What is 2.1 with the one repeating as a fraction? with work if possible

2.1¯¯¯=219
2.
1
¯
=
2
1
9
Showing the work

Let X equal our decimal number (fraction)
Equation 1:
X=2.1¯¯¯(1)
(1)X=2.
1
¯

Since we have 1 digits in our repeating decimal
group, create a second equation by multiplying
both sides by 101 = 10
Equation 2:
10X=21.1¯¯¯(2)
(2)10X=21.
1
¯

Subtracting equation (1) from (2)
10XX9X===21.1...2.1...19
10X=21.1...X=2.1...9X=19
We get
9X=19
9
X
=
19
Solving for X
X=199
X
=
19
9
Simplifying the improper fraction,
=219
=
2
1
9
In conclusion,
2.1¯¯¯=219
2.
1
¯
=
2
1/9