3. Use common denominators to find each quotient. \( \begin{array}{llll}\text { a) } \frac{7}{12} \div \frac{1}{4} & \text { b) } \frac{3}{5} \div \frac{11}{10} & \text { c) } \frac{5}{2} \div \frac{1}{3} & \text { d) } \frac{5}{6} \div \frac{9}{8}\end{array} \)
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To solve these division problems using common denominators, remember that dividing by a fraction is equivalent to multiplying by its reciprocal. Here’s how you can approach each one: First, to find \( \frac{7}{12} \div \frac{1}{4} \): Convert \( \frac{1}{4} \) to a common denominator with \( \frac{7}{12} \). The common denominator is 12, so \( \frac{1}{4} = \frac{3}{12} \). Now, instead of dividing, you can multiply: \[ \frac{7}{12} \div \frac{1}{4} = \frac{7}{12} \times \frac{12}{3} = \frac{7 \times 12}{12 \times 3} = \frac{7}{3} \] For \( \frac{3}{5} \div \frac{11}{10} \): Change \( \frac{3}{5} \) to have a common denominator with \( \frac{11}{10} \). The common denominator is 10, so \( \frac{3}{5} = \frac{6}{10} \). Then: \[ \frac{3}{5} \div \frac{11}{10} = \frac{6}{10} \times \frac{10}{11} = \frac{6}{11} \] Next, for \( \frac{5}{2} \div \frac{1}{3} \): Instead of finding a common denominator, you can directly multiply: \[ \frac{5}{2} \div \frac{1}{3} = \frac{5}{2} \times \frac{3}{1} = \frac{15}{2} \] Lastly, for \( \frac{5}{6} \div \frac{9}{8} \): This is similar; you don’t need a common denominator, just multiply by the reciprocal: \[ \frac{5}{6} \div \frac{9}{8} = \frac{5}{6} \times \frac{8}{9} = \frac{40}{54} \text{ (which simplifies to } \frac{20}{27}) \] So the final answers are: - a) \( \frac{7}{3} \) - b) \( \frac{6}{11} \) - c) \( \frac{15}{2} \) - d) \( \frac{20}{27} \)
