First, convert the dimensions to inches to have uniform units:
10 \frac{3}{4} \text{ feet} = 10.75 \text{ feet} \times 12 \frac{\text{inches}}{\text{foot}} = 129 \text{ inches}
The area of the backsplash in square inches:
24.5 \text{ inches} \times 129 \text{ inches} = 3156 \text{ square inches}
The area of one tile:
12 \text{ inches} \times 12 \text{ inches} = 144 \text{ square inches}
Number of tiles required:
\frac{3156 \text{ square inches}}{144 \text{ square inches/tile}} \approx 21.9167
Since you can't use a fraction of a tile, round up to the nearest whole number:
22 \text{ tiles}