To determine if \( 5 \frac{1}{7} \) is a rational number between \( 5 \) and \( 5 \frac{1}{4} \), we can first convert everything to improper fractions to make comparisons easier.
First, convert \( 5 \frac{1}{7} \) to an improper fraction: \[ 5 \frac{1}{7} = \frac{5 \times 7 + 1}{7} = \frac{35 + 1}{7} = \frac{36}{7} \]
Next, 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:
- \( 5 = \frac{5}{1} \)
- \( 5 \frac{1}{4} = \frac{21}{4} \)
- \( 5 \frac{1}{7} = \frac{36}{7} \)
Now let's convert \( \frac{5}{1} \) to a fraction with a denominator of 28 (which is a common denominator of 1 and 4) to easily compare:
\[ \frac{5}{1} = \frac{5 \times 28}{1 \times 28} = \frac{140}{28} \]
Converting \( \frac{21}{4} \) to a denominator of 28: \[ \frac{21}{4} = \frac{21 \times 7}{4 \times 7} = \frac{147}{28} \]
Now convert \( \frac{36}{7} \) into a fraction with a denominator of 28: \[ \frac{36}{7} = \frac{36 \times 4}{7 \times 4} = \frac{144}{28} \]
Now we have:
- \( 5 = \frac{140}{28} \)
- \( 5 \frac{1}{7} = \frac{144}{28} \)
- \( 5 \frac{1}{4} = \frac{147}{28} \)
We can see that: \[ \frac{140}{28} < \frac{144}{28} < \frac{147}{28} \]
This confirms that \( 5 \frac{1}{7} \) (or \( \frac{36}{7} \)) is indeed a rational number between \( 5 \) and \( 5 \frac{1}{4} \).