What is 15.8131313... as a Fraction?

15.813 equals 3131198 as a fraction.

How do you turn 15.813 repeating into a fraction?

Detailed Answer:

Step 1: To convert 15.813 repeating into a fraction, begin writing this simple equation:

n = 15.813 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (13) and 1 digit in the non-repeating part (8). Multiply both sides by 10³ = 1000.

1000 × n = 15813.13 (equation 2)

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

1000 × n = 15813.13
  10 × n = 15.813
 990 × n = 15655

15655990 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 5 (the GCF - greatest common factor).

n = 15655990 = 15655 ÷ 5990 ÷ 5 = 3131198. So,

15.813 = 3131198 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 3131198 is also equal to 15161198 when expressed as a mixed number.

The repeating decimal 15.813 (vinculum notation) has a repeated block length of 2. It is also represented as 15.8131313... (ellipsis notation) which equals approximately 15.81313131313 (decimal approximation)(*).

The recurring decimal 15.813 can be written as a ratio of two integers having 3131 as the numerator and 198 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.