Q3(i):

Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (i) $0.\overline{6}$

Step 1: Variable Assignment and Expansion

To express the non-terminating, repeating decimal $0.\overline{6}$ as a rational number in the form $\frac{p}{q}$, we begin by assigning it to a variable. Let $x$ represent the given value.

$x = 0.\overline{6}$

Expanding the vinculum (the overline indicating the repeating sequence), we write the number as an infinite decimal series:

$x = 0.6666\dots \quad \text{--- (Equation 1)}$

Step 2: Multiplication by a Power of 10

[Per the fundamental algorithm for converting repeating decimals to fractions], we must shift the decimal point to the right by exactly the number of digits in the repeating block. This aligns the infinite repeating tails so they can be algebraically eliminated.

In the decimal $0.\overline{6}$, there is exactly one repeating digit (the digit 6). Therefore, we multiply both sides of Equation 1 by $10^1$ (which is $10$).

$10 \cdot x = 10 \cdot (0.6666\dots)$

$10x = 6.6666\dots \quad \text{--- (Equation 2)}$

Step 3: Algebraic Subtraction to Eliminate the Repeating Tail

Next, we subtract Equation 1 from Equation 2. Because the infinite sequence of $6$s to the right of the decimal point is identical in both equations, subtracting them will perfectly cancel out the fractional part, leaving an integer.

This yields the simplified linear equation:

$9x = 6$

Step 4: Solving for $x$ and Simplification

Isolate $x$ by dividing both sides of the equation by $9$:

$x = \frac{6}{9}$

[By the Fundamental Theorem of Arithmetic], fractions must be expressed in their simplest form. We find the Greatest Common Divisor (GCD) of the numerator ($6$) and the denominator ($9$).

• Prime factorization of $6 = 2 \times 3$
• Prime factorization of $9 = 3 \times 3$
• $\text{GCD}(6, 9) = 3$

Dividing both the numerator and the denominator by $3$:

$x = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}$

Geometric Verification

The fraction $\frac{2}{3}$ represents two parts out of three equal subdivisions of a whole. When $2$ is divided by $3$ using long division, the quotient is exactly $0.666\dots$, confirming our algebraic derivation.

Final Solution: The repeating decimal $0.\overline{6}$ expressed in the form $\frac{p}{q}$ is $\frac{2}{3}$, where $p=2$ and $q=3$ are integers, and $q \neq 0$.