What is 10.135714571457145714... as a Fraction?

10.135714 equals 10134701999900 as a fraction.

How do you turn 10.135714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.135714 repeating into a fraction, begin writing this simple equation:

n = 10.135714 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (5714) and 2 digits in the non-repeating part (13). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 10135714.5714 (equation 2)

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

1000000 × n = 10135714.5714
    100 × n = 10.135714
 999900 × n = 10134701

n = 10134701999900 (answer)

10.135714 = 10134701999900 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 10134701999900 is also equal to 10135701999900 when expressed as a mixed number.

The repeating decimal 10.135714 (vinculum notation) has a repeated block length of 4. It is also represented as 10.13571457145714... (ellipsis notation) which equals approximately 10.1357145714571457145714 (decimal approximation)(*).

The recurring decimal 10.135714 can be written as a ratio of two integers having 10134701 as the numerator and 999900 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.