Question

Find two rational numbers and two irrational numbers between 3/5 and 7/2.

Answer 1

$\large\underline{\sf{Solution-}}$

To find a rational number between two numbers, the method used is called Arithmetic mean Method.

Let assume that a and b are two rational numbers, then rational number between a and b is given by

$\rm :\longmapsto\:Rational \: number \: between \: a \: and \: b = \dfrac{a + b}{2}$

Here, According to given question,

$\rm :\longmapsto\:a = \dfrac{3}{5} = 0.6$

and

$\rm :\longmapsto\:b = \dfrac{7}{2} = 3.5$

So,

$\rm :\longmapsto\:Rational \: number \: between \: 0.6 \: and \: 3.5$

$\rm \: = \: \: \dfrac{0.6 + 3.5}{2}$

$\rm \: = \: \: \dfrac{4.1}{2}$

$\rm \: = \: \: 2.05$

Now, to find second rational number,

$\rm :\longmapsto\:a = \dfrac{3}{5} = 0.6$

and

$\rm :\longmapsto\:b = 2.05$

Thus,

$\rm :\longmapsto\:Rational \: number \: between \: 0.6 \: and \: 2.05$

$\rm \: = \: \: \dfrac{0.6 + 2.05}{2}$

$\rm \: = \: \: \dfrac{2.65}{2}$

$\rm \: = \: \: 1.325$

So,

2 rational number between 0.6 and 3.5 are 2.05 and 1.325

Now,

To find an irrational number between two numbers, the method used is called Geometric mean Method.

Let assume that a and b are two rational numbers, then irrational number between a and b is given by

$\rm :\longmapsto\:Irrational \: number \: between \: a \: and \: b = \sqrt{ab}$

Here, According to given question,

$\rm :\longmapsto\:a = \dfrac{3}{5} = 0.6$

and

$\rm :\longmapsto\:b = \dfrac{7}{2} = 3.5$

Therefore,

$\rm :\longmapsto\:Irrational \: number \: between \: 0.6 \: and \: 3.5$

$\rm \: = \: \: \sqrt{(0.6)(3.5)}$

$\rm \: = \: \: \sqrt{2.1}$

To find second irrational number,

$\rm :\longmapsto\:a = \dfrac{3}{5} = 0.6$

and

$\rm :\longmapsto\:b = \sqrt{2.1}$

Therefore,

$\rm :\longmapsto\:Irrational \: number \: between \: 0.6 \: and \: \sqrt{2.1}$

$\rm \: = \: \: \sqrt{0.6 \times \sqrt{2.1} }$

$\rm \: = \: \: \sqrt{\dfrac{3}{5} \times \sqrt{2.1} }$

$\rm \: = \: \: \sqrt{\dfrac{3 \sqrt{2.1} }{6} }$

Alternative Method :-

To find rational number and irrational numbers between 2 rational number is decimal representation method.

Rational number are those numbers whom decimal representation is terminating or non - terminating but repeating.

So, 2 rational numbers between 0.6 and 3.5 is 1, 2.

And

Irrational number are those numbers whom decimal representation is neither terminating nor repeating.

So, 2 irrational number between 0.6 and 3.5 are

1.01001000100001...

and

1.02002000200002...