The ants Maricota and Nandinha are strolling on a balcony whose floor is made up of rectangular tiles measuring $4 \mathrm{~cm}$ in width by $6 \mathrm{~cm}$ in length. Maricota starts from point $M$ and Nandinha from $N$, both walking only along the sides of the rectangles, following the path indicated in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_01_b9d1398cc05f7e479435g-1.jpg?height=317&width=500&top_left_y=1252&top_left_x=869)

(a) The two meet after walking the same distance. What was that distance?

(b) Where did they meet?
(a) The journey from $M$ to $N$ consists of 14 lengths and 12 widths of the tiles, so its length is:

$$
14 \times 6 + 12 \times 4 = 84 + 48 = 132 \text{ cm}
$$

Since the ants travel the same distance, each one must walk $132 \div 2 = 66 \text{ cm}$.

(b) Let's follow the path taken by Maricota from the beginning, until she completes $66 \text{ cm}$:

$$
\begin{aligned}
& \underbrace{2 \text{ lengths}}_{2 \times 6 = 12} + \underbrace{1 \text{ width}}_{4 + 12 = 16} + \underbrace{3 \text{ lengths}}_{18 + 16 = 34} + \underbrace{2 \text{ widths}}_{8 + 34 = 42} + \\
& \underbrace{2 \text{ lengths}}_{12 + 42 = 54} + \underbrace{1 \text{ width}}_{4 + 54 = 58} + \underbrace{1 \text{ length}}_{6 + 58 = 64} + \underbrace{1/2 \text{ width}}_{2 + 64 = 66}
\end{aligned}
$$

Maricota's path to the meeting point is indicated in the figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_b9d1398cc05f7e479435g-3.jpg?height=471&width=1013&top_left_y=2055&top_left_x=687)