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