$315 \frac{7}{8} \times 1 \frac{1}{4} \times 3 \frac{1}{2}$

Step 1 ：Convert each mixed number into an improper fraction. So, $315 \frac{7}{8}$ becomes $\frac{2527}{8}$, $1 \frac{1}{4}$ becomes $\frac{5}{4}$, and $3 \frac{1}{2}$ becomes $\frac{7}{2}$.

Step 2 ：Next, multiply the three fractions together. This gives us $\frac{2527}{8} \times \frac{5}{4} \times \frac{7}{2} = \frac{88445}{64}$.

Step 3 ：Then, simplify the resulting fraction. This gives us a whole part of 1381 and a fraction part of $\frac{61}{64}$.

Step 4 ：Finally, convert the simplified fraction back to a mixed number. This gives us the final answer: $\boxed{1381 \frac{61}{64}}$.