Question

Which of the following rational numbers lies between irrational numbers $2-\sqrt{3}$ and $5-\sqrt{3}$ ?

• A. $\frac{3}{5}$
• B. $\frac{47}{15}$
• C. $\mathrm{2.3333....}$
• D. All of the above.

Answer 1

The correct option is D All of the above.

We need to find the approximate values of $2-\sqrt{3}\&5-\sqrt{3}$
$\text{We know}:2.89<3.00<3.24\Rightarrow (1.7{)}^2<3.00<(1.8{)}^2\Rightarrow 1.7<\sqrt{3}<1.8\Rightarrow -1.7>-\sqrt{3}>-1.8\Rightarrow 2-1.7>2-\sqrt{3}>2-1.8\Rightarrow 0.3>2-\sqrt{3}>0.2$

$\text{We know}:2.89<3.00<3.24\Rightarrow (1.7{)}^2<3.00<(1.8{)}^2\Rightarrow 1.7<\sqrt{3}<1.8\Rightarrow -1.7>-\sqrt{3}>-1.8\Rightarrow 5-1.7>5-\sqrt{3}>5-1.8\Rightarrow 3.3>5-\sqrt{3}>3.2$

Hence, with this approximation we can estimate that all the rational number between 0.3 and 3.2 are greater than $2-\sqrt{3}$ and less than $5-\sqrt{3}$.

$\frac{3}{5}=0.6$ lies between given irrational numbers.
$\frac{47}{15}=\mathrm{3.133...}$ lies between given irrational numbers.
$\mathrm{2.333....}$ lies between given irrational numbers.

Below are the representation of every rational number between given irrational numbers.
Since, all lies between irrational number $2-\sqrt{3}$ and $5-\sqrt{3}$.