Which one of the following fractions is greater than $\frac{2}{7}$ and less than $\frac{7}{9}$?
The lower bound is $\frac{2}{7} \approx 0.2857$.
The upper bound is $\frac{7}{9} \approx 0.7778$.
Let’s test option B, $\frac{11}{17} \approx 0.6471$.
Is $\frac{11}{17} > \frac{2}{7}$? Compare $11 \times 7$ with $17 \times 2$. $77 > 34$. Yes, $\frac{11}{17} > \frac{2}{7}$.
Is $\frac{11}{17} < \frac{7}{9}$? Compare $11 \times 9$ with $17 \times 7$. $99 < 119$. Yes, $\frac{11}{17} < \frac{7}{9}$.
Since $\frac{11}{17}$ satisfies both conditions, it is the correct answer.
Let's quickly check other options:
A) $\frac{15}{19} \approx 0.7895$. Is $\frac{15}{19} < \frac{7}{9}$? Compare $15 \times 9 = 135$ with $19 \times 7 = 133$. $135 > 133$, so $\frac{15}{19} > \frac{7}{9}$. Incorrect.
C) $\frac{11}{14} \approx 0.7857$. Is $\frac{11}{14} < \frac{7}{9}$? Compare $11 \times 9 = 99$ with $14 \times 7 = 98$. $99 > 98$, so $\frac{11}{14} > \frac{7}{9}$. Incorrect.
D) $\frac{17}{21} \approx 0.8095$. Is $\frac{17}{21} < \frac{7}{9}$? Compare $17 \times 9 = 153$ with $21 \times 7 = 147$. $153 > 147$, so $\frac{17}{21} > \frac{7}{9}$. Incorrect.
$\frac{2}{7} \approx 0.28$
$\frac{7}{9} \approx 0.78$
A) $\frac{15}{19} \approx \frac{15}{20} = 0.75$. Closer check needed. (Actual: 0.789)
B) $\frac{11}{17} \approx \frac{11}{16} = 0.6875$ or $\frac{10}{16} = 0.625$. Looks promising. (Actual: 0.647)
C) $\frac{11}{14} \approx \frac{11}{15} \approx 0.73$. Closer check needed. (Actual: 0.786)
D) $\frac{17}{21} \approx \frac{17}{20} = 0.85$. Too high. (Actual: 0.810)