Fractions

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Formula

Explanation

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Worked solution

1. The team has 85 players and 16 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, and each player receives exactly one kit only if they are in a group assigned to a single kit.
2. To find the fraction of the team that receives exactly one kit, determine how many players are in a group that uses one kit. Since 16 kits are distributed and each kit is used by a group of players, the number of players per kit must be a divisor of 85 that divides 85 evenly when 16 kits are assigned.
3. Let the number of players per kit be k. Then 16 kits × k players per kit = 85 players. So 16k = 85. Solving for k gives k = 85 / 16. But k must be an integer because kits are not split and each player must be in exactly one group.
4. Since 85 is not divisible by 16, not every kit can be used by the same number of players. Therefore, the kits are distributed so that some groups have more players than others, but each kit is used by a group of players, and no kit is split.
5. The question asks for the fraction of the team that receives exactly one kit. This means: how many players are in a group that uses exactly one kit? Since each kit is used by a group, and a player receives a kit only if they are in a group assigned to a single kit, the number of players who receive exactly one kit is equal to the number of kits, because each kit is assigned to a group, and only one player per kit can receive exactly one kit if the group size is 1.
6. But if a group uses one kit, and the group has more than one player, then no player in that group receives exactly one kit. Only if a group has exactly one player, then that player receives exactly one kit. So the number of players who receive exactly one kit is equal to the number of groups of size 1.
7. Let x be the number of groups of size 1. Then the remaining 16 - x kits are used by groups of size ≥2. The total number of players is 85. So: x × 1 + (16 - x) × (average group size ≥2) = 85.
8. Let the sum of players in the non-singleton groups be 85 - x. These 16 - x groups must contain 85 - x players. Since each of these groups has at least 2 players, 85 - x ≥ 2(16 - x). Solve this inequality: 85 - x ≥ 32 - 2x → 85 - 32 ≥ -x → 53 ≥ -x → x ≥ -53. Always true, so not helpful.
9. Try to find integer values of x such that 85 - x is divisible by (16 - x), and 85 - x ≥ 2(16 - x). Try x = 1: 85 - 1 = 84, 16 - 1 = 15, 84 / 15 = 5.6 → not integer. x = 2: 83 / 14 ≈ 5.93 → no. x = 3: 82 / 13 ≈ 6.31 → no. x = 4: 81 / 12 = 6.75 → no. x = 5: 80 / 11 ≈ 7.27 → no. x = 6: 79 / 10 = 7.9 → no. x = 7: 78 / 9 = 8.666 → no. x = 8: 77 / 8 = 9.625 → no. x = 9: 76 / 7 ≈ 10.857 → no. x = 10: 75 / 6 = 12.5 → no. x = 11: 74 / 5 = 14.8 → no. x = 12: 73 / 4 = 18.25 → no. x = 13: 72 / 3 = 24 → integer. So x = 13 is possible.
10. Check x = 13: 13 groups of 1 player → 13 players. Remaining 16 - 13 = 3 kits. Remaining players: 85 - 13 = 72. 72 / 3 = 24. So 3 groups of 24 players each. Valid. Total players: 13 + 3×24 = 13 + 72 = 85. Correct.
11. So 13 players receive exactly one kit (each in a group of size 1). The fraction of the team that receives exactly one kit is 13 / 85. Check if this fraction is in simplest form: GCD(13, 85). 13 is prime. 85 = 5×17. 13 does not divide 85. So GCD = 1. Therefore, 13/85 is already in simplest form.
12. The fraction of the team that receives exactly one kit is 13/85. This matches the requirement: no kit is split, all 85 players are assigned, and only 13 players are in groups of size 1, so they receive exactly one kit.

Reference answer: 13/85

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