If $x$ is a positive number, then which of the following fractions has the greatest value?

### Concept & Logic This problem analyzes variables inside simple algebraic fractions. A fraction is maximized when its numerator is larger than its denominator (improper fraction value $> 1$). ### Step-by-Step Solution * **Analyze Option (a):** $\frac{x}{x} = 1$ (Always exactly equal to 1 for any positive number). * **Analyze Option (b):** $\frac{x}{x+1}$ (Since the numerator is smaller than the denominator, this value is always $< 1$). * **Analyze Option (c):** $\frac{x+1}{x} = 1 + \frac{1}{x}$ (Since the numerator is strictly greater than the denominator, this value is always $> 1$). * **Analyze Option (d):** $\frac{x+2}{x+3}$ (Since the numerator is smaller than the denominator, this value is always $< 1$). * **Conclusion:** Option (c) is the only fraction expression that produces a value strictly greater than 1, making it the greatest value among all choices. ### Exam Strategy & Shortcut Plug in a basic integer value for $x$, such as $x = 1$: Option (a): $\frac{1}{1} = 1$ Option (b): $\frac{1}{2} = 0.5$ Option (c): $\frac{2}{1} = 2$ Option (d): $\frac{3}{4} = 0.75$ Comparing the absolute values shows that 2 is unambiguously the greatest value. ### Common Pitfall Overcomplicating the variable structure with derivative inequalities can lead to wasted time. Simple classification into proper ($<1$) and improper ($>1$) states resolves the expression relationship immediately. ### Final Answer **Therefore, the correct answer is (x + 1) / x.**