To determine which of the given options is a rational number between \(5\) and \(5 \frac{1}{4}\), we first need to convert \(5\) and \(5 \frac{1}{4}\) to improper fractions for easier comparison.
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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 want to find numbers that fall between \(\frac{20}{4}\) and \(\frac{21}{4}\).
Next, we evaluate each of the options provided:
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\(5 \frac{2}{5}\): \[ 5 \frac{2}{5} = \frac{5 \times 5 + 2}{5} = \frac{25 + 2}{5} = \frac{27}{5} \] To compare \(\frac{27}{5}\) with \(\frac{20}{4}\) and \(\frac{21}{4}\), we find a common denominator (which is \(20\)): \[ \frac{27}{5} = \frac{27 \times 4}{5 \times 4} = \frac{108}{20} \] \[ \frac{20}{4} = \frac{100}{20}, \quad \frac{21}{4} = \frac{105}{20} \] Since \(\frac{100}{20} < \frac{108}{20} > \frac{105}{20}\), \(5 \frac{2}{5}\) is not between \(5\) and \(5 \frac{1}{4}\).
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\(5 \frac{1}{3}\): \[ 5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3} \] Converting to a common denominator: \[ \frac{16}{3} = \frac{16 \times 4}{3 \times 4} = \frac{64}{12} \quad \text{(wrong choice of common denominator here; instead use } 12\text{)} \] Should actually calculate with common denominator of \(12\): \[ \frac{20}{4} = \frac{60}{12}, \quad \frac{21}{4} = \frac{63}{12} \] \(\frac{64}{12}\) is again greater than \(5 \frac{1}{4}\), hence out.
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\(\sqrt{26}\): The square root of 25 is 5 and the square root of 36 is 6. Therefore, \(\sqrt{26}\) is somewhere between these values, but we need a specific range to check: \[ \sqrt{25} < \sqrt{26} < \sqrt{36} \Rightarrow 5 < \sqrt{26} < 6 \]
Approximation indicates \(5.1 < \sqrt{26} < 5.2\). Therefore, we approximate:
Check by squaring back: \[ \(5.1^2 = 26.01 > 26 \quad & \quad 5.0^2 = 25 < 26\)
\] It is between \(5\) and \(5 \frac{1}{4} = 5.25\). -
\(5 \frac{1}{7}\): \[ 5 \frac{1}{7} = \frac{5 \times 7 + 1}{7} = \frac{35 + 1}{7} = \frac{36}{7} \] Again converting: \[ \frac{36}{7} = \frac{36 \times 4}{7 \times 4} = \frac{144}{28} > \(\frac{20}{4} = 20\) > 5.25 \]
Thus, the only rational number in the interval we found was \(\sqrt{26}\) falling in between \(5\) and \(5\frac{1}{4}\). Therefore, the final answer is:
\[ \sqrt{26} \]