Question

Question #48afa

Answer 1

The fraction of the repeating `2.bar(31)` decimal is `229/99`.

Explanation:

Step 1 :

Let `x` be equal to the repeating decimal you are trying to convert to a fraction. Here, `x` is `2.31` or

`x = 2.31313131...`

as an equation.

Step 2:

Examine the repeating decimal to find the repeating digit(s). The repeating digits here are `31`.

Step 3:

Place the repeating digit(s) to the left of the decimal point. Move the repeating digits `[31]` to the left of the decimal point by multiplying `100` (to move both digits to the left) to both sides of the equation in step 1. Thus,

`100x=231.313131...`

Step 4:

Place the repeating digit(s) to the right of the decimal point. Look at the equation in step 1 again. In this example, the repeating digit is already to the right, so there is nothing else to do.

`x = 2.31313131`

Step 5:

Your two equations are:

`100x=231.313131`

`x = 2.31313131`

Subtract the left sides of the two equations.

`100x - x= 99x`

Then, subtract the right sides of the two equations

`231.313131-2.31313131=228.99999969`

As you subtract, just make sure that the difference is positive for both sides

`=> 99x=228.99999969`

Then round both sides up to whole numbers. In this case, `99` is already a whole number so only `228` needs to be rounded to `229` because the digit after it is `9` (Because `9` is greater or equal to `5`, the unit digit `8` of `228` needs to plus itself by `1`).

`=> 99x=229`

Step 6:

Now, you can find the fraction `x` of the repeating decimal `2.31` by dividing `99` to both sides of the equations

`99x =229`

`=> x=229/99`