What is 13.1428571428571428571428571428571428571... as a Fraction?

13.1428571 equals 927 as a fraction.

How do you turn 13.1428571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.1428571 repeating into a fraction, begin writing this simple equation:

n = 13.1428571 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (428571) and 1 digit in the non-repeating part (1). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 131428571.428571 (equation 2)

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

10000000 × n = 131428571.428571
      10 × n = 13.1428571
 9999990 × n = 131428440

1314284409999990 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 = 1314284409999990 = 131428440 ÷ 14285709999990 ÷ 1428570 = 927. So,

13.1428571 = 927 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 927 is also equal to 1317 when expressed as a mixed number.

The repeating decimal 13.1428571 (vinculum notation) has a repeated block length of 6. It is also represented as 13.1428571428571428571... (ellipsis notation) which equals approximately 13.1428571428571428571428571428571 (decimal approximation)(*).

The recurring decimal 13.1428571 can be written as a ratio of two integers having 92 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.