What is 12.32857285728572857... as a Fraction?

12.32857 equals 61636749995 as a fraction.

How do you turn 12.32857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.32857 repeating into a fraction, begin writing this simple equation:

n = 12.32857 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (2857) and 1 digit in the non-repeating part (3). Multiply both sides by 10⁵ = 100000.

100000 × n = 1232857.2857 (equation 2)

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

100000 × n = 1232857.2857
    10 × n = 12.32857
 99990 × n = 1232734

123273499990 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 2 (the GCF - greatest common factor).

n = 123273499990 = 1232734 ÷ 299990 ÷ 2 = 61636749995. So,

12.32857 = 61636749995 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 61636749995 is also equal to 121642749995 when expressed as a mixed number.

The repeating decimal 12.32857 (vinculum notation) has a repeated block length of 4. It is also represented as 12.3285728572857... (ellipsis notation) which equals approximately 12.328572857285728572857 (decimal approximation)(*).

The recurring decimal 12.32857 can be written as a ratio of two integers having 616367 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.