What is 11.181538153815381538... as a Fraction?

11.181538 equals 55902149995 as a fraction.

How do you turn 11.181538 repeating into a fraction?

Detailed Answer:

Step 1: To convert 11.181538 repeating into a fraction, begin writing this simple equation:

n = 11.181538 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (1538) and 2 digits in the non-repeating part (18). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 11181538.1538 (equation 2)

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

1000000 × n = 11181538.1538
    100 × n = 11.181538
 999900 × n = 11180420

11180420999900 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 20 (the GCF - greatest common factor).

n = 11180420999900 = 11180420 ÷ 20999900 ÷ 20 = 55902149995. So,

11.181538 = 55902149995 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 55902149995 is also equal to 11907649995 when expressed as a mixed number.

The repeating decimal 11.181538 (vinculum notation) has a repeated block length of 4. It is also represented as 11.18153815381538... (ellipsis notation) which equals approximately 11.1815381538153815381538 (decimal approximation)(*).

The recurring decimal 11.181538 can be written as a ratio of two integers having 559021 as the numerator and 49995 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.