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, as indicated in the figure. Maricota starts from point $M$, Nandinha starts from $N$, and both walk only along the edges of the tiles, following the path indicated in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-022.jpg?height=321&width=511&top_left_y=1878&top_left_x=1318)

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

(b) At what point did they meet?
(a) The journey from $M$ to $N$ comprises 14 lengths and 12 widths of the tiles. Therefore, its length is $14 \times 6 + 12 \times 4 = 84 + 48 = 132 \mathrm{~cm}$.

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

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

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

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

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-071.jpg?height=474&width=1110&top_left_y=254&top_left_x=610)