To divide the fraction \( \frac{7}{8} \) by the mixed number \( 4 \frac{1}{5} \), we first need to convert the mixed number into an improper fraction.
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Convert \( 4 \frac{1}{5} \) to an improper fraction:
\[ 4 \frac{1}{5} = 4 \times 5 + 1 = 20 + 1 = 21 \]
So, \( 4 \frac{1}{5} = \frac{21}{5} \).
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Now we can rewrite the division as follows:
\[ \frac{7}{8} \div \frac{21}{5} \]
Division by a fraction is the same as multiplying by its reciprocal:
\[ \frac{7}{8} \times \frac{5}{21} \]
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Now, multiply the fractions:
\[ \frac{7 \times 5}{8 \times 21} = \frac{35}{168} \]
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Next, we simplify \( \frac{35}{168} \). To do this, we can find the greatest common divisor (GCD) of 35 and 168.
- The factors of 35 are \( 1, 5, 7, 35 \)
- The factors of 168 are \( 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 28, 42, 84, 168 \)
The common factor is 7.
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Now, divide both the numerator and denominator by their GCD, 7:
\[ \frac{35 \div 7}{168 \div 7} = \frac{5}{24} \]
So, the final simplified quotient of \( \frac{7}{8} \div 4 \frac{1}{5} \) is
\[ \frac{5}{24} \]