What is 0.121428142814281428... as a Fraction?

0.121428 equals 1011883325 as a fraction.

How do you turn 0.121428 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.121428 repeating into a fraction, begin writing this simple equation:

n = 0.121428 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (1428) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 121428.1428 (equation 2)

Step 3: Now subtract equation 1 from equation 2 to cancel the repeating block (or repetend) out.

1000000 × n = 121428.1428
    100 × n = 0.121428
 999900 × n = 121416

121416999900 could be the answer, but it still can be put in the simplest form, i.e., reduced.

To simplify this fraction, divide the numerator and denominator by 12 (the GCF - greatest common factor).

n = 121416999900 = 121416 ÷ 12999900 ÷ 12 = 1011883325. So,

0.121428 = 1011883325 as the lowest possible fraction.

The repeating decimal 0.121428 (vinculum notation) has a repeated block length of 4. It is also represented as 0.12142814281428... (ellipsis notation) which equals approximately 0.1214281428142814281428 (decimal approximation)(*).

The recurring decimal 0.121428 can be written as a ratio of two integers having 10118 as the numerator and 83325 as the denominator. So, it is a rational number (named after ratio). It can be shown that a number is rational if its decimal representation is repeating or terminating.

(*) At present, there is no single universally accepted notation or phrasing for repeating decimals.