The problem asks to express the repeating decimal $0.\overline{14}$ in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

1. Problem Description

The problem asks to express the repeating decimal $0.\overline{14}$ in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

2. Solution Steps

Let $x = 0.\overline{14} = 0.141414...$

Multiply both sides by 100:

$100x = 14.141414...$

Subtract $x$ from $100x$:

$100x - x = 14.141414... - 0.141414...$

$99x = 14$

Divide both sides by 99:

$x = \frac{14}{99}$

Since 14 and 99 are integers and $99 \neq 0$, the fraction $\frac{14}{99}$ is in the required form.

3. Final Answer

$\frac{14}{99}$