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The problem on the image deals with the concept of rate of change between the price of a book and the number of books sold.

Here is a summary of the key points:

1. **Initial situation**:
   - Price of the book: \$100
   - Books sold per month: 60 books

2. **After price increase**:
   - New price of the book: \$110
   - Books sold per month: 40 books

### Calculation of the rate of change
The rate of change ($\Delta y / \Delta x$) represents how the number of books sold changes with respect to the change in price.

Let:
- $x$ represent the price of the book.
- $y$ represent the number of books sold.

We are given two points:
- Initial point: $(100, 60)$ (price = \$100, books sold = 60)
- Final point: $(110, 40)$ (price = \$110, books sold = 40)

The formula for rate of change is:

\[
\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the values:

\[
\frac{\Delta y}{\Delta x} = \frac{40 - 60}{110 - 100} = \frac{-20}{10} = -2
\]

This means that for every \$1 increase in price, 2 fewer books are sold.

### Analysis of the options:

- **Option A**: $\frac{40 - 60}{110 - 100} = 2$
  - Incorrect: This incorrectly calculates the rate of change as positive 2, instead of negative 2.

- **Option B**: $\frac{40 - 60}{110 - 100} = -2$
  - Correct: The rate of change is correctly calculated as $-2$, indicating that for each \$1 increase in price, 2 fewer books are sold.

- **Option C**: $\frac{40 - 60}{110 - 100} = \frac{1}{2}$
  - Incorrect: This erroneously calculates the rate of change as $\frac{1}{2}$, which does not reflect the correct relationship between price and books sold.

- **Option D**: $\frac{40 - 60}{110 - 100} = -2$
  - Correct: This matches the correct interpretation of the rate of change.

### Conclusion:
The employee's claim that the average rate of change is $\frac{1}{2}$ is incorrect because the correct rate of change is $-2$. This means that for every 1 peso increase in price, 2 fewer books are sold.

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Do you want more details or have any other questions? Here are five related questions to expand on this:

1. How is the average rate of change related to the slope of a line?
2. What would the situation look like graphically, with price as the x-axis and books sold as the y-axis?
3. Could the bookstore use this information to predict future sales at different prices?
4. How does a negative rate of change affect the interpretation of real-world data?
5. What is the difference between the average rate of change and instantaneous rate of change in this context?

**Tip**: Always check the units in your rate of change to ensure they reflect the situation accurately!

Cuando el precio de venta de un libro es $100 (pesos mexicanos), se venden al mes 60 libros. Al aumentar el precio a $110, se venden al mes 40 libros. Un empleado de esta librería dice que la razón promedio es -1/2; esto es INCORRECTO porque:

Explore the problem of determining the correct rate of change when the price of a book increases from $100 to $110, causing a decrease in monthly sales from 60 to 40 books. Learn why the employee's claim that the rate is -1/2 is incorrect, and understand the proper calculation method for the rate of change.

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