What is 12.9714714714... as a Fraction?

12.9714 equals 8639666 as a fraction.

How do you turn 12.9714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.9714 repeating into a fraction, begin writing this simple equation:

n = 12.9714 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (714) and 1 digit in the non-repeating part (9). Multiply both sides by 10⁴ = 10000.

10000 × n = 129714.714 (equation 2)

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

10000 × n = 129714.714
   10 × n = 12.9714
 9990 × n = 129585

1295859990 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 15 (the GCF - greatest common factor).

n = 1295859990 = 129585 ÷ 159990 ÷ 15 = 8639666. So,

12.9714 = 8639666 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 8639666 is also equal to 12647666 when expressed as a mixed number.

The repeating decimal 12.9714 (vinculum notation) has a repeated block length of 3. It is also represented as 12.9714714714... (ellipsis notation) which equals approximately 12.9714714714714714 (decimal approximation)(*).

The recurring decimal 12.9714 can be written as a ratio of two integers having 8639 as the numerator and 666 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.