What is 12.5714257142571425714257142... as a Fraction?

12.57142 equals 125713099999 as a fraction.

How do you turn 12.57142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.57142 repeating into a fraction, begin writing this simple equation:

n = 12.57142 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (57142), so multiply both sides by 10⁵ = 100000.

100000 × n = 1257142.57142 (equation 2)

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

100000 × n = 1257142.57142
     1 × n = 12.57142
 99999 × n = 1257130

n = 125713099999 (answer)

12.57142 = 125713099999 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 125713099999 is also equal to 125714299999 when expressed as a mixed number.

The repeating decimal 12.57142 (vinculum notation) has a repeated block length of 5. It is also represented as 12.571425714257142... (ellipsis notation) which equals approximately 12.5714257142571425714257142 (decimal approximation)(*).

The recurring decimal 12.57142 can be written as a ratio of two integers having 1257130 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.