Question

Valeria and Autumn are reading the same book. At the beginning of the month, Valeria was on page 20 and Autumn was on page 40. Valeria will read 20 pages per day and Autumn will read 15 pages per day. Let VV represent the page of the book that Valeria is on at the end of tt days into the month, and let AA represent the page of the book that Autumn is on at the end of tt days into the month. Graph each function and determine after how many days, t, commat, Valeria and Autumn will be on the same page of the book.

Answer 1

To find out when Valeria and Autumn will be on the same page of the book, we can start by defining the functions for the pages they are each on after \( t \) days.

1. **Valeria's page function, \( V(t) \)**:
- She starts on page 20 and reads 20 pages per day.
- The function can be represented as:
\[
V(t) = 20 + 20t
\]

2. **Autumn's page function, \( A(t) \)**:
- She starts on page 40 and reads 15 pages per day.
- The function can be represented as:
\[
A(t) = 40 + 15t
\]

Next, we set these two functions equal to find out when they will be on the same page:
\[
20 + 20t = 40 + 15t
\]

Now, we'll solve for \( t \):

1. Subtract \( 15t \) from both sides:
\[
20 + 20t - 15t = 40
\]
\[
20 + 5t = 40
\]

2. Subtract 20 from both sides:
\[
5t = 20
\]

3. Divide both sides by 5:
\[
t = 4
\]

Thus, Valeria and Autumn will be on the same page after 4 days.

### Verification

We can check what page they will be on after 4 days:

- For Valeria:
\[
V(4) = 20 + 20(4) = 20 + 80 = 100
\]

- For Autumn:
\[
A(4) = 40 + 15(4) = 40 + 60 = 100
\]

Both will be on page 100 after 4 days.

### Conclusion

Valeria and Autumn will be on the same page of the book after \( t = 4 \) days.