9.1: Number Talk: Greater Than 1 or Less Than ? Decide whether each quotient is greater than 1 or less than 1. \[ \begin{array}{l} \frac{1}{2} \div \frac{1}{4} \\ 1 \div \frac{3}{4} \\ \frac{2}{3} \div \frac{2}{8} \\ 2 \frac{2}{8} \div 2 \frac{3}{5} \end{array} \]

To determine whether each quotient is greater than 1 or less than 1, we will calculate each expression step by step. 1. **Calculate \(\frac{1}{2} \div \frac{1}{4}\)**: - Dividing by a fraction is the same as multiplying by its reciprocal. - Thus, \(\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2\). - Since \(2 > 1\), this quotient is **greater than 1**. 2. **Calculate \(1 \div \frac{3}{4}\)**: - Again, we multiply by the reciprocal: \(1 \div \frac{3}{4} = 1 \times \frac{4}{3} = \frac{4}{3}\). - Since \(\frac{4}{3} > 1\), this quotient is **greater than 1**. 3. **Calculate \(\frac{2}{3} \div \frac{2}{8}\)**: - We multiply by the reciprocal: \(\frac{2}{3} \div \frac{2}{8} = \frac{2}{3} \times \frac{8}{2} = \frac{2 \times 8}{3 \times 2} = \frac{8}{3}\). - Since \(\frac{8}{3} > 1\), this quotient is **greater than 1**. 4. **Calculate \(2 \frac{2}{8} \div 2 \frac{3}{5}\)**: - First, convert the mixed numbers to improper fractions: - \(2 \frac{2}{8} = 2 + \frac{2}{8} = \frac{16}{8} + \frac{2}{8} = \frac{18}{8} = \frac{9}{4}\). - \(2 \frac{3}{5} = 2 + \frac{3}{5} = \frac{10}{5} + \frac{3}{5} = \frac{13}{5}\). - Now, divide: \(\frac{9}{4} \div \frac{13}{5} = \frac{9}{4} \times \frac{5}{13} = \frac{9 \times 5}{4 \times 13} = \frac{45}{52}\). - Since \(\frac{45}{52} < 1\), this quotient is **less than 1**. ### Summary of Results: - \(\frac{1}{2} \div \frac{1}{4}\) is **greater than 1**. - \(1 \div \frac{3}{4}\) is **greater than 1**. - \(\frac{2}{3} \div \frac{2}{8}\) is **greater than 1**. - \(2 \frac{2}{8} \div 2 \frac{3}{5}\) is **less than 1**.