What is 12.12714714714... as a Fraction?

12.12714 equals 20191716650 as a fraction.

How do you turn 12.12714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.12714 repeating into a fraction, begin writing this simple equation:

n = 12.12714 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (714) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁵ = 100000.

100000 × n = 1212714.714 (equation 2)

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

100000 × n = 1212714.714
   100 × n = 12.12714
 99900 × n = 1211502

121150299900 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 6 (the GCF - greatest common factor).

n = 121150299900 = 1211502 ÷ 699900 ÷ 6 = 20191716650. So,

12.12714 = 20191716650 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 20191716650 is also equal to 12211716650 when expressed as a mixed number.

The repeating decimal 12.12714 (vinculum notation) has a repeated block length of 3. It is also represented as 12.12714714714... (ellipsis notation) which equals approximately 12.12714714714714714 (decimal approximation)(*).

The recurring decimal 12.12714 can be written as a ratio of two integers having 201917 as the numerator and 16650 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.