Yes, \( 5 \frac{1}{7} \) is a rational number between \( 5 \) and \( 5 \frac{1}{4} \).
To clarify:
- \( 5 \) can be represented as \( 5 \frac{0}{1} \).
- \( 5 \frac{1}{4} \) can be converted to an improper fraction:
\[ 5 \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4}. \]
Now, we convert \( 5 \frac{1}{7} \) to an improper fraction:
\[ 5 \frac{1}{7} = \frac{35}{7} + \frac{1}{7} = \frac{36}{7}. \]
Next, we compare the fractions \( \frac{36}{7} \) and \( \frac{21}{4} \) by finding a common denominator. The least common multiple (LCM) of 7 and 4 is 28:
- Convert \( \frac{36}{7} \) to a fraction with a denominator of 28:
\[ \frac{36}{7} = \frac{36 \times 4}{7 \times 4} = \frac{144}{28}. \]
- Convert \( \frac{21}{4} \) to a fraction with a denominator of 28:
\[ \frac{21}{4} = \frac{21 \times 7}{4 \times 7} = \frac{147}{28}. \]
Now, we see that:
- \( 5 = \frac{5 \times 28}{28} = \frac{140}{28} \)
- \( 5 \frac{1}{4} = \frac{21}{4} = \frac{147}{28} \)
- \( 5 \frac{1}{7} = \frac{36}{7} = \frac{144}{28} \)
Comparing these, we have:
\[ \frac{140}{28} < \frac{144}{28} < \frac{147}{28}, \]
which confirms that
\[ 5 < 5 \frac{1}{7} < 5 \frac{1}{4}. \]
Thus, \( 5 \frac{1}{7} \) is indeed a rational number between \( 5 \) and \( 5 \frac{1}{4} \).