We are asked how we can write a repeating decimal as a fraction. To write repeating decimals as fractions, we need to follow a few steps.

Step $1 .$ Assign a variable to represent the repeating decimal.
Step $2 .$ Write an equation to equate the variable and the decimal.
Step $3 .$ Multiply each side of the equation by $1 0^{d},$ where $d$ is the number of repeating digits in the repeating decimal.
Step $4 .$ Subtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
Step $5 .$ Solve for the variable. Write an equivalent fraction so that the numerator and denominator are integers, if necessary.

Let's take a look at an example. We will consider a repeating decimal

5.4 \overline{3}

and assign variable

x

to it.

x = 5.4 \overline{3}

Now we will multiply each side of the equation by

1 0^{d},

where

d

is the number of repeating digits in the repeating decimal. Since

d = 1

in our example, we will multiply both sides of our equation by

10 .

10 x = 54 . \overline{3}

Next, we will subtract equivalent expressions of the variable and the repeating decimal from each side of the equation. In our case, we will subtract

x

from the left-hand side and

5.4 \overline{3}

from the right-hand side.

10 x - x = 54 . \overline{3} - 5.4 \overline{3}

Let's solve this equation for

x!

10 x - x = 54 . \overline{3} - 5.4 \overline{3}

Subtract terms

9 x = 48.87

$LHS \cdot 10 = RHS \cdot 10$

90 x = 489

$LHS / 90 = RHS / 90$

x = \frac{4 8 9}{9 0}

We found that the repeating decimal

5.4 \overline{3}

can be rewritten as a fraction as

\frac{4 8 9}{9 0} .

This can also be written as

\frac{1 6 3}{3 0},

or

5 \frac{1 3}{3 0} .

Exercise