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Let's break down the problem in your image step by step to find the maximum number of ties they can produce in a day.

### Given Information:
- Clara, Kant, and Leo can make hats and ties, with different times for each:
  - **Clara** takes 15 minutes per hat and 7.65 minutes per tie.
  - **Kant** takes 17 minutes per hat and 11.54 minutes per tie.
  - **Leo** takes 9 minutes per hat and 23.33 minutes per tie.
  
- Each of them works **10.5 hours per day**, which is \( 10.5 \times 60 = 630 \) minutes per day.

- Together, they are supposed to produce a total of **69 hats**. We need to find the maximum number of ties they can produce in a day.

### Step-by-Step Solution:
1. **Total time available per worker:**
   Each person has 630 minutes per day.

2. **Total time spent on hats:**
   We know they must produce a total of 69 hats. Let’s calculate the time each person spends on hats if they divide the hat production among themselves:
   
   Let \( x_C \), \( x_K \), and \( x_L \) be the number of hats produced by Clara, Kant, and Leo, respectively. So:
   \[
   x_C + x_K + x_L = 69
   \]
   The time each person spends on hats is:
   \[
   \text{Time for Clara} = 15x_C, \quad \text{Time for Kant} = 17x_K, \quad \text{Time for Leo} = 9x_L
   \]
   The total time spent on hats for all three is:
   \[
   15x_C + 17x_K + 9x_L
   \]
   
3. **Total time available for ties:**
   Each person has 630 minutes, so the remaining time after making hats is:
   \[
   \text{Remaining time for Clara} = 630 - 15x_C, \quad \text{Remaining time for Kant} = 630 - 17x_K, \quad \text{Remaining time for Leo} = 630 - 9x_L
   \]
   
4. **Maximum number of ties:**
   The number of ties each person can produce is given by:
   \[
   \text{Ties by Clara} = \frac{630 - 15x_C}{7.65}, \quad \text{Ties by Kant} = \frac{630 - 17x_K}{11.54}, \quad \text{Ties by Leo} = \frac{630 - 9x_L}{23.33}
   \]
   We need to maximize the total number of ties:
   \[
   \text{Total ties} = \frac{630 - 15x_C}{7.65} + \frac{630 - 17x_K}{11.54} + \frac{630 - 9x_L}{23.33}
   \]

Next, I'll calculate the values numerically.The maximum number of ties that Clara, Kant, and Leo can jointly produce in a day, while making 69 hats, is approximately **88 ties**.

Would you like more details on any part of the solution?

Here are some related questions:
1. How would the number of ties change if they produced fewer hats?
2. What would happen if Leo took more time to make ties?
3. Can they increase the number of hats if they reduce the number of ties?
4. How does the difference in production times between the workers affect the final output?
5. How would the solution change if the workers could only make an integer number of ties?

**Tip**: When distributing tasks among workers, it's efficient to assign more work to the faster workers to maximize output.

Clara, Kant, and Leo can produce hats and ties. Each of them works 10.5 hours per day. The goal is to produce 69 hats in total. The table provides the time required by each worker to produce one hat and one tie. Find the maximum number of ties they can jointly produce in a day.

Solve the work and time maximization problem where Clara, Kant, and Leo are tasked with producing 69 hats in a day. Calculate the maximum number of ties they can produce based on their individual production times for both hats and ties. Includes step-by-step analysis and optimization approach.

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