What is 10.5714571457145714... as a Fraction?

10.5714 equals 1057049999 as a fraction.

How do you turn 10.5714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.5714 repeating into a fraction, begin writing this simple equation:

n = 10.5714 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (5714), so multiply both sides by 10⁴ = 10000.

10000 × n = 105714.5714 (equation 2)

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

10000 × n = 105714.5714
    1 × n = 10.5714
 9999 × n = 105704

n = 1057049999 (answer)

10.5714 = 1057049999 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 1057049999 is also equal to 1057149999 when expressed as a mixed number.

The repeating decimal 10.5714 (vinculum notation) has a repeated block length of 4. It is also represented as 10.571457145714... (ellipsis notation) which equals approximately 10.57145714571457145714 (decimal approximation)(*).

The recurring decimal 10.5714 can be written as a ratio of two integers having 105704 as the numerator and 9999 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.