Which of the following fractions is the largest?

### Concept & Strategy To compare fractions and determine the largest, the most reliable method is to equalize their denominators. Find the Least Common Multiple (LCM) of all the given denominators and convert each fraction into an equivalent fraction with that LCM as the common denominator. Once all denominators are identical, the fraction possessing the largest numerator is the largest overall fraction. ### Step-by-Step Solution **Given fractions:** * $\frac{7}{8}, \frac{13}{16}, \frac{31}{40}, \frac{63}{80}$ **Calculation:** * Step 1: Identify the denominators and find their LCM. * Denominators: $8, 16, 40, 80$ * Since $80$ is perfectly divisible by $8, 16,$ and $40$, the LCM is simply $80$. * Step 2: Convert each fraction to have a denominator of $80$. * First fraction: $\frac{7}{8} = \frac{7 \times 10}{8 \times 10} = \frac{70}{80}$ * Second fraction: $\frac{13}{16} = \frac{13 \times 5}{16 \times 5} = \frac{65}{80}$ * Third fraction: $\frac{31}{40} = \frac{31 \times 2}{40 \times 2} = \frac{62}{80}$ * Fourth fraction: $\frac{63}{80}$ remains $\frac{63}{80}$ * Step 3: Compare the numerators of the equivalent fractions. * The numerators are $70, 65, 62,$ and $63$. * The largest numerator is $70$. * Step 4: Identify the corresponding original fraction. * The equivalent fraction $\frac{70}{80}$ corresponds to $\frac{7}{8}$. ### Exam Strategy & Shortcut If finding the LCM seems tedious, you can utilize the cross-multiplication method to quickly compare fractions two at a time. * Compare $\frac{7}{8}$ and $\frac{13}{16}$: $7 \times 16 = 112$ and $8 \times 13 = 104$. Since $112 > 104$, $\frac{7}{8}$ is larger. * Compare $\frac{7}{8}$ and $\frac{31}{40}$: $7 \times 40 = 280$ and $8 \times 31 = 248$. Since $280 > 248$, $\frac{7}{8}$ is larger. * Compare $\frac{7}{8}$ and $\frac{63}{80}$: $7 \times 80 = 560$ and $8 \times 63 = 504$. Since $560 > 504$, $\frac{7}{8}$ is the largest overall. ### Common Pitfall A frequent mistake is converting fractions to decimals and making rounding errors, especially under time constraints. While decimal conversion works, equating denominators (or cross-multiplication) is often faster and less prone to simple arithmetic slips when dealing with manageable numbers. ### Final Answer **Therefore, the correct answer is 7/8.**