Sure, let's solve this problem step-by-step:
1. Convert the Mixed Fractions to Improper Fractions:
- Floor length: [tex]\(2 \frac{2}{3}\)[/tex]
- [tex]\(2 \frac{2}{3} = 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3}\)[/tex]
- Floor width: [tex]\(3 \frac{3}{4}\)[/tex]
- [tex]\(3 \frac{3}{4} = 3 + \frac{3}{4} = \frac{12}{4} + \frac{3}{4} = \frac{15}{4}\)[/tex]
- Tile side: [tex]\(1 \frac{2}{3}\)[/tex]
- [tex]\(1 \frac{2}{3} = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}\)[/tex]
2. Calculate the Area of the Floor:
- Length of the floor: [tex]\(\frac{8}{3} \, m\)[/tex]
- Width of the floor: [tex]\(\frac{15}{4} \, m\)[/tex]
- Area of the floor:
[tex]\[
\text{Area of the floor} = \left( \frac{8}{3} \right) \times \left( \frac{15}{4} \right) = \frac{8 \times 15}{3 \times 4} = \frac{120}{12} = 10 \, m^2
\][/tex]
3. Calculate the Area of One Tile:
- Side of the tile: [tex]\(\frac{5}{3} \, m\)[/tex]
- Area of one tile:
[tex]\[
\text{Area of one tile} = \left( \frac{5}{3} \right) \times \left( \frac{5}{3} \right) = \frac{5 \times 5}{3 \times 3} = \frac{25}{9} \approx 2.777777777777778 \, m^2
\][/tex]
4. Calculate the Number of Tiles Required:
- Total area of the floor: [tex]\(10 \, m^2\)[/tex]
- Area of one tile: [tex]\(2.777777777777778 \, m^2\)[/tex]
- Number of tiles:
[tex]\[
\text{Number of tiles} = \left\lceil \frac{\text{Total area of the floor}}{\text{Area of one tile}} \right\rceil = \left\lceil \frac{10}{2.777777777777778} \right\rceil = 4
\][/tex]
Therefore, the number of tiles required to cover the floor is [tex]\( \boxed{4} \)[/tex].
