What is 11.1076923076923076923076923076923076923... as a Fraction?

11.1076923 equals 72265 as a fraction.

How do you turn 11.1076923 repeating into a fraction?

Detailed Answer:

Step 1: To convert 11.1076923 repeating into a fraction, begin writing this simple equation:

n = 11.1076923 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (076923) and 1 digit in the non-repeating part (1). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 111076923.076923 (equation 2)

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

10000000 × n = 111076923.076923
      10 × n = 11.1076923
 9999990 × n = 111076812

1110768129999990 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 153846 (the GCF - greatest common factor).

n = 1110768129999990 = 111076812 ÷ 1538469999990 ÷ 153846 = 72265. So,

11.1076923 = 72265 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 72265 is also equal to 11765 when expressed as a mixed number.

The repeating decimal 11.1076923 (vinculum notation) has a repeated block length of 6. It is also represented as 11.1076923076923076923... (ellipsis notation) which equals approximately 11.1076923076923076923076923076923 (decimal approximation)(*).

The recurring decimal 11.1076923 can be written as a ratio of two integers having 722 as the numerator and 65 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.