A school has 132 football players and 108 netball players. They want to form teams of equal size for both sports. What is the largest possible number of players per team? How many teams can be formed

Number of football players = 132
Number of netball players = 108

Prime factorization of 132:

$$132 = 2 \times 66 = 2 \times 2 \times 33 = 2^2 \times 3 \times 11$$

Prime factorization of 108:

$$108 = 2 \times 54 = 2 \times 2 \times 27 = 2^2 \times 3 \times 3 \times 3 = 2^2 \times 3^3$$

The GCD is the product of the lowest powers of common prime factors:

$$\text{GCD}(132, 108) = 2^2 \times 3^1 = 4 \times 3 = 12$$

The largest possible team size is 12 players.

For football:

$$\text{Number of football teams} = \frac{132}{12} = 11$$

For netball:

$$\text{Number of netball teams} = \frac{108}{12} = 9$$

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)

Check divisibility by 8:

$$1260 \div 8 = 157.5$$

Since the result is not a whole number, 1,260 medals cannot be packed evenly in boxes of 8.

Sum of digits of 1260: $1 + 2 + 6 + 0 = 9$. Since 9 is divisible by 9, 1260 is divisible by 9.

$$1260 \div 9 = 140$$

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)

Total water bottles = 1,600.
Fraction for committee = $\frac{2}{12} = \frac{1}{6}$.
Number of bottles for committee = $\frac{1}{6} \times 1600 = \frac{1600}{6} \approx 266.67$ bottles.

Fraction for players = 0.58.
Number of bottles for players = $0.58 \times 1600 = 928$ bottles.

Fraction for volunteers = $1 - (\text{fraction for committee} + \text{fraction for players})$

$$1 - \left(\frac{1}{6} + 0.58\right) = 1 - \left(\frac{1}{6} + \frac{58}{100}\right)$$

Find a common denominator for 6 and 100, which is 300.

$$\frac{1}{6} = \frac{50}{300}$$ $$\frac{58}{100} = \frac{174}{300}$$

Fraction for volunteers = $1 - \left(\frac{50}{300} + \frac{174}{300}\right) = 1 - \frac{224}{300} = \frac{300}{300} - \frac{224}{300} = \frac{76}{300} = \frac{19}{75}$

Number of bottles for volunteers = $\frac{19}{75} \times 1600 = \frac{30400}{75} \approx 405.33$ bottles.
Rounding to the nearest hundred, 405.33 is closer to 400.

c) The volunteers will receive approximately 400 water bottles.