To solve the expression $1 \frac{5}{9} \div \left( -\frac{2}{3} \right) + \left( -2 \frac{3}{5} \right)$, let's break it down step by step.

Step 1: Convert Mixed Fractions to Improper Fractions

• $1 \frac{5}{9} = \frac{14}{9}$
• $-2 \frac{3}{5} = -\frac{13}{5}$

So the expression becomes: $\frac{14}{9} \div \left( -\frac{2}{3} \right) + \left( -\frac{13}{5} \right)$

Step 2: Perform the Division

Dividing by a fraction is equivalent to multiplying by its reciprocal: $\frac{14}{9} \div \left( -\frac{2}{3} \right) = \frac{14}{9} \times \left( -\frac{3}{2} \right)$ Multiply the numerators and denominators: $\frac{14 \times (-3)}{9 \times 2} = \frac{-42}{18}$ Simplify the fraction: $\frac{-42}{18} = \frac{-7}{3}$

Step 3: Add the Fractions

Now, add $\frac{-7}{3}$ and $\frac{-13}{5}$. To add these, find a common denominator, which is 15: $\frac{-7}{3} = \frac{-35}{15}, \quad \frac{-13}{5} = \frac{-39}{15}$ Now, add the fractions: $\frac{-35}{15} + \frac{-39}{15} = \frac{-74}{15}$

Final Answer:

The result of the expression is $\frac{-74}{15}$ or approximately $-4.93$.

Would you like more details on any part of this?

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Related Questions:

1. How do you convert mixed numbers into improper fractions?
2. What are the steps for dividing fractions?
3. How do you simplify fractions after multiplication?
4. How do you find a common denominator when adding fractions?
5. How do you convert improper fractions back to mixed numbers?

Tip: When dividing by a fraction, always multiply by its reciprocal to simplify the process!