Let's break this expression down step by step:

\[
\frac{3}{7} \times \frac{21}{5} + \frac{5}{9} \div \left(-\frac{10}{3}\right)
\]

### Step 1: Multiply \(\frac{3}{7}\) by \(\frac{21}{5}\)

\[
\frac{3}{7} \times \frac{21}{5} = \frac{3 \times 21}{7 \times 5} = \frac{63}{35} = \frac{9}{5}
\]

### Step 2: Divide \(\frac{5}{9}\) by \(-\frac{10}{3}\)

Dividing by a fraction is the same as multiplying by its reciprocal. So:

\[
\frac{5}{9} \div \left(-\frac{10}{3}\right) = \frac{5}{9} \times \left(-\frac{3}{10}\right) = \frac{5 \times (-3)}{9 \times 10} = \frac{-15}{90} = -\frac{1}{6}
\]

### Step 3: Add the results

Now we add the two results:

\[
\frac{9}{5} + \left(-\frac{1}{6}\right)
\]

To add these fractions, we need a common denominator. The least common denominator of 5 and 6 is 30.

Convert the fractions:

\[
\frac{9}{5} = \frac{54}{30}, \quad -\frac{1}{6} = -\frac{5}{30}
\]

Now add:

\[
\frac{54}{30} + \left(-\frac{5}{30}\right) = \frac{54 - 5}{30} = \frac{49}{30}
\]

### Final Answer:

\[
\frac{49}{30}
\]

Would you like more details or further clarification?

Here are 5 related questions:
1. How do you multiply and divide fractions?
2. What is the process for finding a common denominator?
3. How do you handle negative fractions in operations?
4. How do you simplify complex fractions?
5. What are the steps to convert between mixed numbers and improper fractions?

**Tip:** When dividing fractions, remember that dividing by a fraction is the same as multiplying by its reciprocal.

This problem demonstrates how to solve the fraction expression 3/7 × 21/5 + 5/9 ÷ (-10/3) with detailed steps. It covers multiplying and dividing fractions, as well as adding fractions with different denominators. Suitable for students in Grades 6-8.

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