Let a = $\frac{11}{13}$, b = $\frac{13}{14}$ and c = $\frac{15}{17}$ be three fractions. Which of the following is true ?

Q: Let a = $\frac{11}{13}$, b = $\frac{13}{14}$ and c = $\frac{15}{17}$ be three fractions. Which of the following is true ?

To determine the correct order of the fractions $ a = \frac{11}{13} $, $ b = \frac{13}{14} $, and $ c = \frac{15}{17} $, we need to compare their values. The correct answer is given as Option 3: $\frac{11}{13} < \frac{15}{17} < \frac{13}{14}$. Let's justify why this is correct and why the other options are incorrect. ### Explanation for Option 3: 1. **Comparing $\frac{11}{13}$ and $\frac{15}{17}$:** - Cross-multiply to compare: $11 \times 17 = 187$ and $15 \times 13 = 195$. - Since $187 < 195$, we have $\frac{11}{13} < \frac{15}{17}$. 2. **Comparing $\frac{15}{17}$ and $\frac{13}{14}$:** - Cross-multiply to compare: $15 \times 14 = 210$ and $13 \times 17 = 221$. - Since $210 < 221$, we have $\frac{15}{17} < \frac{13}{14}$. Combining these results, we have $\frac{11}{13} \frac{11}{13}$ as shown by cross-multiplying: $13 \times 13 = 169$ and $11 \times 14 = 154$, where $169 > 154$. 2. **Option 2: $\frac{15}{17} < \frac{13}{14} < \frac{11}{13}$** - This option suggests $\frac{15}{17} < \frac{13}{14}$, which is true, but it also suggests $\frac{13}{14} < \frac{11}{13}$, which is incorrect as shown above. 3. **Option 4: $\frac{11}{13} < \frac{13}{14} < \frac{15}{17}$** - While $\frac{11}{13} < \frac{13}{14}$ is true, $\frac{13}{14} < \frac{15}{17}$ is incorrect. We have already established that $\frac{15}{17} < \frac{13}{14}$. In conclusion, the correct order of the fractions is $\frac{11}{13} < \frac{15}{17} < \frac{13}{14}$, which supports Option 3 as the correct answer.

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Accepted Answer

To determine the correct order of the fractions $ a = \frac{11}{13} $, $ b = \frac{13}{14} $, and $ c = \frac{15}{17} $, we need to compare their values. The correct answer is given as Option 3: $\frac{11}{13} < \frac{15}{17} < \frac{13}{14}$. Let's justify why this is correct and why the other options are incorrect. ### Explanation for Option 3: 1. **Comparing $\frac{11}{13}$ and $\frac{15}{17}$:** - Cross-multiply to compare: $11 \times 17 = 187$ and $15 \times 13 = 195$. - Since $187 < 195$, we have $\frac{11}{13} < \frac{15}{17}$. 2. **Comparing $\frac{15}{17}$ and $\frac{13}{14}$:** - Cross-multiply to compare: $15 \times 14 = 210$ and $13 \times 17 = 221$. - Since $210 < 221$, we have $\frac{15}{17} < \frac{13}{14}$. Combining these results, we have $\frac{11}{13} < \frac{15}{17} < \frac{13}{14}$, which confirms that Option 3 is correct. ### Explanation for why other options are incorrect: 1. **Option 1: $\frac{13}{14} < \frac{11}{13} < \frac{15}{17}$** - We already established that $\frac{11}{13} < \frac{15}{17}$, but $\frac{13}{14}$ is not less than $\frac{11}{13}$. In fact, $\frac{13}{14} > \frac{11}{13}$ as shown by cross-multiplying: $13 \times 13 = 169$ and $11 \times 14 = 154$, where $169 > 154$. 2. **Option 2: $\frac{15}{17} < \frac{13}{14} < \frac{11}{13}$** - This option suggests $\frac{15}{17} < \frac{13}{14}$, which is true, but it also suggests $\frac{13}{14} < \frac{11}{13}$, which is incorrect as shown above. 3. **Option 4: $\frac{11}{13} < \frac{13}{14} < \frac{15}{17}$** - While $\frac{11}{13} < \frac{13}{14}$ is true, $\frac{13}{14} < \frac{15}{17}$ is incorrect. We have already established that $\frac{15}{17} < \frac{13}{14}$. In conclusion, the correct order of the fractions is $\frac{11}{13} < \frac{15}{17} < \frac{13}{14}$, which supports Option 3 as the correct answer.