Which of the following fractions is the largest? $\frac{3}{2}, \frac{7}{3}, \frac{5}{4}, \frac{7}{2}$

### Concept & Logic To compare fractions efficiently, you can convert them into decimal values or compare their numerators and denominators if they share common traits. Since the denominators here are small ($2, 3, 4$), converting them directly to decimals is the fastest approach. ### Step-by-Step Solution * **Step 1:** Convert each fraction into its decimal equivalent. * $\frac{3}{2} = 1.5$ * $\frac{7}{3} = 2.33...$ * $\frac{5}{4} = 1.25$ * $\frac{7}{2} = 3.5$ * **Step 2:** Compare the resulting decimal values. * The sequence from smallest to largest is: $1.25 < 1.5 < 2.33 < 3.5$. * **Step 3:** Identify the largest decimal and its corresponding fraction. * The largest decimal is $3.5$, which corresponds to $\frac{7}{2}$. ### Exam Strategy & Shortcut You can instantly eliminate $\frac{3}{2}$ and $\frac{5}{4}$ by observing the numerators and denominators. $\frac{7}{2}$ is exactly half of $7$, meaning it is larger than $3$. The other fractions ($\frac{3}{2}$, $\frac{7}{3}$, $\frac{5}{4}$) are all clearly smaller than $3$. No calculation is needed if you rely on basic estimation. ### Common Pitfall Students sometimes resort to finding a common denominator (LCM of $2, 3, 4, 2$ is $12$) to cross-compare. While technically correct, cross-multiplication or finding LCMs takes much longer than simple decimal conversion or logical elimination for small numbers. ### Final Answer Therefore, the correct answer is $\frac{7}{2}$.