Q3:

Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.

Given Variables & Initial Setup

We are tasked with finding exactly five rational numbers that lie strictly between the following two given rational numbers:

• Lower bound: $a = \frac{3}{5}$
• Upper bound: $b = \frac{4}{5}$
• Required number of rational numbers: $n = 5$

[Per the Density Property of Rational Numbers, between any two distinct rational numbers, there exist infinitely many rational numbers. To find a specific finite set of rational numbers spaced evenly between two fractions with identical denominators, we utilize the $(n + 1)$ scaling method.]

Step 1: Determining the Scaling Factor

To find $n$ rational numbers between two fractions that already share a common denominator, we must amplify the fractions by multiplying both the numerator and the denominator by a scaling factor of $(n + 1)$.

Given $n = 5$:

$\text{Scaling Factor} = n + 1 = 5 + 1 = 6$

Step 2: Creating Equivalent Fractions

We apply the scaling factor to both the lower and upper bounds to generate equivalent fractions with a larger common denominator. This expands the "gap" between the numerators, allowing us to identify integers between them.

For the lower bound $a$:

$a = \frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}$

For the upper bound $b$:

$b = \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30}$

[By the Fundamental Property of Fractions, multiplying the numerator and denominator by the same non-zero integer preserves the value of the rational number.]

Step 3: Identifying the Intermediate Rational Numbers

We now look for five rational numbers strictly between $\frac{18}{30}$ and $\frac{24}{30}$. Since the denominators are identical, we simply increment the numerator by $1$ starting from $18$ up to $23$:

$\frac{18}{30} < \frac{19}{30} < \frac{20}{30} < \frac{21}{30} < \frac{22}{30} < \frac{23}{30} < \frac{24}{30}$

Thus, the five intermediate rational numbers are:

$\frac{19}{30}, \frac{20}{30}, \frac{21}{30}, \frac{22}{30}, \frac{23}{30}$

Step 4: Simplification to Lowest Terms (Optional but Recommended)

To adhere to standard mathematical conventions, we reduce each intermediate fraction to its simplest form by dividing the numerator and denominator by their Greatest Common Divisor (GCD).

• $\frac{19}{30}$ (Already in simplest form, as 19 is prime)
• $\frac{20}{30} = \frac{20 \div 10}{30 \div 10} = \frac{2}{3}$
• $\frac{21}{30} = \frac{21 \div 3}{30 \div 3} = \frac{7}{10}$
• $\frac{22}{30} = \frac{22 \div 2}{30 \div 2} = \frac{11}{15}$
• $\frac{23}{30}$ (Already in simplest form, as 23 is prime)

Visual Representation: Number Line

The following high-precision SVG illustrates the exact spatial distribution of these rational numbers on a number line.

Final Solution: The five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$ are $\frac{19}{30}$, $\frac{20}{30}$, $\frac{21}{30}$, $\frac{22}{30}$, and $\frac{23}{30}$. (Expressed in simplest form: $\frac{19}{30}$, $\frac{2}{3}$, $\frac{7}{10}$, $\frac{11}{15}$, and $\frac{23}{30}$).