To multiply the mixed numbers \(5 \frac{1}{3}\) and \(1 \frac{5}{6}\), we first convert them into improper fractions.
-
Convert \(5 \frac{1}{3}\) to an improper fraction: \[ 5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3} \]
-
Convert \(1 \frac{5}{6}\) to an improper fraction: \[ 1 \frac{5}{6} = \frac{1 \times 6 + 5}{6} = \frac{6 + 5}{6} = \frac{11}{6} \]
Now we will multiply the two improper fractions: \[ \frac{16}{3} \times \frac{11}{6} = \frac{16 \times 11}{3 \times 6} = \frac{176}{18} \]
Next, we simplify \(\frac{176}{18}\). We find the greatest common divisor (GCD) of 176 and 18. The GCD is 2. Therefore, we divide both the numerator and the denominator by 2: \[ \frac{176 \div 2}{18 \div 2} = \frac{88}{9} \]
Now convert \(\frac{88}{9}\) back to a mixed number. To do this, we divide 88 by 9: \[ 88 \div 9 = 9 \quad \text{(with a remainder of 7)} \] So we can write it as: \[ \frac{88}{9} = 9 \frac{7}{9} \]
Thus, the final answer is: \[ \boxed{9 \frac{7}{9}} \]