Discussion
Let \(a = \frac{11}{13}\), \(b = \frac{13}{14}\) and \(c = \frac{15}{17}\) be three fractions. Which of the following is true ?
Answer: C
The correct order is \(\frac{11}{13} \lt \frac{15}{17} \lt \frac{13}{14}\).
Each fraction is 1 minus a small part, so compare the parts left over: the fraction with the smaller gap from 1 is the larger.
\[ \begin{aligned} 1 - \tfrac{11}{13} &= \tfrac{2}{13} \approx 0.154 \\ 1 - \tfrac{15}{17} &= \tfrac{2}{17} \approx 0.118 \\ 1 - \tfrac{13}{14} &= \tfrac{1}{14} \approx 0.071 \end{aligned} \]
So \(\frac{11}{13}\) is the smallest and \(\frac{13}{14}\) the largest. In decimals they are about 0.846, 0.882 and 0.929.
The option \(\frac{11}{13} \lt \frac{13}{14} \lt \frac{15}{17}\) assumes that a fraction with bigger numbers is bigger, which is not true; the other two put \(\frac{11}{13}\) or \(\frac{13}{14}\) in the wrong place.
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