1. The problem asks for the best estimate of the difference between $6\frac{7}{8}$ and $2\frac{1}{8}$. 2. First, convert the mixed numbers to improper fractions or decimals for easier subtraction. 3. Convert $6\frac{7}{8}$: $$6 + \frac{7}{8} = \frac{48}{8} + \frac{7}{8} = \frac{55}{8} = 6.875$$ 4. Convert $2\frac{1}{8}$: $$2 + \frac{1}{8} = \frac{16}{8} + \frac{1}{8} = \frac{17}{8} = 2.125$$ 5. Subtract the two values: $$6.875 - 2.125 = 4.75$$ 6. The difference is $4.75$, which is closest to $5$ among the given options. 7. Therefore, the best estimate of the difference is $5$.