What is 11.11076923076923076923076923076923076923... as a Fraction?

11.11076923 equals 3611325 as a fraction.

How do you turn 11.11076923 repeating into a fraction?

Detailed Answer:

Step 1: To convert 11.11076923 repeating into a fraction, begin writing this simple equation:

n = 11.11076923 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (076923) and 2 digits in the non-repeating part (11). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 1111076923.076923 (equation 2)

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

100000000 × n = 1111076923.076923
      100 × n = 11.11076923
 99999900 × n = 1111075812

111107581299999900 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 307692 (the GCF - greatest common factor).

n = 111107581299999900 = 1111075812 ÷ 30769299999900 ÷ 307692 = 3611325. So,

11.11076923 = 3611325 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 3611325 is also equal to 1136325 when expressed as a mixed number.

The repeating decimal 11.11076923 (vinculum notation) has a repeated block length of 6. It is also represented as 11.11076923076923076923... (ellipsis notation) which equals approximately 11.11076923076923076923076923076923 (decimal approximation)(*).

The recurring decimal 11.11076923 can be written as a ratio of two integers having 3611 as the numerator and 325 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.