Question

Which of the following irrational numbers lie between $\frac{5}{11}$ and $\frac{7}{11}$

• A. $2\sqrt{2}$
• B. $\frac{\sqrt{3}}{3}$
• C. $\sqrt{5}-\frac{17}{10}$
• D. None of the above.

Answer 1

Answer: (A), (C)

$\sqrt{5}-\frac{17}{10}$

On converting given fractions in to decimals, we get:
$\frac{5}{11}=\mathrm{0.454545...}\&\frac{7}{11}=\mathrm{0.636363...}$

$\underset{–}{\text{Option(a)}}2\sqrt{2}$
$1.4<\sqrt{2}<1.5\Rightarrow 1.4\times 2<2\sqrt{2}<2\times 1.5\Rightarrow 2.8<2\sqrt{2}<3$
Hence, it does not lie between $\frac{5}{11}\&\frac{7}{11}$

$\underset{–}{\text{Option(b)}}\frac{\sqrt{3}}{3}$
$1.7<\sqrt{3}<1.8\Rightarrow 1.7÷3<\frac{\sqrt{3}}{3}<1.8÷3\Rightarrow 0.5\overline{6}<\frac{\sqrt{3}}{3}<0.6$
Hence, it lies between $\frac{5}{11}\&\frac{7}{11}$

$\underset{–}{\text{Option(C)}}\sqrt{5}-\frac{17}{10}$
$2.2<\sqrt{5}<2.3\Rightarrow 2.2-1.7<\sqrt{5}-1.7<2.3-1.7\Rightarrow 0.5<\sqrt{5}-1.7<0.6$
Hence, it lies between $\frac{5}{11}\&\frac{7}{11}$
other options lie outside the given rational numbers