Question

# Find three different irrational numbers between the rational numbers $\dfrac{5}{7}$ and $\dfrac{9}{{11}}$.A.$0.75$, $0.7$, $0.76$B.$0.75757575$,$0.7$, $0.76$C.$0.75075007500075000075...$, $0.7670767007670007670000767...$, $0.808008000800008...$D.None of these

Hint: First we will use the definition of irrational numbers are those numbers which are non-recurring and non-terminating numbers. Then we will convert both the rational numbers in decimal form and find which option satisfies being an irrational number and lies between them.

We are given the rational numbers $\dfrac{5}{7}$ and $\dfrac{9}{{11}}$.
We know that the rational numbers are those numbers which can be written in the form of $\dfrac{p}{q}$, where $p$ is numerator, $q$ is denominator, $q \ne 0$ and both are integers.
We also know that irrational numbers are numbers that can not be represented in the rational number form, they are non-recurring and non-terminating decimal numbers.
Rewriting the rational number $\dfrac{5}{7}$ into decimal form, we get
$\Rightarrow \dfrac{5}{7} = 0.714285714285714285...$
Rewriting the rational number $\dfrac{9}{{11}}$ into decimal form, we get
$\Rightarrow \dfrac{9}{{11}} = 0.81818181...$
Considering option A,
Since the three numbers are terminating rational numbers and $0.7$ does not lie between the two given rational numbers, option A is incorrect.
Considering option B,
Since the three numbers are recurring and terminating rational numbers and $0.7$ does not lie between the two given rational numbers, option B is also incorrect.
Considering option C,
All the three irrational numbers $0.75075007500075000075...$, $0.7670767007670007670000767...$, $0.808008000800008...$ lies between the given rational number.
Hence, option C is correct.

Note: In solving these types of questions, students must know that there are infinite irrational numbers between two rational numbers. Hence, one can find as many irrational numbers as we want between the rational numbers. The answer is just one among them.