To determine which of the equations is true, we need to calculate each side of the equations separately and compare them.
1. \(6 \frac{2}{5} \div 3 \frac{1}{6}\)
Convert to improper fractions:
- \(6 \frac{2}{5} = \frac{32}{5}\)
- \(3 \frac{1}{6} = \frac{19}{6}\)
Now perform the division: \[ \frac{32}{5} \div \frac{19}{6} = \frac{32}{5} \times \frac{6}{19} = \frac{32 \cdot 6}{5 \cdot 19} = \frac{192}{95} \]
Now let's check the right side: \[ \frac{32}{5} \cdot \frac{6}{19} = \frac{192}{95} \]
Both sides are equal, so this equation is true.
2. \(10 \frac{1}{3} \div 3 \frac{2}{5}\)
Convert to improper fractions:
- \(10 \frac{1}{3} = \frac{31}{3}\)
- \(3 \frac{2}{5} = \frac{17}{5}\)
Perform the division: \[ \frac{31}{3} \div \frac{17}{5} = \frac{31}{3} \times \frac{5}{17} = \frac{31 \cdot 5}{3 \cdot 17} = \frac{155}{51} \]
Now check the right side: \[ \frac{3}{31} \cdot \frac{2}{17} = \frac{6}{527} \]
The two sides are not equal.
3. \(9 \frac{3}{7} \div 4 \frac{3}{4}\)
Convert to improper fractions:
- \(9 \frac{3}{7} = \frac{66}{7}\)
- \(4 \frac{3}{4} = \frac{19}{4}\)
Perform the division: \[ \frac{66}{7} \div \frac{19}{4} = \frac{66}{7} \times \frac{4}{19} = \frac{66 \cdot 4}{7 \cdot 19} = \frac{264}{133} \]
Now check the right side: \[ \frac{66}{7} \cdot \frac{19}{4} = \frac{1254}{28} \quad \text{(not equal)} \]
4. \(8 \frac{2}{9} \div 3 \frac{4}{5}\)
Convert to improper fractions:
- \(8 \frac{2}{9} = \frac{74}{9}\)
- \(3 \frac{4}{5} = \frac{19}{5}\)
Perform the division: \[ \frac{74}{9} \div \frac{19}{5} = \frac{74}{9} \times \frac{5}{19} = \frac{370}{171} \]
Now check the right side: \[ \frac{9}{74} \cdot \frac{5}{19} = \frac{45}{1406} \]
The two sides are not equal.
Conclusion:
The only true equation among the provided options is: \[ 6 \frac{2}{5} \div 3 \frac{1}{6} = \frac{32}{5} \cdot \frac{6}{19} \]