Let's solve the expression step by step using properties of rational numbers.

The given expression is:

$\frac{3}{5} \times \frac{-5}{6} + \frac{5}{2} + \frac{-5}{6} \times \frac{7}{5}$

Step 1: Simplify each multiplication term

1. First multiplication: $\frac{3}{5} \times \frac{-5}{6} = \frac{3 \times -5}{5 \times 6} = \frac{-15}{30} = \frac{-1}{2}$

2. Second multiplication: $\frac{-5}{6} \times \frac{7}{5} = \frac{-5 \times 7}{6 \times 5} = \frac{-35}{30} = \frac{-7}{6}$

Step 2: Rewrite the expression with the simplified terms

$\frac{-1}{2} + \frac{5}{2} + \frac{-7}{6}$

Step 3: Combine the terms

Now, we need to add the fractions. To do this, we first find a common denominator for all terms. The denominators are 2, 2, and 6. The least common denominator (LCD) is 6.

1. Convert the fractions to have a denominator of 6: $\frac{-1}{2} = \frac{-3}{6}, \quad \frac{5}{2} = \frac{15}{6}$

2. The expression becomes: $\frac{-3}{6} + \frac{15}{6} + \frac{-7}{6}$

Step 4: Add the numerators

Now that all terms have the same denominator, we can add them: $\frac{-3 + 15 - 7}{6} = \frac{5}{6}$

Final Answer:

The value of the expression is:

$\frac{5}{6}$

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Would you like more details or explanations?

Related Questions:

1. How can I apply the distributive property in rational expressions?
2. What is the best method to find the least common denominator (LCD) in fraction addition?
3. Can rational numbers always be simplified to fractions in their simplest form?
4. How can multiplication of negative rational numbers affect the result?
5. How do I deal with mixed numbers and improper fractions in complex expressions?

Tip: Always check if fractions can be simplified before performing operations like addition or multiplication.