If $\frac{144}{0.144} = \frac{14.4}{x}$, then the value of $x$ is

## Concept & Logic This problem tests proportional reasoning and the simplification of decimal fractions. The key is to simplify the known ratio before solving for the unknown. $$ \frac{a}{b} = \frac{c}{d} \implies d = \frac{b \times c}{a} $$ ## Step-by-Step Solution * **Given:** $\frac{144}{0.144} = \frac{14.4}{x}$ * **Calculation:** First, simplify the left side of the equation. $\frac{144}{0.144} = \frac{144}{\frac{144}{1000}} = 144 \times \frac{1000}{144} = 1000$ Now, substitute this back into the equation: $1000 = \frac{14.4}{x}$ Rearrange to solve for $x$: $x = \frac{14.4}{1000}$ Shift the decimal point three places to the left: $x = 0.0144$ ## Exam Strategy & Shortcut Observe the powers of 10. The left side $\frac{144}{0.144}$ is simply $10^3 = 1000$. Therefore, $x$ must be $14.4$ shifted three decimal places smaller. Mentally shifting $14.4$ left by three places instantly gives $0.0144$. ## Common Pitfall A frequent error is placing the decimal point incorrectly during division by 1000, leading to $0.144$ or $1.44$. Always remember that dividing by $10^n$ moves the decimal point $n$ places to the left. ## Final Answer **Therefore, the correct answer is 0.0144.**