Q3(ii):
Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (ii) $0.4\overline{7}$

Solution :

Initial Setup & Variable Assignment

We are tasked with converting the mixed recurring decimal $0.4\overline{7}$ into a rational number of the form $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$.

Let the given number be represented by the variable $x$:

$x = 0.47777\dots$ (Equation 1)

[By definition of the vinculum (bar) notation, only the digit $7$ repeats infinitely, while the digit $4$ is a non-repeating decimal component].

Step 1: Isolating the Repeating Decimal Block

To eliminate the infinite repeating sequence, we must first align the decimal point immediately before the repeating block. Since there is exactly one non-repeating digit ($4$) after the decimal point, we multiply both sides of Equation 1 by $10^1$.

$10 \cdot x = 10 \cdot (0.47777\dots)$

$10x = 4.7777\dots$ (Equation 2)

[Per the properties of base-10 positional notation, multiplying by $10$ shifts the decimal point one place to the right].

Step 2: Shifting the Decimal Past the Repeating Block

Next, we need a second equation where the fractional part is identical to the fractional part of Equation 2 (which is $.7777\dots$). Since the repeating block consists of exactly one digit ($7$), we multiply Equation 2 by $10^1$.

$10 \cdot 10x = 10 \cdot (4.7777\dots)$

$100x = 47.7777\dots$ (Equation 3)

Step 3: Algebraic Elimination of the Infinite Decimal

We now subtract Equation 2 from Equation 3. Because the infinite repeating decimal tails ($.7777\dots$) are perfectly aligned, they subtract to zero, leaving only integers.

$100x$	$=$	$47.7777\dots$
$- \quad 10x$	$=$	$- \quad 4.7777\dots$
$90x$	$=$	$43.0000\dots$

[By the Subtraction Property of Equality, if $a = b$ and $c = d$, then $a - c = b - d$].

Step 4: Solving for the Rational Form $\frac{p}{q}$

We now have a simple linear equation with integer coefficients:

$90x = 43$

Isolating $x$ by dividing both sides by $90$:

$x = \frac{43}{90}$

We must verify that this fraction is in its simplest form. The numerator, $43$, is a prime number. The denominator, $90$, is not a multiple of $43$ (since $43 \times 2 = 86$). Therefore, the greatest common divisor $\text{GCD}(43, 90) = 1$. The fraction is irreducible, and both $p = 43$ and $q = 90$ are integers with $q \neq 0$.

Final Solution: The mixed recurring decimal $0.4\overline{7}$ expressed in the rational form $\frac{p}{q}$ is $\frac{43}{90}$.

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