Question

What is a rational number with a denominator of 7 that is between 7–√
and 8–√
. Write your answer as an improper fraction

Answer 1

so whats the answer

Answer 2

so 7/33

Answer 3

To find a rational number with a denominator of 7 that lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \), we first need to evaluate the decimal approximations of \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \).

1. **Calculate \( 7 - \sqrt{7} \)**:
- The approximate value of \( \sqrt{7} \) is about \( 2.6457513110645906 \).
- So, \( 7 - \sqrt{7} \approx 7 - 2.6457513110645906 \approx 4.3542486889354094 \).

2. **Calculate \( 8 - \sqrt{8} \)**:
- The approximate value of \( \sqrt{8} \) is about \( 2.8284271247461903 \).
- So, \( 8 - \sqrt{8} \approx 8 - 2.8284271247461903 \approx 5.1715728752538097 \).

Now we need a rational number in the form of an improper fraction with a denominator of 7 that lies between approximately \( 4.3542486889354094 \) and \( 5.1715728752538097 \).

To make this simpler, we can express the numbers between 4 and 5 in terms of fractions with a denominator of 7. This means we are looking for fractions like:

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

Where \( n \) is an integer.

The integers that correspond to rational numbers between these bounds multiplied by 7:

- \( 4.354 \times 7 \approx 30.478 \)
- \( 5.171 \times 7 \approx 36.197 \)

We can round these bounds to their nearest integers:
- Lower bound: \( 30 \)
- Upper bound: \( 36 \)

The integers between 30 and 36 are \( 31, 32, 33, 34, 35, 36 \). Now we convert these into improper fractions with a denominator of 7:

- \( \frac{31}{7} \approx 4.42857 \) (valid)
- \( \frac{32}{7} \approx 4.57143 \) (valid)
- \( \frac{33}{7} \approx 4.71429 \) (valid)
- \( \frac{34}{7} \approx 4.85714 \) (valid)
- \( \frac{35}{7} \approx 5.00000 \) (valid)
- \( \frac{36}{7} \approx 5.14286 \) (valid)

Therefore, any of these fractions—\( \frac{31}{7}, \frac{32}{7}, \frac{33}{7}, \frac{34}{7}, \frac{35}{7}, \frac{36}{7} \)—are valid answers.

For instance, if you take **\( \frac{33}{7} \)**:

\[
\frac{33}{7} \approx 4.71429
\]

This rational number is between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \).

Thus, one possible answer is:

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

Answer 4

The answer is:

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

This fraction has a denominator of 7 and lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \).

Answer 5

It seems like there may be confusion regarding the fraction format. The rational number between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \) with a denominator of 7 is indeed:

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

This means that for a valid rational number expressed with a denominator of 7, the numerator is 33.

So, it is not \( \frac{7}{33} \); rather, the correct answer is \( \frac{33}{7} \).