To multiply the mixed numbers \( 3 \frac{3}{4} \) and \( 8 \frac{2}{9} \), we first need to convert them into improper fractions.

1. Convert \( 3 \frac{3}{4} \) to an improper fraction:
\[ 3 \frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4} \]

2. Convert \( 8 \frac{2}{9} \) to an improper fraction:
\[ 8 \frac{2}{9} = \frac{8 \times 9 + 2}{9} = \frac{72 + 2}{9} = \frac{74}{9} \]

Next, we multiply the improper fractions:
\[ \frac{15}{4} \times \frac{74}{9} = \frac{15 \times 74}{4 \times 9} = \frac{1110}{36} \]

Now, we simplify the fraction \( \frac{1110}{36} \):

3. Find the greatest common divisor (GCD) of 1110 and 36:
   - The prime factorization of 1110 is \( 2 \times 3 \times 5 \times 37 \)
   - The prime factorization of 36 is \( 2^2 \times 3^2 \)
   - The common factors are \( 2 \) and \( 3 \)
   - Therefore, the GCD is \( 2 \times 3 = 6 \)

4. Simplify the fraction by dividing both the numerator and the denominator by the GCD:
\[ \frac{1110 \div 6}{36 \div 6} = \frac{185}{6} \]

Finally, we can convert the improper fraction back to a mixed number:
\[ \frac{185}{6} = 30 \frac{5}{6} \]

So, \( 3 \frac{3}{4} \times 8 \frac{2}{9} = 30 \frac{5}{6} \).

Would you like more details or have any questions? 

Here are some related questions:

1. How do you convert a mixed number to an improper fraction?
2. What is the method to multiply two fractions?
3. How do you simplify a fraction?
4. What is the greatest common divisor (GCD)?
5. How do you find the GCD of two numbers?
6. How do you convert an improper fraction back to a mixed number?
7. What are prime factorizations and how are they used in finding the GCD?
8. Why is it important to simplify fractions?

**Tip:** When multiplying fractions, always simplify them if possible to make calculations easier.

Learn how to multiply the mixed numbers 3 3/4 and 8 2/9 step-by-step with detailed explanations. This problem covers conversion to improper fractions, multiplication of fractions, simplification of fractions using GCD, and conversion back to mixed numbers. Suitable for students in Grades 6-8.

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