To solve the equation \( \frac{9}{2} \div \frac{5}{4} \) (which represents 4 and one-half divided by 1 and one-fourth), we first convert the mixed numbers into improper fractions.
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Convert \( 4 \frac{1}{2} \) to an improper fraction: \[ 4 \frac{1}{2} = 4 \times 2 + 1 = 8 + 1 = \frac{9}{2} \]
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Convert \( 1 \frac{1}{4} \) to an improper fraction: \[ 1 \frac{1}{4} = 1 \times 4 + 1 = 4 + 1 = \frac{5}{4} \]
Now we can perform the division of two fractions: \[ \frac{9}{2} \div \frac{5}{4} = \frac{9}{2} \times \frac{4}{5} \]
Next, we multiply the fractions: \[ \frac{9 \times 4}{2 \times 5} = \frac{36}{10} \]
Now we simplify \( \frac{36}{10} \): \[ \frac{36}{10} = \frac{18}{5} \quad (\text{dividing both the numerator and denominator by 2}) \]
Thus, in the form \( a \frac{b}{c} \):
- \( a = 3 \) (the whole number part),
- \( b = 3 \) (the numerator of the fractional part),
- \( c = 5 \) (the denominator of the fractional part).
The correct values are:
- \( a = 3 \)
- \( b = 3 \)
- \( c = 5 \)
Thus, the answer is: \[ \boxed{a = 3, b = 3, c = 5} \]