What is 8.135757... as a Fraction?

8.1357 equals 6712825 as a fraction.

How do you turn 8.1357 repeating into a fraction?

Detailed Answer:

Step 1: To convert 8.1357 repeating into a fraction, begin writing this simple equation:

n = 8.1357 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (57) and 2 digits in the non-repeating part (13). Multiply both sides by 10⁴ = 10000.

10000 × n = 81357.57 (equation 2)

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

10000 × n = 81357.57
  100 × n = 8.1357
 9900 × n = 80544

805449900 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 12 (the GCF - greatest common factor).

n = 805449900 = 80544 ÷ 129900 ÷ 12 = 6712825. So,

8.1357 = 6712825 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 6712825 is also equal to 8112825 when expressed as a mixed number.

The repeating decimal 8.1357 (vinculum notation) has a repeated block length of 2. It is also represented as 8.13575757... (ellipsis notation) which equals approximately 8.135757575757 (decimal approximation)(*).

The recurring decimal 8.1357 can be written as a ratio of two integers having 6712 as the numerator and 825 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.