What is 10.165714257142571425714257142... as a Fraction?

10.1657142 equals 508280634999950 as a fraction.

How do you turn 10.1657142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 10.1657142 repeating into a fraction, begin writing this simple equation:

n = 10.1657142 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (57142) and 2 digits in the non-repeating part (16). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 101657142.57142 (equation 2)

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

10000000 × n = 101657142.57142
     100 × n = 10.1657142
 9999900 × n = 101656126

1016561269999900 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 2 (the GCF - greatest common factor).

n = 1016561269999900 = 101656126 ÷ 29999900 ÷ 2 = 508280634999950. So,

10.1657142 = 508280634999950 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 508280634999950 is also equal to 108285634999950 when expressed as a mixed number.

The repeating decimal 10.1657142 (vinculum notation) has a repeated block length of 5. It is also represented as 10.16571425714257142... (ellipsis notation) which equals approximately 10.165714257142571425714257142 (decimal approximation)(*).

The recurring decimal 10.1657142 can be written as a ratio of two integers having 50828063 as the numerator and 4999950 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.