Answer : The sum of the given expression is, [tex]\frac{3}{4}[/tex]
Step-by-step explanation :
As we are given the expression:
[tex]0.\bar{5}+0.19\bar{4}[/tex]
Let us assumed that,
[tex]x=0.\bar{5}[/tex]
or,
[tex]x=0.5555.......[/tex] ..........(1)
To shifting the one decimal place to the right, we are multiplying the equation by 10 on both side, we get:
[tex]10x=5.5555.......[/tex] ...........(2)
Now subtracting equation 1 from 2, we get:
[tex]9x=5[/tex]
[tex]x=\frac{5}{9}[/tex] ............(A)
and,
Let us assumed that,
[tex]x=0.19\bar{4}[/tex]
or,
[tex]x=0.194444.......[/tex] ..........(1)
To shifting the two decimal place to the right, we are multiplying the equation by 100 on both side, we get:
[tex]100x=19.4444.......[/tex] ...........(2)
To shifting the one more decimal place to the right, we are multiplying the equation by 10 on both side, we get:
[tex]1000x=194.4444.......[/tex] ...........(3)
Now subtracting equation 2 from 3, we get:
[tex]900x=175[/tex]
[tex]x=\frac{175}{900}[/tex] ............(B)
Now adding equation A and B, we get:
[tex]\frac{5}{9}+\frac{175}{900}[/tex]
[tex]\Rightarrow \frac{500}{900}+\frac{175}{900}[/tex]
[tex]\Rightarrow \frac{675}{900}[/tex]
[tex]\Rightarrow \frac{3}{4}[/tex]
Thus, the sum of the given expression is, [tex]\frac{3}{4}[/tex]
