What is 11.111414... as a Fraction?

11.1114 equals 1100039900 as a fraction.

How do you turn 11.1114 repeating into a fraction?

Detailed Answer:

Step 1: To convert 11.1114 repeating into a fraction, begin writing this simple equation:

n = 11.1114 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (14) and 2 digits in the non-repeating part (11). Multiply both sides by 10⁴ = 10000.

10000 × n = 111114.14 (equation 2)

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

10000 × n = 111114.14
  100 × n = 11.1114
 9900 × n = 110003

n = 1100039900 (answer)

11.1114 = 1100039900 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 1100039900 is also equal to 1111039900 when expressed as a mixed number.

The repeating decimal 11.1114 (vinculum notation) has a repeated block length of 2. It is also represented as 11.11141414... (ellipsis notation) which equals approximately 11.111414141414 (decimal approximation)(*).

The recurring decimal 11.1114 can be written as a ratio of two integers having 110003 as the numerator and 9900 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.