What is 11.11142857142857142857142857142857142857142857... as a Fraction?

11.11142857 equals 3889350 as a fraction.

How do you turn 11.11142857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 11.11142857 repeating into a fraction, begin writing this simple equation:

n = 11.11142857 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (142857) and 2 digits in the non-repeating part (11). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 1111142857.142857 (equation 2)

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

100000000 × n = 1111142857.142857
      100 × n = 11.11142857
 99999900 × n = 1111141746

111114174699999900 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 285714 (the GCF - greatest common factor).

n = 111114174699999900 = 1111141746 ÷ 28571499999900 ÷ 285714 = 3889350. So,

11.11142857 = 3889350 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 3889350 is also equal to 1139350 when expressed as a mixed number.

The repeating decimal 11.11142857 (vinculum notation) has a repeated block length of 6. It is also represented as 11.11142857142857142857... (ellipsis notation) which equals approximately 11.11142857142857142857142857142857 (decimal approximation)(*).

The recurring decimal 11.11142857 can be written as a ratio of two integers having 3889 as the numerator and 350 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.