What is 13.168571857185718571... as a Fraction?

13.168571 equals 87781766660 as a fraction.

How do you turn 13.168571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.168571 repeating into a fraction, begin writing this simple equation:

n = 13.168571 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (8571) and 2 digits in the non-repeating part (16). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 13168571.8571 (equation 2)

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

1000000 × n = 13168571.8571
    100 × n = 13.168571
 999900 × n = 13167255

13167255999900 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 = 13167255999900 = 13167255 ÷ 15999900 ÷ 15 = 87781766660. So,

13.168571 = 87781766660 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 87781766660 is also equal to 131123766660 when expressed as a mixed number.

The repeating decimal 13.168571 (vinculum notation) has a repeated block length of 4. It is also represented as 13.16857185718571... (ellipsis notation) which equals approximately 13.1685718571857185718571 (decimal approximation)(*).

The recurring decimal 13.168571 can be written as a ratio of two integers having 877817 as the numerator and 66660 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.