What is 0.20714285714285714285714285714285714285... as a Fraction?

0.20714285 equals 29140 as a fraction.

How do you turn 0.20714285 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.20714285 repeating into a fraction, begin writing this simple equation:

n = 0.20714285 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (714285) and 2 digits in the non-repeating part (20). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 20714285.714285 (equation 2)

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

100000000 × n = 20714285.714285
      100 × n = 0.20714285
 99999900 × n = 20714265

2071426599999900 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 = 2071426599999900 = 20714265 ÷ 71428599999900 ÷ 714285 = 29140. So,

0.20714285 = 29140 as the lowest possible fraction.

The repeating decimal 0.20714285 (vinculum notation) has a repeated block length of 6. It is also represented as 0.20714285714285714285... (ellipsis notation) which equals approximately 0.20714285714285714285714285714285 (decimal approximation)(*).

The recurring decimal 0.20714285 can be written as a ratio of two integers having 29 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.