What is 10.12142857142857142857142857142857142857142857... as a Fraction?

10.12142857 equals 1417140 as a fraction.

How do you turn 10.12142857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.12142857 repeating into a fraction, begin writing this simple equation:

n = 10.12142857 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (142857) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 1012142857.142857 (equation 2)

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

100000000 × n = 1012142857.142857
      100 × n = 10.12142857
 99999900 × n = 1012141845

101214184599999900 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 = 101214184599999900 = 1012141845 ÷ 71428599999900 ÷ 714285 = 1417140. So,

10.12142857 = 1417140 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 1417140 is also equal to 1017140 when expressed as a mixed number.

The repeating decimal 10.12142857 (vinculum notation) has a repeated block length of 6. It is also represented as 10.12142857142857142857... (ellipsis notation) which equals approximately 10.12142857142857142857142857142857 (decimal approximation)(*).

The recurring decimal 10.12142857 can be written as a ratio of two integers having 1417 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.