For each of the following, write three fractions equal to the given fraction. \( \begin{array}{ll}\text { a. } \frac{1}{3} & \text { b. } \frac{6}{7} \\ \text { c. } \frac{-9}{10} & \text { d. } \frac{a}{8} \\ \frac{6}{7}=\frac{\square}{21} \\ \text { (Simplify your answer.) } \\ \text { Enter the correct numerator to match the new denominator. } \\ \frac{6}{7}=\frac{\square}{63} \text { (Simplify your answer.) } \\ \text { Enter the correct numerator to match the new denominator. } \\ \frac{6}{7}=\frac{\square}{42} \text { (Simplify your answer.) }\end{array} \)

To find equivalent fractions, we can use the property that if two fractions are equal, then their cross products are equal. This means that if we have a fraction \( \frac{a}{b} \), we can find an equivalent fraction \( \frac{c}{d} \) by ensuring that \( a \cdot d = b \cdot c \). Let's solve each part step by step. ### a. For \( \frac{1}{3} \): 1. Multiply both the numerator and denominator by the same number to find equivalent fractions. - \( \frac{1 \cdot 2}{3 \cdot 2} = \frac{2}{6} \) - \( \frac{1 \cdot 3}{3 \cdot 3} = \frac{3}{9} \) - \( \frac{1 \cdot 4}{3 \cdot 4} = \frac{4}{12} \) ### b. For \( \frac{6}{7} \): 1. To find equivalent fractions with different denominators: - For \( \frac{6}{7} = \frac{\square}{21} \): - We can find the numerator by setting up the equation: \( 6 \cdot 21 = 7 \cdot x \). - Solving gives \( x = \frac{6 \cdot 21}{7} = 18 \). - For \( \frac{6}{7} = \frac{\square}{63} \): - Setting up the equation: \( 6 \cdot 63 = 7 \cdot x \). - Solving gives \( x = \frac{6 \cdot 63}{7} = 54 \). - For \( \frac{6}{7} = \frac{\square}{42} \): - Setting up the equation: \( 6 \cdot 42 = 7 \cdot x \). - Solving gives \( x = \frac{6 \cdot 42}{7} = 36 \). ### c. For \( \frac{-9}{10} \): 1. Similar to the previous fractions, we can find equivalent fractions: - \( \frac{-9 \cdot 2}{10 \cdot 2} = \frac{-18}{20} \) - \( \frac{-9 \cdot 3}{10 \cdot 3} = \frac{-27}{30} \) - \( \frac{-9 \cdot 4}{10 \cdot 4} = \frac{-36}{40} \) ### d. For \( \frac{a}{8} \): 1. We can express equivalent fractions by multiplying: - \( \frac{a \cdot 2}{8 \cdot 2} = \frac{2a}{16} \) - \( \frac{a \cdot 3}{8 \cdot 3} = \frac{3a}{24} \) - \( \frac{a \cdot 4}{8 \cdot 4} = \frac{4a}{32} \) ### Summary of Equivalent Fractions: - a. \( \frac{1}{3} = \frac{2}{6}, \frac{3}{9}, \frac{4}{12} \) - b. \( \frac{6}{7} = \frac{18}{21}, \frac{54}{63}, \frac{36}{42} \) - c. \( \frac{-9}{10} = \frac{-18}{20}, \frac{-27}{30}, \frac{-36}{40} \) - d. \( \frac{a}{8} = \frac{2a}{16}, \frac{3a}{24}, \frac{4a}{32} \)