To add \(9\) wholes and \(5\) sevenths with \(1\) whole and \(1\) fourth, we can break the problem into two parts: the whole numbers and the fractional parts.
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Whole Numbers: \[ 9 + 1 = 10 \]
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Fractional Parts: We need to add \( \frac{5}{7} \) and \( \frac{1}{4} \). To do that, we have to find a common denominator. The least common multiple of \(7\) and \(4\) is \(28\).
Converting each fraction: \[ \frac{5}{7} = \frac{5 \times 4}{7 \times 4} = \frac{20}{28} \] \[ \frac{1}{4} = \frac{1 \times 7}{4 \times 7} = \frac{7}{28} \]
Now we can add the converted fractions: \[ \frac{20}{28} + \frac{7}{28} = \frac{27}{28} \]
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Combine Whole Numbers and Fractions: Now, we combine the results: \[ 10 + \frac{27}{28} = 10 \frac{27}{28} \]
Thus, the result of \(9\) wholes and \(5\) sevenths plus \(1\) whole and \(1\) fourth is: \[ \boxed{10 \frac{27}{28}} \]