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:

$$4.354\cdot 7\approx 30.478\text{and}5.172\cdot 7\approx 36.204$$

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:

$$\frac{33}{7}$$