To find a rational number between 5 and 5 1/4, we need to convert 5 1/4 into an improper fraction.
5 1/4 can be expressed as:
\[ 5 + \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4} \]
Now, we have the range from 5 (which is \(\frac{20}{4}\)) to \(\frac{21}{4}\).
Now, let's evaluate the options:
A. \( 5 \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3}\)
To compare \(\frac{16}{3}\) and \(\frac{20}{4}\): \[ \frac{16}{3} \approx 5.33 \quad (\text{greater than 5}) \]
To compare \(\frac{16}{3}\) with \(\frac{21}{4}\): \[ \frac{16}{3} \approx 5.33 \quad (\text{less than } 5.25 \text{ or } \frac{21}{4}) \]
So, \(5 \frac{1}{3}\) is between 5 and 5 1/4.
B. \( 5 \frac{1}{7} = \frac{35}{7} + \frac{1}{7} = \frac{36}{7}\)
To compare \(\frac{36}{7}\): \[ \frac{36}{7} \approx 5.14 \quad (\text{greater than 5}) \]
And comparing with \(\frac{21}{4}\): \[ \frac{36}{7} \approx 5.14 \quad (\text{less than } 5.25 \text{ or } \frac{21}{4}) \]
So, \(5 \frac{1}{7}\) is also between 5 and 5 1/4.
C. \(\sqrt{26} \approx 5.1 \quad (\text{greater than 5} \text{ but less than } 5.25)\)
D. \( 5 \frac{2}{5} = \frac{25}{5} + \frac{2}{5} = \frac{27}{5}\)
To compare \(\frac{27}{5} = 5.4\), which is greater than \(5 \frac{1}{4}\).
So, the rational numbers between 5 and 5 1/4 among the options are A, B, and C. However, the question likely expects a single response as an answer.
The simplest rational number between 5 and 5 1/4 among the provided options is:
B. 5 1/7.