Reasoning — Mathematical Operations

Replace each symbol with its respective operation: 8 ÷ 4 − 2 × 6 + 3. This becomes 2 − 12 + 3 = 10.

Given symbols and their meanings: 'A' = '+' 'B' = '−' 'C' = '×' 'D' = '÷' The expression is: 24 D 6 A 15 B 5 C 2 Substituting the symbols: 24 ÷ 6 + 15 − 5 × 2 Following the BODMAS/PEMDAS rule: 1. Division: 24 ÷ 6 = 4 Expression becomes: 4 + 15 − 5 × 2 2. Multiplication: 5 × 2 = 10 Expression becomes: 4 + 15 − 10 3. Addition: 4 + 15 = 19 Expression becomes: 19 − 10 4. Subtraction: 19 − 10 = 9 Thus, the value of the expression is 9.

Given equation: 18 + 6 ÷ 3 - 2 × 4 = 18 Let's first calculate the current value of the left side: 18 + (6 ÷ 3) - (2 × 4) = 18 + 2 - 8 = 20 - 8 = 12 The current value (12) is not equal to the target value (18). We need to interchange signs to make it correct. Let's check the options: 1. Interchange '+' and '÷': 18 ÷ 6 + 3 - 2 × 4 = 3 + 3 - 8 = 6 - 8 = -2. (Incorrect) 2. Interchange '-' and '×': 18 + 6 ÷ 3 × 2 - 4 = 18 + 2 × 2 - 4 = 18 + 4 - 4 = 22 - 4 = 18. (Correct) 3. Interchange '+' and '-': 18 - 6 ÷ 3 + 2 × 4 = 18 - 2 + 8 = 16 + 8 = 24. (Incorrect) 4. Interchange '÷' and '×': 18 + 6 × 3 - 2 ÷ 4 = 18 + 18 - 0.5 = 36 - 0.5 = 35.5. (Incorrect) Therefore, interchanging '-' and '×' makes the equation correct.

Let's analyze the given examples to find the pattern: 1. 5 * 3 = 34 Here, 5² + 3² = 25 + 9 = 34. The pattern seems to be A * B = A² + B². 2. 6 * 2 = 40 Using the same pattern: 6² + 2² = 36 + 4 = 40. This confirms the pattern. Now, apply the pattern to find 7 * 1: 7 * 1 = 7² + 1² = 49 + 1 = 50 Thus, 7 * 1 = 50.

Given symbols and their meanings: 'P' = '÷' 'Q' = '×' 'R' = '+' 'S' = '−' The expression is: 48 P 12 R 15 S 6 Q 3 Substituting the symbols: 48 ÷ 12 + 15 − 6 × 3 Following the BODMAS/PEMDAS rule: 1. Division: 48 ÷ 12 = 4 Expression becomes: 4 + 15 − 6 × 3 2. Multiplication: 6 × 3 = 18 Expression becomes: 4 + 15 − 18 3. Addition: 4 + 15 = 19 Expression becomes: 19 − 18 4. Subtraction: 19 − 18 = 1 Thus, the value of the expression is 1.

Given equation: 18 ÷ 3 + 5 × 2 - 7 = 15 Let's first calculate the current value of the left side: (18 ÷ 3) + (5 × 2) - 7 = 6 + 10 - 7 = 16 - 7 = 9 The current value (9) is not equal to the target value (15). We need to interchange two numbers. Let's check the options: 1. Interchange 3 and 5: 18 ÷ 5 + 3 × 2 - 7 = 3.6 + 6 - 7 = 9.6 - 7 = 2.6. (Incorrect) 2. Interchange 2 and 7: 18 ÷ 3 + 5 × 7 - 2 = 6 + 35 - 2 = 41 - 2 = 39. (Incorrect) 3. Interchange 18 and 7: 7 ÷ 3 + 5 × 2 - 18 = 2.33 (approx) + 10 - 18 = 12.33 - 18 = -5.67 (approx). (Incorrect) 4. Interchange 5 and 7: 18 ÷ 3 + 7 × 2 - 5 = 6 + 14 - 5 = 20 - 5 = 15. (Correct) Therefore, interchanging 5 and 7 makes the equation correct.

Given equation: 25 * 5 * 3 * 10 * 2 = 15 We need to replace the '*' signs with the operators from the options to make the equation true. Let's check each option: 1. ÷, +, -, ×: 25 ÷ 5 + 3 - 10 × 2 = 5 + 3 - 20 = 8 - 20 = -12. (Incorrect) 2. +, -, ×, ÷: 25 + 5 - 3 × 10 ÷ 2 = 25 + 5 - (3 × 10) ÷ 2 = 25 + 5 - 30 ÷ 2 = 25 + 5 - 15 = 30 - 15 = 15. (Correct) 3. ×, ÷, +, -: 25 × 5 ÷ 3 + 10 - 2 = 125 ÷ 3 + 10 - 2 = 41.66 (approx) + 10 - 2 = 49.66 (approx). (Incorrect) 4. -, ×, ÷, +: 25 - 5 × 3 ÷ 10 + 2 = 25 - (5 × 3) ÷ 10 + 2 = 25 - 15 ÷ 10 + 2 = 25 - 1.5 + 2 = 23.5 + 2 = 25.5. (Incorrect) Therefore, the combination '+, -, ×, ÷' makes the equation correct.

Let's analyze the given examples to find the pattern: 1. 12 $ 3 = 36 Here, 12 × 3 = 36. The pattern seems to be A $ B = A × B. 2. 15 $ 5 = 75 Using the same pattern: 15 × 5 = 75. This confirms the pattern. Now, apply the pattern to find 20 $ 4: 20 $ 4 = 20 × 4 = 80 Thus, 20 $ 4 = 80.

Given symbols and their meanings: '+' means '×' '−' means '÷' '×' means '−' '÷' means '+' The expression is: 16 ÷ 8 − 4 + 2 × 5 Substituting the symbols: 16 + 8 ÷ 4 × 2 − 5 Following the BODMAS/PEMDAS rule: 1. Division: 8 ÷ 4 = 2 Expression becomes: 16 + 2 × 2 − 5 2. Multiplication: 2 × 2 = 4 Expression becomes: 16 + 4 − 5 3. Addition: 16 + 4 = 20 Expression becomes: 20 − 5 4. Subtraction: 20 − 5 = 15 Thus, the value of the expression is 15.

Given equation: 36 * 6 * 2 * 18 * 3 = 27 We need to replace the '*' signs with the operators from the options to make the equation true. Let's check each option: 1. ÷, +, -, ×: 36 ÷ 6 + 2 - 18 × 3 = 6 + 2 - 54 = 8 - 54 = -46. (Incorrect) 2. ×, ÷, +, -: 36 × 6 ÷ 2 + 18 - 3 = 216 ÷ 2 + 18 - 3 = 108 + 18 - 3 = 126 - 3 = 123. (Incorrect) 3. ÷, ×, +, -: 36 ÷ 6 × 2 + 18 - 3 = 6 × 2 + 18 - 3 = 12 + 18 - 3 = 30 - 3 = 27. (Correct) 4. +, -, ×, ÷: 36 + 6 - 2 × 18 ÷ 3 = 36 + 6 - 36 ÷ 3 = 36 + 6 - 12 = 42 - 12 = 30. (Incorrect) Therefore, the combination '÷, ×, +, -' makes the equation correct.

Given rule for the operator '@': If A > B, then A @ B = A + B If A ≤ B, then A @ B = A × B We need to find the value of (10 @ 5) @ (3 @ 7). Step 1: Evaluate (10 @ 5) Here, A = 10 and B = 5. Since A (10) > B (5), we use the rule A @ B = A + B. 10 @ 5 = 10 + 5 = 15. Step 2: Evaluate (3 @ 7) Here, A = 3 and B = 7. Since A (3) ≤ B (7), we use the rule A @ B = A × B. 3 @ 7 = 3 × 7 = 21. Step 3: Evaluate the final expression (15 @ 21) Now we have 15 @ 21. Here, A = 15 and B = 21. Since A (15) ≤ B (21), we use the rule A @ B = A × B. 15 @ 21 = 15 × 21 To calculate 15 × 21: 15 × 20 = 300 15 × 1 = 15 300 + 15 = 315 Thus, the value of (10 @ 5) @ (3 @ 7) is 315.

Given equation: 24 * 4 * 5 * 2 = 14 We need to replace the '*' signs with the operators from the options to make the equation true. Let's check each option: 1. ÷, +, -: 24 ÷ 4 + 5 - 2 = 6 + 5 - 2 = 11 - 2 = 9. (Incorrect) 2. +, -, ×: 24 + 4 - 5 × 2 = 28 - 10 = 18. (Incorrect) 3. ×, ÷, -: 24 × 4 ÷ 5 - 2 = 96 ÷ 5 - 2 = 19.2 - 2 = 17.2. (Incorrect) 4. -, ×, ÷: 24 - 4 × 5 ÷ 2 = 24 - (4 × 5) ÷ 2 = 24 - 20 ÷ 2 = 24 - 10 = 14. (Correct) Therefore, the combination '-, ×, ÷' makes the equation correct.

Given symbols and their meanings: 'L' = '+' 'M' = '−' 'N' = '×' 'P' = '÷' The expression is: 28 P 7 N 12 L 15 M 10 Substituting the symbols: 28 ÷ 7 × 12 + 15 − 10 Following the BODMAS/PEMDAS rule: 1. Division: 28 ÷ 7 = 4 Expression becomes: 4 × 12 + 15 − 10 2. Multiplication: 4 × 12 = 48 Expression becomes: 48 + 15 − 10 3. Addition: 48 + 15 = 63 Expression becomes: 63 − 10 4. Subtraction: 63 − 10 = 53 Thus, the value of the expression is 53.

Let's analyze the given examples to find the pattern for the operator '#': 1. 4 # 2 = 26 If we try (A × 5) + (B × 3): (4 × 5) + (2 × 3) = 20 + 6 = 26. This matches. 2. 6 # 3 = 39 Using the same pattern: (6 × 5) + (3 × 3) = 30 + 9 = 39. This also matches. The pattern is confirmed: A # B = (A × 5) + (B × 3). Now, apply the pattern to find 8 # 4: 8 # 4 = (8 × 5) + (4 × 3) = 40 + 12 = 52 Thus, 8 # 4 = 52.

Given symbols and their meanings: 'a' = 'add' (+) 'b' = 'subtract' (-) 'c' = 'multiply' (×) 'd' = 'divide' (÷) 'e' = 'exponentiate' (power of) The expression is: 10 c (5 a 3) d 4 b (2 e 3) Following the order of operations (BODMAS/PEMDAS), we first solve the expressions within parentheses: 1. (5 a 3): Substitute 'a' with '+' 5 + 3 = 8 2. (2 e 3): Substitute 'e' with 'exponentiate' 2³ = 2 × 2 × 2 = 8 Now substitute these values back into the main expression: 10 c 8 d 4 b 8 Next, substitute the remaining symbols with their mathematical operators: 10 × 8 ÷ 4 - 8 Now, follow BODMAS/PEMDAS for the remaining operations: 1. Multiplication: 10 × 8 = 80 Expression becomes: 80 ÷ 4 - 8 2. Division: 80 ÷ 4 = 20 Expression becomes: 20 - 8 3. Subtraction: 20 - 8 = 12 Thus, the value of the expression is 12.