offered combinations. Conventional operator precedence applies: × and ÷ before + and −; equality '=' separates the left and right sides once it appears. Concept / Approach: Interpret each option as the ordered triple of symbols to be inserted between (16, 6, 4, 24). For example, 'x ÷ =' means 16 × 6 ÷ 4 = 24. Evaluate the arithmetic strictly by precedence and check if the resulting statement is true. Step-by-Step Solution: Try option 'x ÷ =': 16 × 6 ÷ 4 = 24.Compute left side: 16 × 6 = 96; 96 ÷ 4 = 24.Therefore 24 = 24, which is true.Check a counterexample to confirm uniqueness: 'x = ÷' would read 16 × 6 = 4 ÷ 24 ⇒ 96 = 1/6, false. Verification / Alternative check: You can also verify by moving all terms to one side and confirming it equals zero, but directly checking equality with precedence is faster here. Why Other Options Are Wrong: '÷ = x': 16 ÷ 6 = 4 × 24 ⇒ 2.666… = 96, false. 'x = ÷': 16 × 6 = 4 ÷ 24 ⇒ 96 = 0.0416…, false. '= ÷ ÷': 16 = 6 ÷ 4 ÷ 24 ⇒ 16 = 1/16, false. 'None of these': unnecessary because a valid option exists. Common Pitfalls: Ignoring operator precedence (e.g., computing from left to right only) or misplacing the equality sign in your mental parsing. Always read the option strictly as the three ordered replacements. Final Answer: x ÷ =

Question 13

Fill operators to make a true statement — Replace the asterisks with the correct signs so that the equality holds.  9 * 3 * 3 * 3 * 6

Accepted Answer

Introduction / Context: We must choose a sequence of four symbols that, when inserted between the five numbers (9, 3, 3, 3, 6), yields a true equation under standard arithmetic precedence. These puzzles test careful operator placement and precedence awareness. Given Data / Assumptions: Expression skeleton: 9 * 3 * 3 * 3 * 6. Each option provides four symbols; the last one is '=' to assert equality to the final number. Standard precedence: × and ÷ before + and −. Concept / Approach: Interpret each option literally as the four replacements in order. For example, '÷ x - =' gives 9 ÷ 3 × 3 − 3 = 6. Evaluate with precedence (first ÷ and ×), then addition/subtraction, and finally compare both sides. Step-by-Step Solution: Using option '÷ x - =': compute left side.9 ÷ 3 = 3; 3 × 3 = 9; 9 − 3 = 6.Thus 6 = 6, the statement is true. Verification / Alternative check: Briefly test another option to confirm it fails. For instance, 'x + - =': 9 × 3 + 3 − 3 = 27 + 3 − 3 = 27 ≠ 6. Why Other Options Are Wrong: '+ - x =': 9 + 3 − 3 × 3 = 9 + 3 − 9 = 3 ≠ 6. '- + + =': 9 − 3 + 3 + 3 = 12 ≠ 6. 'x + - =': 27 + 3 − 3 = 27 ≠ 6. 'None of these': not needed because a valid sequence exists. Common Pitfalls: Overriding precedence (e.g., doing addition before multiplication) or reading the symbols out of order. Always apply the operators exactly where the asterisks are, in the given order. Final Answer: ÷ x - =

Question 14

Insert signs to satisfy a zero result — Choose the correct ordered signs for the blanks in:  8 * 6 * 96 * 2 = 0

Accepted Answer

Introduction / Context: Here, we must create an expression equal to zero by selecting a specific sequence of three operators. Because × and ÷ dominate precedence, structuring the middle subtraction to balance two multiplicative terms is a common path to zero. Given Data / Assumptions: Template: 8 * 6 * 96 * 2 = 0. We replace the three asterisks with the ordered symbols from one option. Standard precedence applies. Concept / Approach: Try to form A − B where A and B are equal. Choosing 'x - ÷' yields 8 × 6 − 96 ÷ 2, which is promising because 8 × 6 = 48 and 96 ÷ 2 = 48, giving 48 − 48 = 0. Step-by-Step Solution: Select 'x - ÷': 8 × 6 − 96 ÷ 2.Compute: 8 × 6 = 48; 96 ÷ 2 = 48.Difference: 48 − 48 = 0.Therefore the equation holds. Verification / Alternative check: Check one alternative: 'x ÷ -' ⇒ 8 × 6 ÷ 96 − 2 = 48 ÷ 96 − 2 = 0.5 − 2 ≠ 0. Why Other Options Are Wrong: '- x ÷': 8 − 6 × 96 ÷ 2 is dominated by the large product term and won’t be zero. '÷ - x': 8 ÷ 6 − 96 × 2 is strongly negative, not zero. 'None of these': unnecessary because a correct option exists. Common Pitfalls: Evaluating strictly left-to-right and ignoring precedence; you must compute × and ÷ before the subtraction to see the intended cancellation. Final Answer: x - ÷

Question 15

Make the equation correct by swapping numbers — Which interchange of numbers makes the statement true?  8 × 20 ÷ 3 + 9 − 5 = 38

Accepted Answer

Introduction / Context: Interchange problems ask which pair of numbers, when swapped, turns a false arithmetic statement into a true one. You must respect operator precedence when reevaluating the expression after each hypothetical swap. Given Data / Assumptions: Original expression: 8 × 20 ÷ 3 + 9 − 5. Goal value: 38. Allowed operation: swap exactly one of the given pairs of numbers in the options. Concept / Approach: Compute the expression after each candidate swap, honoring ×/÷ precedence. We look for a combination that lands exactly at 38. Step-by-Step Solution: Test (3, 5) swap ⇒ 8 × 20 ÷ 5 + 9 − 3.Evaluate: 8 × 20 = 160; 160 ÷ 5 = 32; 32 + 9 = 41; 41 − 3 = 38.Hence the swap (3, 5) makes the equation true. Verification / Alternative check: Check a distractor quickly: swap (8, 9) ⇒ 9 × 20 ÷ 3 + 8 − 5 = 60 + 3 = 63, not 38. Why Other Options Are Wrong: (3, 9): 8 × 20 ÷ 9 + 3 − 5 ≈ 11.78, not 38. (3, 8): 3 × 20 ÷ 8 + 9 − 5 = 7.5 + 9 − 5 = 11.5. (8, 9): evaluates to 63 as shown. 'None of these': a valid swap does exist. Common Pitfalls: Applying operations strictly left-to-right without precedence or swapping a number but forgetting to apply it at both of its original positions if it appears multiple times (here, each appears once). Final Answer: 3, 5

Question 16

Make the equation true by swapping signs — Which interchange of signs makes the statement correct?  (18 ÷ 9) + 3 × 5 = 45

Accepted Answer

Introduction / Context: Sign-interchange problems ask you to swap two operator symbols everywhere they occur to make a false equation true. Parentheses and operator precedence must still be respected. Given Data / Assumptions: Original: (18 ÷ 9) + 3 × 5 = 17, which is not 45. We may interchange a specified pair of signs throughout the expression. Standard precedence applies after the swap. Concept / Approach: Try swapping '+' and '÷' wherever they appear. The division inside parentheses becomes addition, and the addition between terms becomes division, potentially rebalancing the expression. Step-by-Step Solution: Swap '+' ↔ '÷': Expression becomes (18 + 9) ÷ 3 × 5.Evaluate: 27 ÷ 3 = 9; 9 × 5 = 45.Therefore, after the swap, the equation equals 45 and is correct. Verification / Alternative check: Check another option: swapping '×' and '÷' yields (18 × 9) + 3 ÷ 5 = 162.6…, not 45. Why Other Options Are Wrong: 'x ÷': makes the value ≫ 45. Swapping numbers ('18 and 5' or '3 and 9') does not achieve 45 under precedence. 'None of these': unnecessary because '+ ÷' works. Common Pitfalls: Swapping only one occurrence instead of all occurrences, or discarding parentheses impact when recalculating. Final Answer: + ÷

Question 17

Make the equation true by swapping signs — Which interchange of signs makes the equality hold?  5 + 6 ÷ 3 − 12 × 2 = 17

Accepted Answer

Introduction / Context: We are to interchange two operator symbols everywhere they appear so that the evaluated expression equals 17. Because × and ÷ have higher precedence, interchanging them can drastically change the computed value and is a natural candidate to test. Given Data / Assumptions: Original expression: 5 + 6 ÷ 3 − 12 × 2. Goal: value 17. We evaluate with standard precedence after performing the chosen global sign swap. Concept / Approach: Swap '÷' and '×' globally. This changes the division into multiplication and the multiplication into division, potentially balancing the large product term against the rest to reach 17. Step-by-Step Solution: After swapping '÷' ↔ '×': 5 + 6 × 3 − 12 ÷ 2.Compute: 6 × 3 = 18; 12 ÷ 2 = 6.Now evaluate: 5 + 18 − 6 = 17.Hence the equation is satisfied. Verification / Alternative check: Swapping '+' and '×' gives 5 × 6 ÷ 3 − 12 + 2 = 10 − 12 + 2 = 0, not 17; so those distractors fail. Why Other Options Are Wrong: '+ and x': produces 0. '+ and ÷': yields 5 ÷ 6 × 3 − 12 + 2 ≠ 17. '+ and -': merely swaps the final additions/subtractions and cannot fix the large product term. 'None of these': unnecessary. Common Pitfalls: Forgetting to apply the swap to all occurrences or disregarding precedence when re-evaluating the new expression. Final Answer: ÷ and x

Question 18

Operator remapping puzzle — If '+' means '−', '−' means '×', '×' means '÷', and '÷' means '+', evaluate:  16 × 2 ÷ 25 + 7 − 4

Accepted Answer

Introduction / Context: In 'coded-operator' problems, each printed symbol stands for a different real operation. The task is to translate the expression into the true operations first, and only then evaluate it with standard precedence. Doing arithmetic directly on the fake symbols leads to wrong results. Given Data / Assumptions: Mappings: '+' → '−'; '−' → '×'; '×' → '÷'; '÷' → '+'. Printed expression: 16 × 2 ÷ 25 + 7 − 4. Standard precedence after translation. Concept / Approach: Replace each symbol token-by-token using the mapping to obtain the real expression. Then compute ×/÷ first, followed by +/−. Keep numbers in the same order; only the operators change. Step-by-Step Solution: Translate: '×' → '÷', '÷' → '+', '+' → '−', '−' → '×'.Thus: 16 ÷ 2 + 25 − 7 × 4.Compute precedence: 16 ÷ 2 = 8; 7 × 4 = 28.Combine: 8 + 25 − 28 = 33 − 28 = 5. Verification / Alternative check: Re-check each mapped operator carefully; a single mis-map (e.g., treating '−' as '+' by mistake) changes the outcome drastically. Why Other Options Are Wrong: 104, 22, and 4 stem from computing on the printed symbols or misapplying precedence. The correct, fully translated and evaluated result is 5. Common Pitfalls: Applying the mapping to numbers (never do this), or changing evaluation order to left-to-right instead of using precedence. Final Answer: 5

Question 19

Find the true equation under remapped signs — With '−' ≡ '÷', '+' ≡ '×', '÷' ≡ '−', and '×' ≡ '+', which option becomes a correct equality?

Accepted Answer

Introduction / Context: Each printed operator represents a different real operation. To test which printed equation is true, translate every operator according to the given mapping and then evaluate with normal precedence. Only one choice will balance exactly. Given Data / Assumptions: Mapping: '−' → '÷'; '+' → '×'; '÷' → '−'; '×' → '+'. We must check options (a)–(d) under this mapping. Use standard precedence after translation. Concept / Approach: Translate each option's left-hand side (LHS) into the real arithmetic expression and evaluate. Compare with the right-hand side (RHS). The correct option is the one for which LHS equals RHS exactly. Step-by-Step Solution (for correct option b): Printed: 49 - 7 + 3 ÷ 5 x 8 = 24.Translate ops: '-' → '÷', '+' → '×', '÷' → '−', 'x' → '+'.Real LHS: 49 ÷ 7 × 3 − 5 + 8.Compute precedence: 49 ÷ 7 = 7; 7 × 3 = 21.Then: 21 − 5 + 8 = 16 + 8 = 24 = RHS. Verification / Alternative check (show a failure): Option a translates to 49 × 7 ÷ 3 + 5 − 8 ≈ 111.33, not 20; thus it is false. Why Other Options Are Wrong: They evaluate to values different from their RHS after proper translation and precedence. Only option (b) balances exactly at 24. Common Pitfalls: Forgetting that the mapping applies to every operator occurrence or ignoring precedence after translation. Translate first, then compute. Final Answer: 49 - 7 + 3 ÷ 5 x 8 = 24

Question 20

Select the correct combination of mathematical signs to replace each ‘‘*’’ so that the equation becomes true. Fill the three positions in order for: 6 * 4 * 12 * 12

Accepted Answer

Introduction / Context: This is a sign-substitution puzzle: each asterisk must be replaced by a mathematical sign (in the sequence suggested by an option) so that the resulting statement is a true equality. Given Data / Assumptions: Template: 6 * 4 * 12 * 12 Options provide an ordered triple of signs to place at the three starred positions. Standard operator meanings and standard order of operations apply. Concept / Approach: Try each option by substituting signs left to right and check whether the resulting statement evaluates to a true equation. Step-by-Step Solution: Option D → 6 x 4 - 12 = 12Compute: 6 × 4 = 24; 24 − 12 = 12; equality holds.Other options either do not yield an equality or give false statements (e.g., 6 ÷ 4 − 12 = 12 is false). Verification / Alternative check: Re-evaluating with precedence confirms only Option D balances the equation. Why Other Options Are Wrong: They evaluate to unequal LHS/RHS values (e.g., Option A gives −10.5 = 12, false). Common Pitfalls: Forgetting that one of the inserted signs can be ‘‘=’’ and must appear in the correct third position. Final Answer: x, -, = produces a true statement: 6 × 4 − 12 = 12.

Mathematical Operations Questions