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Read the problem situation and identify the independent and dependent variables. Then, convert the information to coordinate points and use the formula $m=\frac{y_2-y_1}{x_2-x_1}$ to answer the question.



Samantha loves reading and is participating in a read-a-thon to raise money for a charity. She plans to read 15 books during the 90-day read-a-thon. During the first 30 days, she reads 7 books. What is the unit rate of the number of remaining days she has to read each book to meet her goal?


  1. Identify the independent and dependent variables:
  • The independent variable is [the number of books / the number of remaining days].
  • The dependent variable is [the number of books / the number of remaining days].


  1. Convert the information to coordinate points and use the formula $m=\frac{y_2-y_1}{x_2-x_1}$ to answer the question:

What is the unit rate of the number of remaining days she has to read each book to meet her goal?

Verified Answer :

  • Identify the independent variable as the number of books and the dependent variable as the number of days.
  • Define the progress points as \((7, 30)\) and \((15, 90)\).
  • Apply the rate of change formula \(m = \frac{90 - 30}{15 - 7}\).
  • Calculate the unit rate: \(\boxed{7.5}\) days per book.

Explanation

  1. Problem Analysis and Variable Identification

We need to analyze Samantha's read-a-thon progress to determine the unit rate of days per book for the remainder of her goal.
The total goal is 15 books in 90 days. After 30 days, Samantha has read 7 books.
To find the remaining goal:

  • Remaining books: \(15 - 7 = 8\) books.
  • Remaining days: \(90 - 30 = 60\) days.

In this context, the number of books read affects the time spent, making the 'number of books' the independent variable and the 'number of days' the dependent variable.

  1. Coordinate Points Representation

We can represent the situation with two coordinate points \((x, y)\), where \(x\) is the number of books and \(y\) is the number of days.

  • At the start (day 0), she had read 0 books. Thus, the first point is \((0, 0)\).
  • At day 30, she has read 7 books. The point is \((7, 30)\).
  • At the final goal (day 90), she will have read 15 books. The point is \((15, 90)\).

To find the rate of change for the remaining portion, we use the points \((7, 30)\) and \((15, 90)\).

  1. Calculating the Unit Rate

The unit rate \(m\) is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
\[m = \frac{90 - 30}{15 - 7} = \frac{60}{8}\]
Performing the division:
\[\frac{60}{8} = 7.5\]
This means Samantha has a unit rate of 7.5 days per book to finish her goal.

  1. Conclusion

The independent variable is the number of books. The dependent variable is the number of days. Based on our calculation, the unit rate is 7.5 days per book.
Final answer: \(\boxed{7.5}\).

Examples

This concept of unit rates is essential in planning projects, such as budgeting for a home renovation or scheduling study time for exams. By understanding how many hours or days you need to dedicate per task, you can effectively manage your time to meet deadlines, just like Samantha calculated the days per book to reach her reading goal.

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