What is 4.17857142857142857142857142857142857142... as a Fraction?

4.17857142 equals 11728 as a fraction.

How do you turn 4.17857142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 4.17857142 repeating into a fraction, begin writing this simple equation:

n = 4.17857142 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (857142) and 2 digits in the non-repeating part (17). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 417857142.857142 (equation 2)

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

100000000 × n = 417857142.857142
      100 × n = 4.17857142
 99999900 × n = 417856725

41785672599999900 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 3571425 (the GCF - greatest common factor).

n = 41785672599999900 = 417856725 ÷ 357142599999900 ÷ 3571425 = 11728. So,

4.17857142 = 11728 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 11728 is also equal to 4528 when expressed as a mixed number.

The repeating decimal 4.17857142 (vinculum notation) has a repeated block length of 6. It is also represented as 4.17857142857142857142... (ellipsis notation) which equals approximately 4.17857142857142857142857142857142 (decimal approximation)(*).

The recurring decimal 4.17857142 can be written as a ratio of two integers having 117 as the numerator and 28 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.