What is 13.1118181818... as a Fraction?

13.1118 equals 144231100 as a fraction.

How do you turn 13.1118 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.1118 repeating into a fraction, begin writing this simple equation:

n = 13.1118 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (18) and 2 digits in the non-repeating part (11). Multiply both sides by 10⁴ = 10000.

10000 × n = 131118.18 (equation 2)

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

10000 × n = 131118.18
  100 × n = 13.1118
 9900 × n = 129807

1298079900 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 9 (the GCF - greatest common factor).

n = 1298079900 = 129807 ÷ 99900 ÷ 9 = 144231100. So,

13.1118 = 144231100 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 144231100 is also equal to 131231100 when expressed as a mixed number.

The repeating decimal 13.1118 (vinculum notation) has a repeated block length of 2. It is also represented as 13.11181818... (ellipsis notation) which equals approximately 13.111818181818 (decimal approximation)(*).

The recurring decimal 13.1118 can be written as a ratio of two integers having 14423 as the numerator and 1100 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.