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.
ScanSolve AI Answer
You're on a roll — Here's the solution for Item 1: Item 1: a) Step 1: Find the greatest common divisor (GCD) of the number of football players and netball players to determine the largest possible equal team size for both games. Number of football players = 132 Number of netball players = 108 Prime factorization of 132: 132 = 2 × 66 = 2 × 2 × 33 = 2^2 × 3 × 11 Prime factorization of 108: 108 = 2 × 54 = 2 × 2 × 27 = 2^2 × 3 × 3 × 3 = 2^2 × 3^3 The GCD is the product of the lowest powers of common prime factors: GCD(132, 108) = 2^2 × 3^1 = 4 × 3 = 12 The largest possible team size is 12 players. Step 2: Calculate the number of teams for each game using this team size. For football: Number of football teams = (132)/(12) = 11 For netball: Number of netball teams = (108)/(12) = 9 a) i) The number of large equal teams that can be formed for each game is: Football: 11 teams Netball: 9 teams a) ii) The number of players in a netball team will be the largest common team size: 12 players b) Step 1: Determine which box size (8 or 9) can pack 1,260 medals evenly. Check divisibility by 8: 1260 ÷ 8 = 157.5 Since the result is not a whole number, 1,260 medals cannot be packed evenly in boxes of 8. Step 2: Check divisibility by 9: Sum of digits of 1260: 1 + 2 + 6 + 0 = 9. Since 9 is divisible by 9, 1260 is divisible by 9. 1260 ÷ 9 = 140 Since the result is a whole number, 1,260 medals can be packed evenly in boxes of 9. b) They should use boxes of 9 because 1,260 is perfectly divisible by 9 (140 boxes), while it is not divisible by 8. c) Step 1: Calculate the number of bottles for the committee and players. Total water bottles = 1,600. Fraction for committee = (2)/(12) = (1)/(6). Number of bottles for committee = (1)/(6) × 1600 = (1600)/(6) ≈ 266.67 bottles. Fraction for players = 0.58. Number of bottles for players = 0.58 × 1600 = 928 bottles. Step 2: Calculate the fraction of bottles for volunteers. Fraction for volunteers = 1 - (fraction for committee + fraction for players) 1 - ((1)/(6) + 0.58) = 1 - ((1)/(6) + (58)/(100)) Find a common denominator for 6 and 100, which is 300. (1)/(6) = (50)/(300) (58)/(100) = (174)/(300) Fraction for volunteers = 1 - ((50)/(300) + (174)/(300)) = 1 - (224)/(300) = (300)/(300) - (224)/(300) = (76)/(300) = (19)/(75) Step 3: Calculate the number of bottles for volunteers and round to the nearest hundred. Number of bottles for volunteers = (19)/(75) × 1600 = (30400)/(75) ≈ 405.33 bottles. Rounding to the nearest hundred, 405.33 is closer to 400. c) The volunteers will receive approximately 400 water bottles. Send me the next one 📸