What is 13.85714571457145714... as a Fraction?

13.85714 equals 69278849995 as a fraction.

How do you turn 13.85714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.85714 repeating into a fraction, begin writing this simple equation:

n = 13.85714 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (5714) and 1 digit in the non-repeating part (8). Multiply both sides by 10⁵ = 100000.

100000 × n = 1385714.5714 (equation 2)

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

100000 × n = 1385714.5714
    10 × n = 13.85714
 99990 × n = 1385576

138557699990 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 = 138557699990 = 1385576 ÷ 299990 ÷ 2 = 69278849995. So,

13.85714 = 69278849995 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 69278849995 is also equal to 134285349995 when expressed as a mixed number.

The repeating decimal 13.85714 (vinculum notation) has a repeated block length of 4. It is also represented as 13.8571457145714... (ellipsis notation) which equals approximately 13.857145714571457145714 (decimal approximation)(*).

The recurring decimal 13.85714 can be written as a ratio of two integers having 692788 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.