Question

Emily and Lily are reading the same copy of the same book for a summer reading Assignment • Emily has already read 15 pages. Emily reads 25 pages per day until the book is finished. • Lily has already read 30 pages. Lily reads 20 pages per day until the book is finished. One day, the two girls notice that they are on the same page. What page number are they on when they have read the same amount of the book?

Answer 1

We can model the number of pages for Emily and Lily as a linear function.

The y-intercept is the number of pages they have read at time t=0. In this case, the y-intercept for Emiliy is y(0)=15, as she has already read 15 pages when Lily was starting.

The y-intercept for Lily is y(0)=0, as she hasn't read any pages at the moment we start counting time.

The slope is the amount of pages they read per day: 15 pages per day for Emily and 20 pages per day for Lily. As the slope of Lily is greater than the slope for Emily, we can estimate that somewhere Liliy will catch up with Emily.

Then, we start by writing the linear equations for both.

The pages that Emily read in t days can be modeled as:

`y_e=15+15\cdot t=15t+15`

and the pages that Lily read in t days can be modeled as:

`y_l=20\cdot t+0=20t`

We can calculate the time at which both are at the same page by working from the equality ye=yt, that is both are on the same page:

`\begin{gathered} y_e=y_t \\ 15t+15=20t \\ 15=20t-15t \\ 15=5t \\ t=(15)/(5) \\ t=3 \end{gathered}`

They will be on the same page at day t=3.

Then, we can calculate the number of pages at that time replacing t by 3 in any of both equations:

`y_l(3)=25\cdot3=75`

They will catch up in page 75.

Answer:

The page will be page 75 and it will happen on the 3rd day.