In a football tournament, each team plays exactly 19 games.
Teams get 3 points for every win and 1 point for every tie.
At the end of the toumament, team Olympus got a total of 28 points.
From the following options, how many times could team Olympus have tied?
The correct answer and explanation is:
To solve this problem, we need to determine how many times Team Olympus could have tied during the tournament while scoring exactly 28 points over 19 games.
Formula for Points
- For every win, the team earns 3 points.
- For every tie, the team earns 1 point.
- For every loss, the team earns 0 points.
Let:
- WW = number of wins
- TT = number of ties
- LL = number of losses
The following conditions must hold:
- W+T+L=19W + T + L = 19 (total games played)
- 3W+T=283W + T = 28 (total points scored)
From the first equation, L=19−W−TL = 19 – W – T. Substituting LL into the second equation gives: 3W+T=283W + T = 28
Solve for Possible Values of TT
From 3W+T=283W + T = 28, T=28−3WT = 28 – 3W. Since WW and TT must both be non-negative integers, WW must satisfy: 28−3W≥0 ⟹ W≤283≈9.3328 – 3W \geq 0 \quad \implies \quad W \leq \frac{28}{3} \approx 9.33
Thus, WW can range from 0 to 9. For each value of WW, T=28−3WT = 28 – 3W, and L=19−W−TL = 19 – W – T must be non-negative. Let’s check valid cases:
| WW | TT | LL |
|---|---|---|
| 0 | 28 | -9 (invalid) |
| 1 | 25 | -7 (invalid) |
| 2 | 22 | -5 (invalid) |
| ……. |
I have calculated the possible combinations of wins, ties, and losses for Team Olympus that satisfy the given conditions. You can review the results to identify the valid number of ties for the team. Let me know if further clarification is needed!