1. **State the problem:** We need to find the area of a rectangular rug with width $8$ m and height $2 \frac{7}{12}$ m. 2. **Formula:** The area $A$ of a rectangle is given by: $$A = \text{width} \times \text{height}$$ 3. **Convert mixed number to improper fraction:** $$2 \frac{7}{12} = \frac{2 \times 12 + 7}{12} = \frac{24 + 7}{12} = \frac{31}{12}$$ 4. **Calculate the area:** $$A = 8 \times \frac{31}{12} = \frac{8 \times 31}{12} = \frac{248}{12}$$ 5. **Simplify the fraction:** $$\frac{248}{12} = \frac{62}{3} = 20 \frac{2}{3}$$ 6. **Final answer:** The area of the rug is $20 \frac{2}{3}$ square meters.