Q2:

Find six rational numbers between 3 and 4.

Initial Setup & Theoretical Foundation

We are required to find exactly six rational numbers strictly between the integers $3$ and $4$.

[Per the Density Property of Rational Numbers], between any two distinct rational numbers, there exists an infinite number of rational numbers. An integer can always be expressed as a rational number $\frac{p}{q}$ where $q = 1$. Therefore, we can write the given numbers as:

• $a = \frac{3}{1}$
• $b = \frac{4}{1}$

Step 1: Applying the Common Denominator Algorithm

To find $n$ evenly spaced rational numbers between two integers $a$ and $b$, a highly reliable algebraic method is to convert both integers into equivalent fractions sharing a common denominator of $n + 1$.

Given that we need $n = 6$ rational numbers, we calculate our target multiplier:

$n + 1 = 6 + 1 = 7$

We now multiply both the numerator and the denominator of our initial rational expressions by $7$ to maintain equivalence [By the Multiplicative Identity Property, multiplying by $\frac{7}{7}$ is equivalent to multiplying by $1$]:

$3 = \frac{3 \times 7}{1 \times 7} = \frac{21}{7}$

$4 = \frac{4 \times 7}{1 \times 7} = \frac{28}{7}$

Step 2: Identifying the Intermediate Rational Numbers

We have successfully established the boundary values as $\frac{21}{7}$ and $\frac{28}{7}$. Because the denominators are identical, any fraction with a numerator strictly between $21$ and $28$ (over the denominator $7$) will be a rational number lying strictly between $3$ and $4$.

Incrementing the numerator by $1$ sequentially yields exactly six rational numbers:

$\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \frac{27}{7}$

Visual Representation: The Rational Number Line

The following scale demonstrates the exact geometric placement of these rational numbers, dividing the segment between $3$ and $4$ into $7$ equal sub-intervals.

Alternative Method (Decimal Expansion)

While the fractional method is the standard algebraic approach, rational numbers can also be expressed as terminating decimals. Any terminating decimal between $3.0$ and $4.0$ is a valid rational number. For example:

$3.1 = \frac{31}{10}, \quad 3.2 = \frac{32}{10} = \frac{16}{5}, \quad 3.3 = \frac{33}{10}, \quad 3.4 = \frac{34}{10} = \frac{17}{5}, \quad 3.5 = \frac{35}{10} = \frac{7}{2}, \quad 3.6 = \frac{36}{10} = \frac{18}{5}$

Both sets of numbers are mathematically correct due to the infinite density of rational numbers.

Final Solution: Six rational numbers between $3$ and $4$ are $\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \text{ and } \frac{27}{7}$.