This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems: 1. Which of these number sentences are true, and which are false? a) Step 1: Evaluate the left side. 3 × 5 + 3 × 7 = 15 + 21 = 36 Step 2: Evaluate the right side. 3 × 12 = 36 Step 3: Compare both sides. 36 = 36 The statement is true. b) Step 1: Evaluate the left side. 3 × 5 + 3 × 7 = 15 + 21 = 36 Step 2: Evaluate the right side. 6 × 12 = 72 Step 3: Compare both sides. 36 ≠ 72 The statement is false. 2. Find the numbers that make the number sentences true. a) Let the missing number be x. Step 1: Set up the equation. 15x - 11 = 11x + 1 Step 2: Subtract 11x from both sides. 15x - 11x - 11 = 1 4x - 11 = 1 Step 3: Add 11 to both sides. 4x = 1 + 11 4x = 12 Step 4: Divide by 4. x = (12)/(4) x = 3 The number is 3. b) Let the missing number be x. Step 1: Set up the equation. 100 - 5x = 3x - 28 Step 2: Add 5x to both sides. 100 = 3x + 5x - 28 100 = 8x - 28 Step 3: Add 28 to both sides. 100 + 28 = 8x 128 = 8x Step 4: Divide by 8. x = (128)/(8) x = 16 The number is 16. c) Let the missing number be x. Step 1: Set up the equation. 10x + 1500 = 20x + 1250 Step 2: Subtract 10x from both sides. 1500 = 20x - 10x + 1250 1500 = 10x + 1250 Step 3: Subtract 1250 from both sides. 1500 - 1250 = 10x 250 = 10x Step 4: Divide by 10. x = (250)/(10) x = 25 The number is 25. 3. A factory worker is bored with his job, so he decides to make it more interesting while filling his 30 bottles of cordial. He decides to put the lid on every fourth bottle, the flavour in every fifth bottle and the label on every sixth bottle. If he started at the 1st bottle, how many bottles would have: Total bottles = 30. Lids on multiples of 4: \4, 8, 12, 16, 20, 24, 28\ Flavour in multiples of 5: \5, 10, 15, 20, 25, 30\ Label on multiples of 6: \6, 12, 18, 24, 30\ a) Lids on? Step 1: List multiples of 4 up to 30. 4, 8, 12, 16, 20, 24, 28 There are 7 such bottles. The number of bottles with lids is 7. b) Lids and flavours? Step 1: Find bottles that are multiples of both 4 and 5. This means multiples of LCM(4, 5). LCM(4, 5) = 20 Step 2: List multiples of 20 up to 30. 20 There is 1 such bottle. The number of bottles with lids and flavours is 1. c) Flavours and labels? Step 1: Find bottles that are multiples of both 5 and 6. This means multiples of LCM(5, 6). LCM(5, 6) = 30 Step 2: List multiples of 30 up to 30. 30 There is 1 such bottle. The number of bottles with flavours and labels is 1. d) Lids, flavours and labels? Step 1: Find bottles that are multiples of 4, 5, and 6. This means multiples of LCM(4, 5, 6). LCM(4, 5, 6) = LCM(2^2, 5, 2 × 3) = 2^2 × 3 × 5 = 4 × 3 × 5 = 60 Step 2: List multiples of 60 up to 30. There are no multiples of 60 within the first 30 bottles. The number of bottles with lids, flavours, and labels is 0. 4. Solve the following order of operation sums: a) Step 1: Solve the parenthesis. (13 - 6) × 5 = 7 × 5 Step 2: Perform multiplication. 7 × 5 = 35 The answer is 35. b) Step 1: Solve the parenthesis. 18 ÷ (4 + 5) = 18 ÷ 9 Step 2: Perform division. 18 ÷ 9 = 2 The answer is 2. c) Step 1: Perform multiplication. 15 + 10 × 3 = 15 + 30 Step 2: Perform addition. 15 + 30 = 45 The answer is 45. d) Step 1: Solve the parenthesis. (15 + 10) × 3 = 25 × 3 Step 2: Perform multiplication. 25 × 3 = 75 The answer is 75. e) Step 1: Perform division and multiplication from left to right. 21 - 15 ÷ 3 + 7 × 6 = 21 - 5 + 42 Step 2: Perform addition and subtraction from left to right. 21 - 5 + 42 = 16 + 42 16 + 42 = 58 The answer is 58. f) Step 1: Solve the parenthesis. 99 ÷ (4 + 5) + 7 × 7 - 100 ÷ 5 = 99 ÷ 9 + 7 × 7 - 100 ÷ 5 Step 2: Perform divisions and