What is 11.111616161616... as a Fraction?

11.1116 equals 220011980 as a fraction.

How do you turn 11.1116 repeating into a fraction?

Detailed Answer:

Step 1: To convert 11.1116 repeating into a fraction, begin writing this simple equation:

n = 11.1116 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (16) and 2 digits in the non-repeating part (11). Multiply both sides by 10⁴ = 10000.

10000 × n = 111116.16 (equation 2)

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

10000 × n = 111116.16
  100 × n = 11.1116
 9900 × n = 110005

1100059900 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 5 (the GCF - greatest common factor).

n = 1100059900 = 110005 ÷ 59900 ÷ 5 = 220011980. So,

11.1116 = 220011980 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 220011980 is also equal to 112211980 when expressed as a mixed number.

The repeating decimal 11.1116 (vinculum notation) has a repeated block length of 2. It is also represented as 11.11161616... (ellipsis notation) which equals approximately 11.111616161616 (decimal approximation)(*).

The recurring decimal 11.1116 can be written as a ratio of two integers having 22001 as the numerator and 1980 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.