To find the quotient of \( 25 \frac{1}{2} \div 4 \frac{1}{4} \), we first need to convert the mixed numbers into improper fractions.
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Convert \( 25 \frac{1}{2} \): \[ 25 \frac{1}{2} = 25 \times 2 + 1 = 50 + 1 = 51 \] So, \( 25 \frac{1}{2} = \frac{51}{2} \).
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Convert \( 4 \frac{1}{4} \): \[ 4 \frac{1}{4} = 4 \times 4 + 1 = 16 + 1 = 17 \] So, \( 4 \frac{1}{4} = \frac{17}{4} \).
Now we can write the division as: \[ \frac{51}{2} \div \frac{17}{4} \] Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{51}{2} \times \frac{4}{17} \]
Now multiply the fractions: \[ \frac{51 \times 4}{2 \times 17} \]
Calculating the numerator and denominator: \[ 51 \times 4 = 204 \] \[ 2 \times 17 = 34 \]
Thus, we have: \[ \frac{204}{34} \]
Now, simplify the fraction: \[ \frac{204 \div 34}{34 \div 34} = \frac{6}{1} = 6 \]
So, the quotient of \( 25 \frac{1}{2} \div 4 \frac{1}{4} \) is \( \boxed{6} \).