1. **State the problem:** We need to add the fractions $\frac{5}{2}$ and $\frac{39}{4}$. 2. **Formula and rules:** To add fractions, they must have the same denominator. If not, find the least common denominator (LCD). 3. **Find the LCD:** The denominators are 2 and 4. The LCD of 2 and 4 is 4. 4. **Convert fractions:** Convert $\frac{5}{2}$ to have denominator 4 by multiplying numerator and denominator by 2: $$\frac{5}{2} = \frac{5 \times 2}{2 \times 2} = \frac{10}{4}$$ 5. **Add the fractions:** Now add $\frac{10}{4} + \frac{39}{4} = \frac{10 + 39}{4} = \frac{49}{4}$. 6. **Simplify or convert to mixed number:** $\frac{49}{4} = 12 \frac{1}{4}$ because $49 \div 4 = 12$ remainder 1. **Final answer:** $\frac{49}{4}$ or $12 \frac{1}{4}$.