What is 12.175714257142571425714257142... as a Fraction?

12.1757142 equals 4870237399996 as a fraction.

How do you turn 12.1757142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.1757142 repeating into a fraction, begin writing this simple equation:

n = 12.1757142 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (57142) and 2 digits in the non-repeating part (17). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 121757142.57142 (equation 2)

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

10000000 × n = 121757142.57142
     100 × n = 12.1757142
 9999900 × n = 121755925

1217559259999900 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 25 (the GCF - greatest common factor).

n = 1217559259999900 = 121755925 ÷ 259999900 ÷ 25 = 4870237399996. So,

12.1757142 = 4870237399996 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 4870237399996 is also equal to 1270285399996 when expressed as a mixed number.

The repeating decimal 12.1757142 (vinculum notation) has a repeated block length of 5. It is also represented as 12.17571425714257142... (ellipsis notation) which equals approximately 12.175714257142571425714257142 (decimal approximation)(*).

The recurring decimal 12.1757142 can be written as a ratio of two integers having 4870237 as the numerator and 399996 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.