What is 0.208571857185718571... as a Fraction?

0.208571 equals 69517333300 as a fraction.

How do you turn 0.208571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.208571 repeating into a fraction, begin writing this simple equation:

n = 0.208571 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (8571) and 2 digits in the non-repeating part (20). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 208571.8571 (equation 2)

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

1000000 × n = 208571.8571
    100 × n = 0.208571
 999900 × n = 208551

208551999900 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 3 (the GCF - greatest common factor).

n = 208551999900 = 208551 ÷ 3999900 ÷ 3 = 69517333300. So,

0.208571 = 69517333300 as the lowest possible fraction.

The repeating decimal 0.208571 (vinculum notation) has a repeated block length of 4. It is also represented as 0.20857185718571... (ellipsis notation) which equals approximately 0.2085718571857185718571 (decimal approximation)(*).

The recurring decimal 0.208571 can be written as a ratio of two integers having 69517 as the numerator and 333300 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.