What is a rational number with a denominator of 7 that is between 7–√

Question

7
and 8–√
8
. Write your answer as an improper fraction.

Answer 1

To find a rational number with a denominator of 7 that is between \(7 - \sqrt{7}\) and \(8 - \sqrt{8}\), we first need to evaluate the numerical values of these two expressions.

1. Calculate \(7 - \sqrt{7}\):

   \[ \sqrt{7} \approx 2.6457513110645906 \] Therefore, \[ 7 - \sqrt{7} \approx 7 - 2.6457513110645906 \approx 4.3542486889354094 \]

2. Calculate \(8 - \sqrt{8}\):

   \[ \sqrt{8} \approx 2.8284271247461903 \] Therefore, \[ 8 - \sqrt{8} \approx 8 - 2.8284271247461903 \approx 5.1715728752538097 \]

Now, we want to find a rational number with a denominator of 7 that lies within the interval \( (7 - \sqrt{7}, 8 - \sqrt{8}) \), which we have approximated as \( (4.354, 5.172) \).

The rational numbers in the form of improper fractions with a denominator of 7 can be expressed as:

\[ \frac{n}{7} \]

for integers \(n\).

To find \(n\), we set up the inequality:

\[ 4.354 < \frac{n}{7} < 5.172 \]

Multiplying through by 7 gives:

\[ 4.354 \cdot 7 < n < 5.172 \cdot 7 \]

Calculating the bounds:

\[ 4.354 \cdot 7 \approx 30.478 \quad \text{and} \quad 5.172 \cdot 7 \approx 36.204 \]

Thus, we have the inequality:

\[ 30.478 < n < 36.204 \]

The possible integer values for \(n\) are \(31, 32, 33, 34, 35, 36\).

In terms of rational numbers, we can take any of these values:

• For \(n = 31\): \(\frac{31}{7}\)
• For \(n = 32\): \(\frac{32}{7}\)
• For \(n = 33\): \(\frac{33}{7}\)
• For \(n = 34\): \(\frac{34}{7}\)
• For \(n = 35\): \(\frac{35}{7}\)
• For \(n = 36\): \(\frac{36}{7}\)

To ensure that we get a number specifically within our bounds:

Calculating these fractions:

• \(\frac{31}{7} \approx 4.428571\) (valid)
• \(\frac{32}{7} \approx 4.571429\) (valid)
• \(\frac{33}{7} \approx 4.714286\) (valid)
• \(\frac{34}{7} \approx 4.857143\) (valid)
• \(\frac{35}{7} \approx 5.0\) (valid)
• \(\frac{36}{7} \approx 5.142857\) (valid)

So, one of the rational numbers we could use is:

\[ \frac{33}{7} \]

This is a rational number with a denominator of 7 that is indeed between \(7 - \sqrt{7}\) and \(8 - \sqrt{8}\).

Thus, the final answer is:

\[ \frac{33}{7} \]