Cururu is a strange frog that moves only with two types of jumps, the

Type I: $10 \mathrm{~cm}$ to the East and $30 \mathrm{~cm}$ to the North and the

Type II: $20 \mathrm{~cm}$ to the West and $40 \mathrm{~cm}$ to the South.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-024.jpg?height=426&width=628&top_left_y=1235&top_left_x=748)

(a) How does Cururu get to a point located $190 \mathrm{~cm}$ to the East and $950 \mathrm{~cm}$ to the North of his home?

(b) Is it possible for Cururu to get to a point located $180 \mathrm{~cm}$ to the East and $950 \mathrm{~cm}$ to the North of his home?
For every $x$ jumps of type I, the frog moves $10 x \mathrm{~cm}$ to the East and $30 x \mathrm{~cm}$ to the North, and for every $y$ jumps of type II, the frog moves $20 y \mathrm{~cm}$ to the West and $40 y \mathrm{~cm}$ to the South. Thus, after $x$ jumps of type I and $y$ of type II, the frog has moved $10 x-20 y \mathrm{~cm}$ to the East and $30 x-40 y \mathrm{~cm}$ to the North.

(a) Solving

$$
\left\{\begin{array}{l}
10 x-20 y=190 \\
30 x-40 y=950
\end{array}\right.
$$

we find $x=57$ and $y=19$. Therefore, the frog should make 57 jumps of type I and 19 of type II, in any order, to reach a point located $190 \mathrm{~cm}$ to the East and $950 \mathrm{~cm}$ to the North of its home.

(b) Since the number of jumps of each type is an integer, Cururu will only reach the point located $180 \mathrm{~cm}$ to the East and $950 \mathrm{~cm}$ to the North of its home if the system

$$
\left\{\begin{array}{l}
10 x-20 y=180 \\
30 x-40 y=950
\end{array}\right.
$$

has an integer solution. However, the solution to this system is $x=59$ and $y=41 / 2$, which is not an integer. Therefore, Cururu will not be able to reach that point.