What is 11.12142857142857142857142857142857142857142857... as a Fraction?

11.12142857 equals 1557140 as a fraction.

How do you turn 11.12142857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 11.12142857 repeating into a fraction, begin writing this simple equation:

n = 11.12142857 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (142857) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 1112142857.142857 (equation 2)

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

100000000 × n = 1112142857.142857
      100 × n = 11.12142857
 99999900 × n = 1112141745

111214174599999900 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 714285 (the GCF - greatest common factor).

n = 111214174599999900 = 1112141745 ÷ 71428599999900 ÷ 714285 = 1557140. So,

11.12142857 = 1557140 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 1557140 is also equal to 1117140 when expressed as a mixed number.

The repeating decimal 11.12142857 (vinculum notation) has a repeated block length of 6. It is also represented as 11.12142857142857142857... (ellipsis notation) which equals approximately 11.12142857142857142857142857142857 (decimal approximation)(*).

The recurring decimal 11.12142857 can be written as a ratio of two integers having 1557 as the numerator and 140 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.