Question

Find the sum of(0.8+0.45 bar on 5)amd express the results in its simplest form pq

Answer 1

$\red{\underline{\underline{\sf{Correct\:\:Question:}}}}$

Find the sum of (0.8 + $\sf{ 0.45 \overline{5} }$ )and express the results in its simplest form $\red{\frac{p}{q} }$.

$\pink{\underline{\underline{\sf{Answer -}}}}$

$\bf 0.8 = \frac{8}{10} = \frac{4}{5}$

Now,

$\bf 0.45 \overline{5}$

$\bf Let, x = 0.4555555........(i)$

Multiplying 10 to both sides of eq.(i), we get,

$\bf 10x = 4.5555555......(ii)$

Now,

Multiplying 10 to both sides of eq.(ii), we get,

$\bf 100x = 45.555555......(iii)$

So,

Subtracting eq.(ii) from eq. (iii), we get

$\bf 100x = 45.555555......(iii) \\ - 10x = 4.555555.......(ii) \\ \rule{150}1 \\.\:\: 90x = 41.000000....... \\ \rule{150}1$

→ 90x = 41

$\implies \boxed{\boxed{\bf x= \dfrac{41}{90}}}$

Now,

$\bf \dfrac{4}{5} + \dfrac{41}{90}$

$= \bf \dfrac{72+41}{90} = \dfrac{113}{90}$

Hence,

$\bf{ 0.8 + 0.45 \overline{5} = \dfrac{113}{90}}$

Answer 2

$\large \underline{ \rm{\pink{Correct\;\: Question}}} :\\ \\ \small \sf Find \quad the \quad sum \quad of\quad \\ \small \sf (0.8 + 0.4\bar{5}) \quad and \quad express \\ \small\sf the \quad results \quad in \quad in \: \: \: the \: \: \: form \: \: \: \\ \small \bf \frac{p}{q} \\ \\ \large \underline{ \pink{ \rm{Answer }}}:$

0.8 = $\sf\dfrac{8}{10} = \dfrac{4}{5}$

Let

x = 0.455

x = 0.45555555555……...

Assuming as Eq 1

Multiplying with 10 on both sides

10x = 4.5555555…………….…

Assuming as Eq 2

Multiplying with 100 on both sides of Eq 1

100x = 45.5555555....…..

Assuming as Eq 3

Now, Eq 3 - 2

$\: \: \: \: \: \: \: \sf{100x = 45.5555555.......} \\ \underline{ ( - ) \: \sf{ \: \: \: 10x = \: \: 4.55555555.......}} \\ \sf \underline{ \: \: \: \: \: \: \: \: \: \: 90x =41.00000000....... } \\ \\ \longrightarrow \boxed{ \sf \orange{x = \frac{41}{90} }}$

Now,

$\sf\dfrac{4}{5}+\dfrac{41}{90} = \dfrac{113}{90}$

$\therefore\rm\underline {\green{Hence,\;\: 0.8 + 0.4\bar{5} = \dfrac{113}{90}}}$