Question

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

Answer 1

To begin, let's calculate the square roots of 7 and 8:

√7 ≈ 2.646...
√8 ≈ 2.828...

Now, we need to find a rational number with a denominator of 7 that lies between these two values. We know that √8 is greater than √7, so we need to find a rational number that is closer to √7 than √8.

To simplify the calculation, let's consider the decimal representation.

2.646... < x < 2.828...

To find a rational number between these two values, we could use the average:

(2.646... + 2.828...) / 2 = 2.738...

Now, we need to write this as an improper fraction with a denominator of 7.

To do this, multiply both the numerator and denominator by 7 to maintain the same value:

2.738... * 7 = 19.166...

Now, we can write this as the improper fraction:

19.166... / 7

To find the numerator, subtract the whole number part from the decimal:

19.166... - 19 = 0.166...

Now, place the numerator over the denominator to get:

0.166... / 7

Since the decimal repeats indefinitely, we can write it as a fraction:

0.166... = 1/6

Therefore, the rational number with a denominator of 7 that lies between √7 and √8 is 1/6.

So the answer is: 1/6.