1. **State the problem:** We are given that the sum of two numbers is $4 \frac{1}{8}$ and their difference is $\frac{7}{8}$. We need to find the two numbers. 2. **Convert mixed number to improper fraction:** $$4 \frac{1}{8} = \frac{4 \times 8 + 1}{8} = \frac{33}{8}$$ 3. **Set up equations:** Let the two numbers be $x$ and $y$ with $x > y$. $$x + y = \frac{33}{8}$$ $$x - y = \frac{7}{8}$$ 4. **Add the two equations:** $$ (x + y) + (x - y) = \frac{33}{8} + \frac{7}{8} $$ $$ 2x = \frac{40}{8} = 5 $$ 5. **Solve for $x$:** $$ x = \frac{\cancel{2} \times x}{\cancel{2}} = \frac{5}{2} = 2.5 $$ 6. **Substitute $x$ back into the sum equation to find $y$:** $$ 2.5 + y = \frac{33}{8} $$ $$ y = \frac{33}{8} - 2.5 = \frac{33}{8} - \frac{20}{8} = \frac{13}{8} = 1.625 $$ 7. **Final answer:** The two numbers are $\boxed{\frac{5}{2} \text{ and } \frac{13}{8}}$ or $2.5$ and $1.625$ respectively.