To find a rational number between 5 and \(5 \frac{1}{4}\), we first need to express these two numbers in improper fractions.
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Convert \(5\) to an improper fraction: \[ 5 = \frac{5 \times 4}{4} = \frac{20}{4} \]
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Convert \(5 \frac{1}{4}\) to an improper fraction: \[ 5 \frac{1}{4} = \frac{5 \times 4 + 1}{4} = \frac{20 + 1}{4} = \frac{21}{4} \]
Now we have the range between \( \frac{20}{4} \) and \( \frac{21}{4} \).
Next, we evaluate each option:
A. Square root of 26: \[ \sqrt{26} \approx 5.099 \quad \text{(approximately, which is between 5 and 5 1/4)} \]
B. \(5 \frac{2}{5}\): \[ 5 \frac{2}{5} = \frac{5 \times 5 + 2}{5} = \frac{25 + 2}{5} = \frac{27}{5} = 5.4 \quad \text{(this is greater than 5 1/4)} \]
C. \(5 \frac{1}{3}\): \[ 5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3} \approx 5.333 \quad \text{(this is greater than 5 1/4)} \]
D. \(5 \frac{1}{7}\): \[ 5 \frac{1}{7} = \frac{5 \times 7 + 1}{7} = \frac{35 + 1}{7} = \frac{36}{7} \approx 5.143 \quad \text{(this is between 5 and 5 1/4)} \]
So the options that result in rational numbers between 5 and \(5 \frac{1}{4}\) are:
- \( \sqrt{26} \) (approximately \(5.099\))
- \(5 \frac{1}{7} \) (approximately \(5.143\))
Among the provided options, both \(A\) and \(D\) are valid rational numbers between 5 and \(5 \frac{1}{4}\). If you are required to select only one, then the lowest rational number in the range is \(D\).
Therefore, the answer is: D. \(5 \frac{1}{7}\)