What is 13.214615461546154615... as a Fraction?

13.214615 equals 6606647499950 as a fraction.

How do you turn 13.214615 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.214615 repeating into a fraction, begin writing this simple equation:

n = 13.214615 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (4615) and 2 digits in the non-repeating part (21). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 13214615.4615 (equation 2)

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

1000000 × n = 13214615.4615
    100 × n = 13.214615
 999900 × n = 13213294

13213294999900 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 = 13213294999900 = 13213294 ÷ 2999900 ÷ 2 = 6606647499950. So,

13.214615 = 6606647499950 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 6606647499950 is also equal to 13107297499950 when expressed as a mixed number.

The repeating decimal 13.214615 (vinculum notation) has a repeated block length of 4. It is also represented as 13.21461546154615... (ellipsis notation) which equals approximately 13.2146154615461546154615 (decimal approximation)(*).

The recurring decimal 13.214615 can be written as a ratio of two integers having 6606647 as the numerator and 499950 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.