What is 0.16857142857142857142857142857142857142... as a Fraction?

0.16857142 equals 59350 as a fraction.

How do you turn 0.16857142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.16857142 repeating into a fraction, begin writing this simple equation:

n = 0.16857142 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (857142) and 2 digits in the non-repeating part (16). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 16857142.857142 (equation 2)

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

100000000 × n = 16857142.857142
      100 × n = 0.16857142
 99999900 × n = 16857126

1685712699999900 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 285714 (the GCF - greatest common factor).

n = 1685712699999900 = 16857126 ÷ 28571499999900 ÷ 285714 = 59350. So,

0.16857142 = 59350 as the lowest possible fraction.

The repeating decimal 0.16857142 (vinculum notation) has a repeated block length of 6. It is also represented as 0.16857142857142857142... (ellipsis notation) which equals approximately 0.16857142857142857142857142857142 (decimal approximation)(*).

The recurring decimal 0.16857142 can be written as a ratio of two integers having 59 as the numerator and 350 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.