What is 10.8571485714857148571485714... as a Fraction?

10.85714 equals 108570499999 as a fraction.

How do you turn 10.85714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.85714 repeating into a fraction, begin writing this simple equation:

n = 10.85714 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (85714), so multiply both sides by 10⁵ = 100000.

100000 × n = 1085714.85714 (equation 2)

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

100000 × n = 1085714.85714
     1 × n = 10.85714
 99999 × n = 1085704

n = 108570499999 (answer)

10.85714 = 108570499999 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 108570499999 is also equal to 108571499999 when expressed as a mixed number.

The repeating decimal 10.85714 (vinculum notation) has a repeated block length of 5. It is also represented as 10.857148571485714... (ellipsis notation) which equals approximately 10.8571485714857148571485714 (decimal approximation)(*).

The recurring decimal 10.85714 can be written as a ratio of two integers having 1085704 as the numerator and 99999 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.