How many hours must Jill tutor for the amount she brings in to equal the amount she spends?

To determine the number of hours Jill must tutor for the amount she brings in to equal the amount she spends, we need to find the break-even point where her total revenue equals her total costs.

### Definitions:
- **Fixed Costs**: One-time expenses that do not change with the number of hours Jill tutors.  
  - **Advertisements**: $400
- **Variable Costs**: Costs that change with the number of hours Jill tutors.
  - **Facility Fee**: $20 per hour
- **Revenue**: The amount Jill charges her clients per hour.
  - **Revenue per Hour**: $60 per hour

### Break-Even Point Calculation:

To find the break-even point, we set Jill's total revenue equal to her total costs:

1. **Total Costs**: 
   \[
   \text{Total Costs} = \text{Fixed Costs} + \text{Variable Costs per Hour} \times \text{Number of Hours}
   \]
   \[
   \text{Total Costs} = 400 + 20h
   \]

2. **Total Revenue**:
   \[
   \text{Total Revenue} = \text{Revenue per Hour} \times \text{Number of Hours}
   \]
   \[
   \text{Total Revenue} = 60h
   \]

3. **Set Total Revenue equal to Total Costs**:
   \[
   60h = 400 + 20h
   \]

4. **Solve for $h$**:
   \[
   60h - 20h = 400
   \]
   \[
   40h = 400
   \]
   \[
   h = \frac{400}{40}
   \]
   \[
   h = 10
   \]

### Conclusion:

Jill must tutor for **10 hours** for the amount she brings in to equal the amount she spends. This is her break-even point, where her total revenue matches her total costs.

Would you like to explore further details or have any more questions?

### Related Questions:

1. How much profit does Jill make if she tutors for 15 hours?
2. If Jill reduces the hourly rate to $50, how many hours must she tutor to break even?
3. What would be the impact on Jill's break-even point if she had no fixed costs?
4. How does increasing the facility fee to $25 per hour change the break-even hours?
5. How much would Jill need to charge per hour to break even after 5 hours?

### Tip:
To find a break-even point, always set total revenue equal to total costs. This helps determine the minimum effort required to avoid losses.

Learn how to calculate Jill's break-even point for tutoring hours. Understand the relationship between fixed costs, variable costs, and revenue per hour in this scenario. Suitable for high school students studying business or economics.

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