Question

no copied or silly answer​

Answer 1

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

Given rational number is

$\green{\bf :\longmapsto\:1.\overline{48} + 1.4191919...}$

Let we first represent the above sum of two rational numbers in the form of p/q individually.

So, Consider

$\red{\rm :\longmapsto\:1.\overline{48} }$

Let assume that

$\red{\rm :\longmapsto\:x = 1.\overline{48} - - - - (1) }$

On multiply both sides by 100, we get

$\red{\rm :\longmapsto\:100x = 148.\overline{48} - - - - (2) }$

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

$\red{\rm :\longmapsto\:99x = 147 \: }$

$\red{\rm :\longmapsto\:x = \dfrac{147}{99} \: }$

Thus,

$\red{\rm \implies\:\boxed{ \tt{ \: 1.\overline{48} = \frac{147}{99} \: }}}$

Now Consider,

$\purple{\rm :\longmapsto\:1.41919... \: }$

Let we assume that

$\purple{\rm :\longmapsto\:y = 1.41919... \: }$

can be rewritten as

$\purple{\rm :\longmapsto\:y = 1.4 \: \overline {19}\: }$

On multiply both sides by 10, we get

$\purple{\rm :\longmapsto\:10y = 14. \: \overline {19}\: - - - - (3) }$

On multiply by 100, we get

$\purple{\rm :\longmapsto\:1000y = 1419. \: \overline {19}\: - - - - (4) }$

On Subtracting equation (3) from equation (4), we get

$\purple{\rm :\longmapsto\:990y = 1405 }$

$\purple{\rm :\longmapsto\:y = \dfrac{1405}{990} }$

Thus,

$\purple{\rm \implies\:\boxed{ \tt{ \: 1.4191919... = \frac{1405}{990} \: }}}$

Now, Consider,

$\green{\bf :\longmapsto\:1.\overline{48} + 1.4191919...}$

So, on substituting the values evaluated above, we get

$\green{\rm \:  =  \:\dfrac{147}{99} + \dfrac{1405}{990} }$

can be rewritten as

$\green{\bf \:  =  \:\dfrac{1470}{990} + \dfrac{1405}{990} }$

$\green{\bf \:  =  \:\dfrac{1470 + 1405}{990} }$

$\green{\bf \:  =  \:\dfrac{2875}{990} }$

$\green{\bf \:  =  \:\dfrac{575}{198} }$

Hence,

$\green{\rm \implies\:\boxed{ \bf{ \: 1.\overline {48} + 1.4191919... = \dfrac{575}{198} } \: }}$

More to know :-

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

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