What is 23.111616... as a Fraction?

23.1116 equals 457611980 as a fraction.

How do you turn 23.1116 repeating into a fraction?

Detailed Answer:

Step 1: To convert 23.1116 repeating into a fraction, begin writing this simple equation:

n = 23.1116 (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 = 231116.16 (equation 2)

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

10000 × n = 231116.16
  100 × n = 23.1116
 9900 × n = 228805

2288059900 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 = 2288059900 = 228805 ÷ 59900 ÷ 5 = 457611980. So,

23.1116 = 457611980 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 457611980 is also equal to 232211980 when expressed as a mixed number.

The repeating decimal 23.1116 (vinculum notation) has a repeated block length of 2. It is also represented as 23.11161616... (ellipsis notation) which equals approximately 23.111616161616 (decimal approximation)(*).

The recurring decimal 23.1116 can be written as a ratio of two integers having 45761 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.