To find the quotient of \( 8 \frac{1}{4} \) and \( 2 \frac{1}{2} \), we first need to convert these mixed numbers into improper fractions.
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Convert \( 8 \frac{1}{4} \):
- Multiply the whole number (8) by the denominator (4): \( 8 \times 4 = 32 \).
- Add the numerator (1): \( 32 + 1 = 33 \).
- So, \( 8 \frac{1}{4} = \frac{33}{4} \).
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Convert \( 2 \frac{1}{2} \):
- Multiply the whole number (2) by the denominator (2): \( 2 \times 2 = 4 \).
- Add the numerator (1): \( 4 + 1 = 5 \).
- So, \( 2 \frac{1}{2} = \frac{5}{2} \).
Now, we need to divide the two improper fractions:
\[ \frac{33}{4} \div \frac{5}{2} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{33}{4} \times \frac{2}{5} \]
Now, multiply the numerators and the denominators:
\[ \frac{33 \times 2}{4 \times 5} = \frac{66}{20} \]
Next, we simplify \( \frac{66}{20} \):
- Find the greatest common divisor (GCD) of 66 and 20, which is 2.
- Divide both the numerator and the denominator by 2:
\[ \frac{66 \div 2}{20 \div 2} = \frac{33}{10} \]
Thus, the simplified quotient of \( 8 \frac{1}{4} \) and \( 2 \frac{1}{2} \) is
\[ \frac{33}{10}. \]