What is 12.2714285714285714285714285714285714285... as a Fraction?

12.2714285 equals 85970 as a fraction.

How do you turn 12.2714285 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.2714285 repeating into a fraction, begin writing this simple equation:

n = 12.2714285 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (714285) and 1 digit in the non-repeating part (2). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 122714285.714285 (equation 2)

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

10000000 × n = 122714285.714285
      10 × n = 12.2714285
 9999990 × n = 122714163

1227141639999990 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 142857 (the GCF - greatest common factor).

n = 1227141639999990 = 122714163 ÷ 1428579999990 ÷ 142857 = 85970. So,

12.2714285 = 85970 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 85970 is also equal to 121970 when expressed as a mixed number.

The repeating decimal 12.2714285 (vinculum notation) has a repeated block length of 6. It is also represented as 12.2714285714285714285... (ellipsis notation) which equals approximately 12.2714285714285714285714285714285 (decimal approximation)(*).

The recurring decimal 12.2714285 can be written as a ratio of two integers having 859 as the numerator and 70 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.