What is 3.12142857142857142857142857142857142857142857... as a Fraction?

3.12142857 equals 437140 as a fraction.

How do you turn 3.12142857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.12142857 repeating into a fraction, begin writing this simple equation:

n = 3.12142857 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (142857) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 312142857.142857 (equation 2)

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

100000000 × n = 312142857.142857
      100 × n = 3.12142857
 99999900 × n = 312142545

31214254599999900 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 714285 (the GCF - greatest common factor).

n = 31214254599999900 = 312142545 ÷ 71428599999900 ÷ 714285 = 437140. So,

3.12142857 = 437140 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 437140 is also equal to 317140 when expressed as a mixed number.

The repeating decimal 3.12142857 (vinculum notation) has a repeated block length of 6. It is also represented as 3.12142857142857142857... (ellipsis notation) which equals approximately 3.12142857142857142857142857142857 (decimal approximation)(*).

The recurring decimal 3.12142857 can be written as a ratio of two integers having 437 as the numerator and 140 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.