To determine if \(5\frac{1}{7}\) (which is a mixed number) is a rational number between \(5\frac{1}{4}\) (also a mixed number), we can first convert both mixed numbers into improper fractions.
- Convert \(5\frac{1}{7}\) to an improper fraction:
\[ 5\frac{1}{7} = 5 \times 7 + 1 = 35 + 1 = \frac{36}{7} \]
- Convert \(5\frac{1}{4}\) to an improper fraction:
\[ 5\frac{1}{4} = 5 \times 4 + 1 = 20 + 1 = \frac{21}{4} \]
Next, we need to compare \(\frac{36}{7}\) and \(\frac{21}{4}\). To do this, we can find a common denominator. The least common multiple of 7 and 4 is 28.
- Convert \(\frac{36}{7}\) to have the denominator of 28:
\[ \frac{36}{7} = \frac{36 \times 4}{7 \times 4} = \frac{144}{28} \]
- Convert \(\frac{21}{4}\) to have the denominator of 28:
\[ \frac{21}{4} = \frac{21 \times 7}{4 \times 7} = \frac{147}{28} \]
Now, we can see that:
- \(\frac{36}{7} = \frac{144}{28}\)
- \(\frac{21}{4} = \frac{147}{28}\)
- Determine if \(\frac{36}{7}\) is between \(5\frac{1}{4}\) (or \(\frac{21}{4}\)):
Since \(144 < 147\), it follows that:
\[ \frac{36}{7} < \frac{21}{4} \]
Now, we also need to note the value we obtained in fractions:
- \(5\frac{1}{4} = 5.25\) (as a decimal)
- \(5\frac{1}{7} \approx 5.142857\) (as a decimal)
- Therefore, as decimals:
- \(5\frac{1}{7} \approx 5.142857\)
- \(5\frac{1}{4} = 5.25\)
Thus, \(5\frac{1}{7}\) is less than \(5\frac{1}{4}\) but is still greater than 5. Therefore, 5 and one-seventh (\(5\frac{1}{7}\)) is indeed a rational number between 5 and 5 and one fourth (\(5\frac{1}{4}\)).
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