Fractions

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Formula

Explanation

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Worked solution

1. The team has 73 players and 19 identical training kits. Each kit is to be shared equally among a group of players, with no kit split. This means each kit is used by a whole number of players.
2. Let the number of players sharing one kit be k. Then 19 kits are distributed such that each kit is used by k players, so total players covered = 19 × k.
3. Since there are 73 players, and all kits are used fully with no player left out or overcounted, 19 × k = 73 must hold.
4. Solve for k: k = 73 / 19. Calculate 73 ÷ 19. 19 × 3 = 57, 73 − 57 = 16. So 73 / 19 = 3 and 16/19. Thus k = 3 16/19, which is not an integer.
5. Since k must be an integer (each kit is shared by a whole number of players), and 73 is not divisible by 19, not all kits can be used to cover exactly k players with k integer.
6. But the question states that kits are to be shared equally among a group of players, with no kit split. This implies that each kit is used by the same number of players, and all kits are used.
7. So the total number of players must be divisible by the number of kits, if each kit is used by the same number of players. But 73 ÷ 19 is not an integer, so it's impossible to divide 73 players equally among 19 kits with each kit used by the same number of players.
8. However, the question asks: 'What fraction of the team receives exactly one kit?' This implies that some players receive one kit, others may receive more or none, but the kits are distributed such that each kit is used by a group of players, and no kit is split.
9. Since kits are identical and not split, and each kit is used by a group of players, the number of players who receive a kit must be a multiple of the number of kits, but only if each kit is used by the same number of players.
10. But the key is: 'each kit is to be shared equally among a group of players'. This means each kit is used by the same number of players. So if 19 kits are used, and each is used by k players, then total players = 19k. But 73 is not divisible by 19, so 19k = 73 has no integer solution for k.
11. Therefore, it is impossible to distribute 19 identical kits equally among 73 players such that each kit is used by the same number of players and no kit is split.
12. But the question assumes such a distribution is possible. So we must reinterpret: perhaps not all kits are used, or not all players receive a kit. But the question says 'to distribute' 19 kits, so all 19 are used. And 'shared equally' means each kit is used by the same number of players.

Reference answer: 19/73

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