What is 12.7142857142857142857142857142857142857142857... as a Fraction?

12.7142857 equals 897 as a fraction.

How do you turn 12.7142857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.7142857 repeating into a fraction, begin writing this simple equation:

n = 12.7142857 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (142857) and 1 digit in the non-repeating part (7). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 127142857.142857 (equation 2)

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

10000000 × n = 127142857.142857
      10 × n = 12.7142857
 9999990 × n = 127142730

1271427309999990 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 1428570 (the GCF - greatest common factor).

n = 1271427309999990 = 127142730 ÷ 14285709999990 ÷ 1428570 = 897. So,

12.7142857 = 897 as the lowest possible fraction.

As the numerator is greater than the denominator, we have an improper fraction, so we can also express it as a mixed number, thus 897 is also equal to 1257 when expressed as a mixed number.

The repeating decimal 12.7142857 (vinculum notation) has a repeated block length of 6. It is also represented as 12.7142857142857142857... (ellipsis notation) which equals approximately 12.7142857142857142857142857142857 (decimal approximation)(*).

The recurring decimal 12.7142857 can be written as a ratio of two integers having 89 as the numerator and 7 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.