Question
What is a rational number between 5 and 5 1/4? Coices are square root of 26, 5 1/3, 5 2/5, or 5 1/7?
Answers
GPT-4o mini
To find a rational number between 5 and \(5 \frac{1}{4}\), we first convert \(5 \frac{1}{4}\) to an improper fraction:
\[
5 \frac{1}{4} = \frac{5 \times 4 + 1}{4} = \frac{20 + 1}{4} = \frac{21}{4}
\]
So we are looking for a rational number \(x\) such that:
\[
5 < x < \frac{21}{4}
\]
Next, we convert 5 into a fraction:
\[
5 = \frac{20}{4}
\]
Now we can determine the range:
\[
\frac{20}{4} < x < \frac{21}{4}
\]
To evaluate the choices given:
1. **Square root of 26**:
\[
\sqrt{26} \approx 5.099
\]
This value is between 5 and \(5 \frac{1}{4} = 5.25\).
2. **\(5 \frac{1}{3}\)**:
\[
5 \frac{1}{3} = \frac{16}{3} \approx 5.333
\]
This value is greater than \(5 \frac{1}{4} = 5.25\).
3. **\(5 \frac{2}{5}\)**:
\[
5 \frac{2}{5} = \frac{27}{5} = 5.4
\]
This value is also greater than \(5 \frac{1}{4} = 5.25\).
4. **\(5 \frac{1}{7}\)**:
\[
5 \frac{1}{7} = \frac{36}{7} \approx 5.143
\]
This value is between 5 and \(5 \frac{1}{4} = 5.25\).
Now, comparing the values we computed:
- \(\sqrt{26} \approx 5.099\) is between 5 and 5.25.
- \(5 \frac{1}{3} \approx 5.333\) is beyond \(5 \frac{1}{4}\).
- \(5 \frac{2}{5} = 5.4\) is also beyond \(5 \frac{1}{4}\).
- \(5 \frac{1}{7} \approx 5.143\) is between 5 and 5.25.
Therefore, the rational numbers that are between 5 and \(5 \frac{1}{4}\) are:
- \( \sqrt{26} \)
- \( 5 \frac{1}{7} \)
Among the options, both \(\sqrt{26}\) and \(5 \frac{1}{7}\) are correct answers. However, if we are to choose one option, the answer between 5 and \(5 \frac{1}{4}\) is:
**\(5 \frac{1}{7}\)**.
\[
5 \frac{1}{4} = \frac{5 \times 4 + 1}{4} = \frac{20 + 1}{4} = \frac{21}{4}
\]
So we are looking for a rational number \(x\) such that:
\[
5 < x < \frac{21}{4}
\]
Next, we convert 5 into a fraction:
\[
5 = \frac{20}{4}
\]
Now we can determine the range:
\[
\frac{20}{4} < x < \frac{21}{4}
\]
To evaluate the choices given:
1. **Square root of 26**:
\[
\sqrt{26} \approx 5.099
\]
This value is between 5 and \(5 \frac{1}{4} = 5.25\).
2. **\(5 \frac{1}{3}\)**:
\[
5 \frac{1}{3} = \frac{16}{3} \approx 5.333
\]
This value is greater than \(5 \frac{1}{4} = 5.25\).
3. **\(5 \frac{2}{5}\)**:
\[
5 \frac{2}{5} = \frac{27}{5} = 5.4
\]
This value is also greater than \(5 \frac{1}{4} = 5.25\).
4. **\(5 \frac{1}{7}\)**:
\[
5 \frac{1}{7} = \frac{36}{7} \approx 5.143
\]
This value is between 5 and \(5 \frac{1}{4} = 5.25\).
Now, comparing the values we computed:
- \(\sqrt{26} \approx 5.099\) is between 5 and 5.25.
- \(5 \frac{1}{3} \approx 5.333\) is beyond \(5 \frac{1}{4}\).
- \(5 \frac{2}{5} = 5.4\) is also beyond \(5 \frac{1}{4}\).
- \(5 \frac{1}{7} \approx 5.143\) is between 5 and 5.25.
Therefore, the rational numbers that are between 5 and \(5 \frac{1}{4}\) are:
- \( \sqrt{26} \)
- \( 5 \frac{1}{7} \)
Among the options, both \(\sqrt{26}\) and \(5 \frac{1}{7}\) are correct answers. However, if we are to choose one option, the answer between 5 and \(5 \frac{1}{4}\) is:
**\(5 \frac{1}{7}\)**.