Which of the following is a rational number between 5 and 514

Question

?(1 point)
Responses

525
5 Start Fraction 2 over 5 end fraction

26−−√
start root 26 end root

513
5 Start Fraction 1 over 3 end fraction

517

Answer 1

To determine which of the numbers listed is a rational number between 5 and 514, we need to analyze each option.

1. 525: This is greater than 514.
2. \(5 \frac{2}{5}\): This is equal to \(5 + \frac{2}{5} = 5.4\), which is between 5 and 514.
3. \(\sqrt{26}\): This is an irrational number and is approximately 5.1. It is also between 5 and 514, but it is not a rational number.
4. 513: This is between 5 and 514.
5. 517: This is greater than 514.

The rational numbers between 5 and 514 from the options provided are \(5 \frac{2}{5}\) and 513. However, since the question asks for a rational number between 5 and 514 and considering it asks for one response, the best choice is:

\(5 \frac{2}{5}\) (which is approximately 5.4).

Answer 2

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

Answer 3

To find a rational number with a denominator of 7 that is between \(\sqrt{7}\) and \(\sqrt{8}\), we first need to approximate the values of these square roots.

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

   • \(\sqrt{7} \approx 2.6457513110645906\)

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

   • \(\sqrt{8} = 2\sqrt{2} \approx 2.8284271247461903\)

Now, we're looking for a fraction with a denominator of 7 that lies between these two values. A rational number with a denominator of 7 can be expressed as:

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

We need to find an integer \(n\) such that:

\[ 2.6457513110645906 < \frac{n}{7} < 2.8284271247461903 \]

To solve for \(n\), we multiply the inequalities by 7:

\[ 7 \times 2.6457513110645906 < n < 7 \times 2.8284271247461903 \]

Calculating the left side:

\[ 7 \times 2.6457513110645906 \approx 18.520258177452133 \quad (\text{rounded to } 18.52) \]

Calculating the right side:

\[ 7 \times 2.8284271247461903 \approx 19.798989873222332 \quad (\text{rounded to } 19.80) \]

Now we need \(n\) to be an integer that satisfies:

\[ 18.520258 < n < 19.798989 \]

The only integer \(n\) that fits this condition is \(19\).

Thus, the rational number with a denominator of 7 that is between \(\sqrt{7}\) and \(\sqrt{8}\) is:

\[ \frac{19}{7} \]