Question

plz slove the question asap!!​

Answer 1

Answer:

2.2 33333 . . . . and 2.233 are two rational numbers between 2.23 23 23 . . . and 2.24 24 24 . . .

Step-by-step explanation:

We know that possible decimal expansions of rational numbers are ( non terminating but repeating ), ( non terminating ) and ( non repeating ). This means we can choose any number whose decimal expansion is either repeating or terminating.

So the rational numbers between 2.23 23 23. . . and 2.24 24 24 . . . are ∞. You can choose any two as per the requirement.

Answer 2

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

Given rational numbers are

$\red{\boxed{ \rm{2. \overline{23} \: \: and \: \:2. \overline{24} }}}$

Let we first convert these decimal form in to fraction form

Consider,

$\rm :\longmapsto\:2. \overline{23}$

Let we assume that

$\rm :\longmapsto\:x \: = \: 2. \overline{23} - - - (1)$

Multiply by 100 on both sides, we get

$\rm :\longmapsto\:100x = 223. \overline{23} - - - (2)$

On Subtracting equation (1) from equation (2), we get

$\rm :\longmapsto \: 99x = 221$

$\bf\implies \:x = \dfrac{221}{99}$

Hence,

$\red{\boxed{ \bf{ \: \: 2. \overline{23} = \dfrac{221}{99} \: }}}$

Now,

Consider,

$\rm :\longmapsto\:2. \overline{24}$

Let we assume that,

$\rm :\longmapsto\:y \: = \: 2. \overline{24} - - - (1)$

Multiply by 100 on both sides, we get

$\rm :\longmapsto\:100y \: = \: 224. \overline{24} - - - (2)$

On Subtracting equation (1) from equation (2), we get

$\rm :\longmapsto\:99y \: = \: 222$

$\bf\implies \:y = \dfrac{222}{99}$

Hence,

$\red{\boxed{ \bf{ \: \: 2. \overline{24} = \dfrac{222}{99} \: }}}$

Now, we have to find two rational number between

$\rm :\longmapsto\:\dfrac{221}{99} \: \: and \: \: \dfrac{222}{99}$

We use arithmetic mean method to find rational number.

We know,

If a and b are two numbers, then a rational number between a and b

$\rm \: = \: \dfrac{a + b}{2}$

So,

$\rm :\longmapsto\:A \: rational \: number \: between \: \dfrac{221}{99} \: \: and \: \: \dfrac{222}{99}$

$\rm \: = \: \dfrac{1}{2} \bigg(\dfrac{221}{99} + \dfrac{222}{99} \bigg)$

$\rm \: = \: \dfrac{1}{2} \bigg(\dfrac{221 + 222}{99} \bigg)$

$\rm \: = \: \dfrac{1}{2} \bigg(\dfrac{443}{99} \bigg)$

$\rm \: = \: \dfrac{443}{198}$

Now, other rational number

$\rm :\longmapsto\:A \: rational \: number \: between \: \dfrac{221}{99} \: \: and \: \: \dfrac{443}{198}$

$\rm \: = \: \dfrac{1}{2} \bigg(\dfrac{221}{99} + \dfrac{443}{198} \bigg)$

$\rm \: = \: \dfrac{1}{2} \bigg(\dfrac{442 + 443}{198} \bigg)$

$\rm \: = \: \dfrac{1}{2} \bigg(\dfrac{885}{198} \bigg)$

$\rm \: = \: \dfrac{885}{396}$

Hence,

$\boxed{\rm :\longmapsto\:2 \: rational \: number \: between \: \dfrac{221}{99} \: \: and \: \: \dfrac{222}{99}} \\ \\are \\ \\ \boxed{ \bf{ \dfrac{443}{198} \: and \: \dfrac{885}{396}}}$

OR

In decimal form

$\boxed{\rm :\longmapsto\:2 \: rational \: number \: between \: 2. \overline{23} \: \: and \: \: 2. \overline{24}} \\ \\are \\ \\ \boxed{ \bf{ \: 2. 2\overline{37} \: and \:2. 23\overline{48}}}$