2. Every hour, Martina reads an equal number of pages of a book. First, Martina calculated how many pages she should read every hour to finish the book exactly by the given deadline. Then she calculated that she would finish reading the entire book 2.5 hours before the deadline if she read two more pages per hour than planned. If she read four more pages per hour than planned, she would finish the book 4.5 hours before the deadline. How many pages does the book have, and how long does it take Martina to read the entire book?
2. Every hour, Martina reads an equal number of pages of a book. First, Martina calculated how many pages she should read every hour to finish the book exactly by the given deadline. Then, she calculated that she would finish reading the entire book 2.5 hours earlier if she read two more pages per hour than planned. If she read four more pages per hour than planned, she would finish the book 4.5 hours earlier. How many pages does the book have, and how long does Martina need to read the entire book?

## Solution.

Let $x$ be the number of pages Martina reads every hour, and $y$ be the number of hours (deadline) in which she needs to read the entire book. The total number of pages in the book is $x \cdot y$.

If she reads two more pages per hour $(x+2)$, she will finish reading the entire book in

$(y-2.5)$ hours, so we have $(x+2) \cdot(y-2.5)=x \cdot y$.

2 POINTS

i.e., $x y+2 y-2.5 x-5=x y$

$2 y=5+2.5 x$.

1 POINT

If she reads four more pages per hour $(x+4)$, she will finish reading the entire book in

$(y-4.5)$ hours, so we have $(x+4) \cdot(y-4.5)=x \cdot y$.

2 POINTS

i.e., $x y+4 y-4.5 x-18=x y$

$4 y=4.5 x+18$.

1 POINT

If we multiply the first equation by 2, we get an equivalent equation

$4 y=10+5 x$.

1 POINT

By equating the equal values, we get the equation

$5 x+10=18+4.5 x$

thus $x=16$.

1 POINT

For $x=16$, we have $2 y=5+2.5 \cdot 16$

$2 y=45$

$y=22.5$

1 POINT

Martina needs to read $x \cdot y=16 \cdot 22.5=360$ pages of the book in 22.5 hours.

Note 1. The system can also be obtained using the graphical method. Each linear equation is worth 3 POINTS.

![](https://cdn.mathpix.com/cropped/2024_05_30_2e894aa57f8bdfbd97cfg-21.jpg?height=528&width=1262&top_left_y=1798&top_left_x=223)

![](https://cdn.mathpix.com/cropped/2024_05_30_2e894aa57f8bdfbd97cfg-21.jpg?height=385&width=422&top_left_y=1852&top_left_x=226)

$2(y-2.5)=2.5 x$

![](https://cdn.mathpix.com/cropped/2024_05_30_2e894aa57f8bdfbd97cfg-21.jpg?height=457&width=419&top_left_y=1779&top_left_x=1047)

$4(y-4.5)=4.5 x$

Note 2. The time calculated in any unit of time should be evaluated equally.