What is 5.12714285714285714285714285714285714285... as a Fraction?

5.12714285 equals 3589700 as a fraction.

How do you turn 5.12714285 repeating into a fraction?

Detailed Answer:

Step 1: To convert 5.12714285 repeating into a fraction, begin writing this simple equation:

n = 5.12714285 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (714285) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 512714285.714285 (equation 2)

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

100000000 × n = 512714285.714285
      100 × n = 5.12714285
 99999900 × n = 512713773

51271377399999900 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 142857 (the GCF - greatest common factor).

n = 51271377399999900 = 512713773 ÷ 14285799999900 ÷ 142857 = 3589700. So,

5.12714285 = 3589700 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 3589700 is also equal to 589700 when expressed as a mixed number.

The repeating decimal 5.12714285 (vinculum notation) has a repeated block length of 6. It is also represented as 5.12714285714285714285... (ellipsis notation) which equals approximately 5.12714285714285714285714285714285 (decimal approximation)(*).

The recurring decimal 5.12714285 can be written as a ratio of two integers having 3589 as the numerator and 700 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.