Question

Which of the following is a rational number between 5 and 5 1/4?
5 2/5

5 1/3

√26

5 1/7

Answer 1

To determine which of the given options is a rational number between \(5\) and \(5 \frac{1}{4}\), we first need to convert \(5\) and \(5 \frac{1}{4}\) to improper fractions for easier comparison.

1. **Convert \(5\) to an improper fraction:**
\[
5 = \frac{5 \times 4}{4} = \frac{20}{4}
\]

2. **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 want to find numbers that fall between \(\frac{20}{4}\) and \(\frac{21}{4}\).

Next, we evaluate each of the options provided:

1. **\(5 \frac{2}{5}\):**
\[
5 \frac{2}{5} = \frac{5 \times 5 + 2}{5} = \frac{25 + 2}{5} = \frac{27}{5}
\]
To compare \(\frac{27}{5}\) with \(\frac{20}{4}\) and \(\frac{21}{4}\), we find a common denominator (which is \(20\)):
\[
\frac{27}{5} = \frac{27 \times 4}{5 \times 4} = \frac{108}{20}
\]
\[
\frac{20}{4} = \frac{100}{20}, \quad \frac{21}{4} = \frac{105}{20}
\]
Since \(\frac{100}{20} < \frac{108}{20} > \frac{105}{20}\), \(5 \frac{2}{5}\) is not between \(5\) and \(5 \frac{1}{4}\).

2. **\(5 \frac{1}{3}\):**
\[
5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3}
\]
Converting to a common denominator:
\[
\frac{16}{3} = \frac{16 \times 4}{3 \times 4} = \frac{64}{12} \quad \text{(wrong choice of common denominator here; instead use } 12\text{)}
\]
Should actually calculate with common denominator of \(12\):
\[
\frac{20}{4} = \frac{60}{12}, \quad \frac{21}{4} = \frac{63}{12}
\]
\(\frac{64}{12}\) is again greater than \(5 \frac{1}{4}\), hence out.

3. **\(\sqrt{26}\):**
The square root of 25 is 5 and the square root of 36 is 6. Therefore, \(\sqrt{26}\) is somewhere between these values, but we need a specific range to check:
\[
\sqrt{25} < \sqrt{26} < \sqrt{36} \Rightarrow 5 < \sqrt{26} < 6
\]

Approximation indicates \(5.1 < \sqrt{26} < 5.2\). Therefore, we approximate:

Check by squaring back:
\[
\(5.1^2 = 26.01 > 26 \quad \& \quad 5.0^2 = 25 < 26\)
\]
It is between \(5\) and \(5 \frac{1}{4} = 5.25\).

4. **\(5 \frac{1}{7}\):**
\[
5 \frac{1}{7} = \frac{5 \times 7 + 1}{7} = \frac{35 + 1}{7} = \frac{36}{7}
\]
Again converting:
\[
\frac{36}{7} = \frac{36 \times 4}{7 \times 4} = \frac{144}{28} > \(\frac{20}{4} = 20\) > 5.25
\]

Thus, the only rational number in the interval we found was **\(\sqrt{26}\)** falling in between \(5\) and \(5\frac{1}{4}\). Therefore, the final answer is:

\[
\sqrt{26}
\]