Understanding the Question: Finding a Rational Number

The question asks us to find a rational number that lies between the two irrational numbers, \(\sqrt{5}\) and \(\sqrt{7}\). We are given four options, all expressed as mixed numbers. To solve this, we need to:

• Understand what rational and irrational numbers are.
• Estimate the values of \(\sqrt{5}\) and \(\sqrt{7}\).
• Convert the given options into a comparable format (like decimals or improper fractions).
• Determine which option falls within the estimated range.

What are Rational and Irrational Numbers?

• Rational Numbers: Numbers that can be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). Examples: \(\frac{1}{2}\), \(3\) (which is \(\frac{3}{1}\)), \(-0.75\) (which is \(\frac{-3}{4}\)), \(0.333...\) (which is \(\frac{1}{3}\)). Terminating or repeating decimals are rational.
• Irrational Numbers: Numbers that cannot be expressed in the form \(\frac{p}{q}\). Their decimal representations are non-terminating and non-repeating. Examples: \(\sqrt{2}\), \(\pi\), \(\sqrt{5}\), \(\sqrt{7}\). Square roots of non-perfect squares are irrational.

Estimating the Values of \(\sqrt{5}\) and \(\sqrt{7}\)

To find a number between \(\sqrt{5}\) and \(\sqrt{7}\), we first need an idea of their decimal values.

• Consider perfect squares near 5: \(2^2 = 4\) and \(3^2 = 9\). Since \(4 < 5 < 9\), we know that \(\sqrt{4} < \sqrt{5} < \sqrt{9}\), which means \(2 < \sqrt{5} < 3\). \(\sqrt{5}\) is slightly greater than 2. A common approximation is \(\sqrt{5} \approx 2.236\).
• Consider perfect squares near 7: \(2^2 = 4\) and \(3^2 = 9\). Since \(4 < 7 < 9\), we know that \(\sqrt{4} < \sqrt{7} < \sqrt{9}\), which means \(2 < \sqrt{7} < 3\). \(\sqrt{7}\) is between 2 and 3, and is closer to 3 than to 2 (since 7 is closer to 9 than to 4). A common approximation is \(\sqrt{7} \approx 2.646\).

So, we are looking for a rational number between approximately 2.236 and 2.646.

Analyzing the Given Options

Let's convert the mixed number options into decimal form to easily compare them with our estimated range.

• Option 1: \(4\frac{1}{5}\)
  • Conversion: \(4\frac{1}{5} = 4 + \frac{1}{5} = 4 + 0.2 = 4.2\)
  • Comparison: \(4.2 > 2.646\). This number is greater than \(\sqrt{7}\).
• Option 2: \(3\frac{1}{5}\)
  • Conversion: \(3\frac{1}{5} = 3 + \frac{1}{5} = 3 + 0.2 = 3.2\)
  • Comparison: \(3.2 > 2.646\). This number is greater than \(\sqrt{7}\).
• Option 3: \(2\frac{2}{5}\)
  • Conversion: \(2\frac{2}{5} = 2 + \frac{2}{5} = 2 + 0.4 = 2.4\)
  • Comparison: \(2.236 < 2.4 < 2.646\). This number is between \(\sqrt{5}\) and \(\sqrt{7}\).
• Option 4: \(1\frac{1}{5}\)
  • Conversion: \(1\frac{1}{5} = 1 + \frac{1}{5} = 1 + 0.2 = 1.2\)
  • Comparison: \(1.2 < 2.236\). This number is less than \(\sqrt{5}\).

Identifying the Rational Number Between \(\sqrt{5}\) and \(\sqrt{7}\)

Based on our analysis, the number \(2.4\) (which is \(2\frac{2}{5}\)) falls between the estimated values of \(\sqrt{5} \approx 2.236\) and \(\sqrt{7} \approx 2.646\).

Let's verify that \(2\frac{2}{5}\) is indeed a rational number. We can write \(2\frac{2}{5}\) as an improper fraction:

\(2\frac{2}{5} = \frac{(2 \times 5) + 2}{5} = \frac{10 + 2}{5} = \frac{12}{5}\)

Since 12 and 5 are integers and 5 is not zero, \(\frac{12}{5}\) is in the form \(\frac{p}{q}\), making it a rational number.

Conclusion

The rational number between \(\sqrt{5}\) and \(\sqrt{7}\) from the given options is \(2\frac{2}{5}\).

Revision Table: Key Concepts Reviewed

Additional Information: Locating Numbers on the Number Line

Visualizing numbers on a number line helps understand their relative positions. Both rational and irrational numbers can be plotted on the real number line. For example:

• \(1.2\) would be to the right of 1.
• \(\sqrt{5} \approx 2.236\) would be just after 2.2.
• \(2.4\) would be between 2.2 and 2.6.
• \(\sqrt{7} \approx 2.646\) would be just after 2.6.
• \(3.2\) would be just after 3.2.
• \(4.2\) would be just after 4.2.

This confirms that 2.4 is indeed located between \(\sqrt{5}\) and \(\sqrt{7}\).