1. Calculate the tip:
\text{Tip} = 17.85 \times 0.15 = 2.6775
2. Round the tip to the nearest quarter:
2.6775 \approx 2.75
3. Add the rounded tip to the original bill to find the total amount:
\text{Total} = 17.85 + 2.75 = 20.60
Hence, the total amount spent is $\$20.60$.