To evaluate the expression \((-4 \frac{1}{4} + (-4.5)) / 2 \frac{1}{2}\), we follow these steps:
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Convert the mixed numbers to improper fractions or decimals for easier calculations:
- Convert \(-4 \frac{1}{4}\): \[ -4 \frac{1}{4} = -\left(4 \cdot 4 + 1\right) / 4 = -\frac{17}{4} \]
- Convert \(-4.5\) to a fraction: \[ -4.5 = -\frac{9}{2} \]
- Convert \(2 \frac{1}{2}\): \[ 2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{5}{2} \]
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Add the two fractions in the numerator: \[ -\frac{17}{4} + (-\frac{9}{2}) \] First, we need a common denominator. The least common multiple of 4 and 2 is 4. Convert \(-\frac{9}{2}\) to have a denominator of 4: \[ -\frac{9}{2} = -\frac{9 \cdot 2}{2 \cdot 2} = -\frac{18}{4} \] Now the addition becomes: \[ -\frac{17}{4} - \frac{18}{4} = -\frac{17 + 18}{4} = -\frac{35}{4} \]
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Now divide by \(2 \frac{1}{2}\) (which is \(\frac{5}{2}\)): \[ \left(-\frac{35}{4}\right) / \left(\frac{5}{2}\right) \] To divide by a fraction, we multiply by its reciprocal: \[ -\frac{35}{4} \cdot \frac{2}{5} \]
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Perform the multiplication: \[ -\frac{35 \cdot 2}{4 \cdot 5} = -\frac{70}{20} \]
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Simplify the fraction: \[ -\frac{70 \div 10}{20 \div 10} = -\frac{7}{2} \]
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Convert to a mixed number: \(-\frac{7}{2} = -3 \frac{1}{2}\)
So, the final answer is: \[ \boxed{-3 \frac{1}{2}} \]