What is 3.0857142857142857142857142857142857142... as a Fraction?

3.857142 equals 10835 as a fraction.

How do you turn 3.857142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.857142 repeating into a fraction, begin writing this simple equation:

n = 3.857142 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (857142), so multiply both sides by 10⁶ = 1000000.

1000000 × n = 3857142.857142 (equation 2)

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

1000000 × n = 3857142.857142
      1 × n = 3.857142
 999999 × n = 3857139

3857139999999 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 = 3857139999999 = 3857139 ÷ 142857999999 ÷ 142857 = 10835. So,

3.857142 = 10835 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 10835 is also equal to 3335 when expressed as a mixed number.

The repeating decimal 3.857142 (vinculum notation) has a repeated block length of 6. It is also represented as 3.857142857142857142... (ellipsis notation) which equals approximately 3.857142857142857142857142857142 (decimal approximation)(*).

The recurring decimal 3.857142 can be written as a ratio of two integers having 108 as the numerator and 35 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.