To divide the mixed numbers $3 \frac{6}{7}$ by $6 \frac{3}{4}$, we follow these steps:

Step 1: Convert the mixed numbers to improper fractions.

• $3 \frac{6}{7} = \frac{3 \times 7 + 6}{7} = \frac{21 + 6}{7} = \frac{27}{7}$
• $6 \frac{3}{4} = \frac{6 \times 4 + 3}{4} = \frac{24 + 3}{4} = \frac{27}{4}$

Step 2: Perform the division by multiplying by the reciprocal.

$\frac{27}{7} \div \frac{27}{4} = \frac{27}{7} \times \frac{4}{27} = \frac{27 \times 4}{7 \times 27} = \frac{108}{189}$ The $27$s cancel out: $= \frac{4}{7}$

Final Answer:

$3 \frac{6}{7} \div 6 \frac{3}{4} = \frac{4}{7}$

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:

1. How do you convert a mixed number to an improper fraction?
2. What is the reciprocal of a fraction, and why is it used in division?
3. How would the answer change if the operation were multiplication instead of division?
4. How do you simplify fractions after performing arithmetic operations?
5. Can you divide other mixed numbers, like $5 \frac{2}{5}$ by $2 \frac{1}{2}$?

Tip: Always double-check the conversion from mixed numbers to improper fractions to ensure accuracy in your calculations.