What is 13.125714257142571425714257142... as a Fraction?

13.1257142 equals 13125583999990 as a fraction.

How do you turn 13.1257142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.1257142 repeating into a fraction, begin writing this simple equation:

n = 13.1257142 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (57142) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 131257142.57142 (equation 2)

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

10000000 × n = 131257142.57142
     100 × n = 13.1257142
 9999900 × n = 131255830

1312558309999900 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 10 (the GCF - greatest common factor).

n = 1312558309999900 = 131255830 ÷ 109999900 ÷ 10 = 13125583999990. So,

13.1257142 = 13125583999990 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 13125583999990 is also equal to 13125713999990 when expressed as a mixed number.

The repeating decimal 13.1257142 (vinculum notation) has a repeated block length of 5. It is also represented as 13.12571425714257142... (ellipsis notation) which equals approximately 13.125714257142571425714257142 (decimal approximation)(*).

The recurring decimal 13.1257142 can be written as a ratio of two integers having 13125583 as the numerator and 999990 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.