What is 13.0142857142857... as a Fraction?

13.142857 equals 91170 as a fraction.

How do you turn 13.142857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.142857 repeating into a fraction, begin writing this simple equation:

n = 13.142857 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (142857), so multiply both sides by 10⁶ = 1000000.

1000000 × n = 13142857.142857 (equation 2)

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

1000000 × n = 13142857.142857
      1 × n = 13.142857
 999999 × n = 13142844

13142844999999 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 = 13142844999999 = 13142844 ÷ 142857999999 ÷ 142857 = 91170. So,

13.142857 = 91170 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 91170 is also equal to 13170 when expressed as a mixed number.

The repeating decimal 13.142857 (vinculum notation) has a repeated block length of 6. It is also represented as 13.142857142857142857... (ellipsis notation) which equals approximately 13.142857142857142857142857142857 (decimal approximation)(*).

The recurring decimal 13.142857 can be written as a ratio of two integers having 911 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.