What is 10.9857142857142857142857142857142857142... as a Fraction?

10.9857142 equals 76970 as a fraction.

How do you turn 10.9857142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.9857142 repeating into a fraction, begin writing this simple equation:

n = 10.9857142 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (857142) and 1 digit in the non-repeating part (9). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 109857142.857142 (equation 2)

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

10000000 × n = 109857142.857142
      10 × n = 10.9857142
 9999990 × n = 109857033

1098570339999990 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 = 1098570339999990 = 109857033 ÷ 1428579999990 ÷ 142857 = 76970. So,

10.9857142 = 76970 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 76970 is also equal to 106970 when expressed as a mixed number.

The repeating decimal 10.9857142 (vinculum notation) has a repeated block length of 6. It is also represented as 10.9857142857142857142... (ellipsis notation) which equals approximately 10.9857142857142857142857142857142 (decimal approximation)(*).

The recurring decimal 10.9857142 can be written as a ratio of two integers having 769 as the numerator and 70 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.