To solve the expression $1 \frac{2}{15} + \left( -3 \frac{1}{2} \right)$, we need to follow these steps:

Step 1: Convert the mixed numbers into improper fractions.

• For $1 \frac{2}{15}$: $1 \frac{2}{15} = \frac{15}{15} + \frac{2}{15} = \frac{17}{15}$

• For $-3 \frac{1}{2}$: $-3 \frac{1}{2} = -\left( \frac{6}{2} + \frac{1}{2} \right) = -\frac{7}{2}$

Step 2: Find a common denominator to add the fractions.

The least common denominator (LCD) of 15 and 2 is 30.

• Convert $\frac{17}{15}$ to a denominator of 30: $\frac{17}{15} = \frac{17 \times 2}{15 \times 2} = \frac{34}{30}$
• Convert $-\frac{7}{2}$ to a denominator of 30: $-\frac{7}{2} = \frac{-7 \times 15}{2 \times 15} = \frac{-105}{30}$

Step 3: Add the fractions.

$\frac{34}{30} + \frac{-105}{30} = \frac{34 - 105}{30} = \frac{-71}{30}$

Step 4: Convert back to a mixed number (if needed).

$\frac{-71}{30} = -2 \frac{11}{30}$

Final Answer:

$1 \frac{2}{15} + \left( -3 \frac{1}{2} \right) = -2 \frac{11}{30}$

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to expand your knowledge:

1. How do you subtract mixed numbers?
2. What are the steps to simplify improper fractions?
3. How do you convert improper fractions to mixed numbers?
4. Why is finding a common denominator necessary in fraction addition?
5. Can you solve $5 \frac{3}{4} + (-2 \frac{2}{5})$?

Tip: When dealing with mixed numbers, converting them into improper fractions first often simplifies the process of addition or subtraction.