To determine if \( 5 \frac{1}{7} \) is a rational number between \( 5 \) and \( 5 \frac{1}{4} \), we need to compare \( 5 \frac{1}{7} \) and \( 5 \frac{1}{4} \).
First, we convert \( 5 \frac{1}{7} \) and \( 5 \frac{1}{4} \) to improper fractions:
- \( 5 \frac{1}{7} = \frac{5 \times 7 + 1}{7} = \frac{35 + 1}{7} = \frac{36}{7} \)
- \( 5 \frac{1}{4} = \frac{5 \times 4 + 1}{4} = \frac{20 + 1}{4} = \frac{21}{4} \)
Next, we need to convert \( 5 \) to an improper fraction as well:
- \( 5 = \frac{5}{1} = \frac{20}{4} \) (for easier comparison)
Now we compare these fractions:
- \( 5 = \frac{20}{4} \)
- \( 5 \frac{1}{4} = \frac{21}{4} \)
- \( 5 \frac{1}{7} = \frac{36}{7} \)
To compare \( \frac{36}{7} \) and \( \frac{21}{4} \), we can find a common denominator, which is \( 28 \):
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Convert \( \frac{36}{7} \) to a denominator of \( 28 \): \[ \frac{36}{7} = \frac{36 \times 4}{7 \times 4} = \frac{144}{28} \]
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Convert \( \frac{21}{4} \) to a denominator of \( 28 \): \[ \frac{21}{4} = \frac{21 \times 7}{4 \times 7} = \frac{147}{28} \]
Now we can see that:
- \( 5 = \frac{20}{4} = \frac{140}{28} \)
- \( 5 \frac{1}{4} = \frac{21}{4} = \frac{147}{28} \)
- \( 5 \frac{1}{7} = \frac{36}{7} = \frac{144}{28} \)
Now we can summarize:
- \( \frac{140}{28} < \frac{144}{28} < \frac{147}{28} \)
Thus, since \( 5 \frac{1}{7} = \frac{36}{7} \) (or \( \frac{144}{28} \)) is greater than \( 5 \) (or \( \frac{140}{28} \)) and less than \( 5 \frac{1}{4} \) (or \( \frac{147}{28} \)), the statement is true.
Therefore, \( 5 \frac{1}{7} \) is a rational number between \( 5 \) and \( 5 \frac{1}{4} \).