Question

What is a rational number with a denominator of 7 that is between √10 and √11 .
Write your answer as an improper fraction.
HALPPPP

Answer 1

To find a rational number with a denominator of 7 that is between \( \sqrt{10} \) and \( \sqrt{11} \), we first need to approximate the square roots.

Calculating the square roots:
- \( \sqrt{10} \) is approximately \( 3.162 \).
- \( \sqrt{11} \) is approximately \( 3.317 \).

Now we need a rational number with a denominator of 7 that falls within this range. A rational number can be expressed as \( \frac{n}{7} \), where \( n \) is an integer.

To find suitable \( n \):
1. We multiply the approximate values by 7 to find the range for \( n \):
- \( 3.162 \times 7 \approx 22.134 \)
- \( 3.317 \times 7 \approx 23.219 \)

Thus, \( n \) must satisfy:
\[ 22.134 < n < 23.219 \]

The integer values for \( n \) that satisfy this inequality are \( 23 \).

Now, let's express this rational number:
\[
\frac{23}{7}
\]

Checking:
- Convert \( \frac{23}{7} \) back to decimal form:
\[
\frac{23}{7} \approx 3.2857
\]

Now confirm that this value is indeed between \( \sqrt{10} \) and \( \sqrt{11} \):
- \( 3.162 < 3.2857 < 3.317 \) is true.

Therefore, the rational number with a denominator of 7 that is between \( \sqrt{10} \) and \( \sqrt{11} \) is:

\[
\frac{23}{7}
\]