To convert the products of the given problems into mixed numbers, we will follow the steps for multiplication of fractions and mixed numbers.
1. \( \frac{7}{8} \times 3 \frac{1}{2} \)
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First, convert \( 3 \frac{1}{2} \) to an improper fraction: \[ 3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2} \]
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Now multiply the fractions: \[ \frac{7}{8} \times \frac{7}{2} = \frac{7 \times 7}{8 \times 2} = \frac{49}{16} \]
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Convert \( \frac{49}{16} \) to a mixed number:
- Divide \( 49 \) by \( 16 \) to get \( 3 \) remainder \( 1 \): \[ 49 \div 16 = 3 \quad (16 \times 3 = 48 , \text{and} , 49 - 48 = 1) \]
- So, \( \frac{49}{16} = 3 \frac{1}{16} \).
2. \( 1 \frac{2}{3} \times 6 \frac{1}{2} \)
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Convert both mixed numbers to improper fractions: \[ 1 \frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{5}{3} \] \[ 6 \frac{1}{2} = \frac{6 \times 2 + 1}{2} = \frac{13}{2} \]
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Multiply the fractions: \[ \frac{5}{3} \times \frac{13}{2} = \frac{5 \times 13}{3 \times 2} = \frac{65}{6} \]
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Convert \( \frac{65}{6} \) to a mixed number:
- Divide \( 65 \) by \( 6 \) to get \( 10 \) remainder \( 5 \): \[ 65 \div 6 = 10 \quad (6 \times 10 = 60 , \text{and} , 65 - 60 = 5) \]
- So, \( \frac{65}{6} = 10 \frac{5}{6} \).
3. \( 6 \times 2 \frac{1}{7} \)
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Convert \( 2 \frac{1}{7} \) to an improper fraction: \[ 2 \frac{1}{7} = \frac{2 \times 7 + 1}{7} = \frac{15}{7} \]
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Multiply by \( 6 \): \[ 6 \times \frac{15}{7} = \frac{6 \times 15}{1 \times 7} = \frac{90}{7} \]
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Convert \( \frac{90}{7} \) to a mixed number:
- Divide \( 90 \) by \( 7 \) to get \( 12 \) remainder \( 6 \): \[ 90 \div 7 = 12 \quad (7 \times 12 = 84 , \text{and} , 90 - 84 = 6) \]
- So, \( \frac{90}{7} = 12 \frac{6}{7} \).
Summary of Results
- \( \frac{7}{8} \times 3 \frac{1}{2} = 3 \frac{1}{16} \)
- \( 1 \frac{2}{3} \times 6 \frac{1}{2} = 10 \frac{5}{6} \)
- \( 6 \times 2 \frac{1}{7} = 12 \frac{6}{7} \)
These are the final mixed number results for all the products.