What is 10.142857142857142857142857142857142857142857... as a Fraction?

10.142857 equals 717 as a fraction.

How do you turn 10.142857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.142857 repeating into a fraction, begin writing this simple equation:

n = 10.142857 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (142857), so multiply both sides by 10⁶ = 1000000.

1000000 × n = 10142857.142857 (equation 2)

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

1000000 × n = 10142857.142857
      1 × n = 10.142857
 999999 × n = 10142847

10142847999999 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 = 10142847999999 = 10142847 ÷ 142857999999 ÷ 142857 = 717. So,

10.142857 = 717 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 717 is also equal to 1017 when expressed as a mixed number.

The repeating decimal 10.142857 (vinculum notation) has a repeated block length of 6. It is also represented as 10.142857142857142857... (ellipsis notation) which equals approximately 10.142857142857142857142857142857 (decimal approximation)(*).

The recurring decimal 10.142857 can be written as a ratio of two integers having 71 as the numerator and 7 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.