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The task involves organizing the distribution of participants, supplies, and calculating resources based on the given data for a sports day event. Here are the details for each question:

### (a) Distribute the number of participants in each category:
- **Total participants**: 200
- **Football tournament**: \( \frac{2}{5} \times 200 = 80 \) participants
- **Basketball tournament**: \( 0.35 \times 200 = 70 \) participants
- **Track races**: \( 15\% \times 200 = 0.15 \times 200 = 30 \) participants
- **Support activities**: The remaining participants.  
  So, \( 200 - (80 + 70 + 30) = 20 \) participants.

### (b) Estimate the number of cartons of water bottles and boxes of snacks needed for ball games participants:
- **Participants in ball games** (football + basketball): \( 80 + 70 = 150 \) participants.
- **Water bottles**: Each carton has 12 bottles.
  \[
  \frac{150}{12} \approx 12.5 \Rightarrow \text{13 cartons needed.}
  \]
- **Snacks**: Each box contains 8 snacks.
  \[
  \frac{150}{8} = 18.75 \Rightarrow \text{19 boxes needed.}
  \]

### (c) Determine the amount of cloth saved on purchase of t-shirts and caps:
- **Participants needing t-shirts**: 30 for track races, plus 9 additional t-shirts.  
  Total t-shirts: \( 30 + 9 = 39 \).
  - Cloth for t-shirts: \( 39 \times 5 = 195 \) meters.
- **Participants needing caps**: 20 for support activities, plus 6 additional caps.  
  Total caps: \( 20 + 6 = 26 \).
  - Cloth for caps: \( 26 \times 0.75 = 19.5 \) meters.
- **Total cloth without offer**: \( 195 + 19.5 = 214.5 \) meters.
- With the "buy 3, get 1 free" deal:
  - Free t-shirts: \( \frac{39}{4} = 9 \) (free items).
  - Free caps: \( \frac{26}{4} \approx 6 \) (free items).
  - Cloth saved:  
    \[
    (9 \times 5) + (6 \times 0.75) = 45 + 4.5 = 49.5 \text{ meters}.
    \]

Thus, 49.5 meters of cloth are saved.

Would you like more details or have any questions?

### Related questions:
1. How many total items (t-shirts and caps) are purchased including free items?
2. What percentage of the participants are involved in the support activities?
3. How many free caps and t-shirts are given due to the "buy 3, get 1 free" offer?
4. What is the total number of participants in the ball games?
5. How much cloth is used for the remaining caps and t-shirts after applying the offer?

### Tip:
When solving such multi-step problems, breaking down the information into categories (e.g., by participants or items) helps in organizing the solution logically.

A physical education teacher is organizing a sports day event and needs to manage various aspects such as participant numbers, food supplies, and equipment distribution. The 200 participants were distributed into different categories: 2/5 for football, 0.35 for basketball, 15% for track races, and the rest for support activities. Tasks involve distributing participants, calculating resources like water bottles, snacks, and cloth for t-shirts and caps, and applying a 'buy 3, get 1 free' offer.

Organize a sports day event by distributing 200 participants across football, basketball, track races, and support activities. Learn how to calculate the number of water bottles and snacks needed, and determine the cloth saved on t-shirts and caps with a 'buy 3, get 1 free' deal.

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