1. **Problem statement:** Evaluate the expression $7 \frac{3}{8} + 4 \frac{1}{8}$ using number lines and rational number addition. 2. **Formula and rules:** To add mixed numbers, first convert them to improper fractions or add whole parts and fractional parts separately. 3. **Convert mixed numbers to improper fractions:** $$7 \frac{3}{8} = \frac{7 \times 8 + 3}{8} = \frac{56 + 3}{8} = \frac{59}{8}$$ $$4 \frac{1}{8} = \frac{4 \times 8 + 1}{8} = \frac{32 + 1}{8} = \frac{33}{8}$$ 4. **Add the improper fractions:** $$\frac{59}{8} + \frac{33}{8} = \frac{59 + 33}{8} = \frac{92}{8}$$ 5. **Simplify the fraction:** $$\frac{92}{8} = \frac{\cancel{4} \times 23}{\cancel{4} \times 2} = \frac{23}{2}$$ 6. **Convert back to mixed number:** $$\frac{23}{2} = 11 \frac{1}{2}$$ 7. **Answer:** $$7 \frac{3}{8} + 4 \frac{1}{8} = 11 \frac{1}{2}$$