What is 11.6857142857142857142857142857142857142... as a Fraction?

11.6857142 equals 40935 as a fraction.

How do you turn 11.6857142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 11.6857142 repeating into a fraction, begin writing this simple equation:

n = 11.6857142 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (857142) and 1 digit in the non-repeating part (6). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 116857142.857142 (equation 2)

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

10000000 × n = 116857142.857142
      10 × n = 11.6857142
 9999990 × n = 116857026

1168570269999990 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 285714 (the GCF - greatest common factor).

n = 1168570269999990 = 116857026 ÷ 2857149999990 ÷ 285714 = 40935. So,

11.6857142 = 40935 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 40935 is also equal to 112435 when expressed as a mixed number.

The repeating decimal 11.6857142 (vinculum notation) has a repeated block length of 6. It is also represented as 11.6857142857142857142... (ellipsis notation) which equals approximately 11.6857142857142857142857142857142 (decimal approximation)(*).

The recurring decimal 11.6857142 can be written as a ratio of two integers having 409 as the numerator and 35 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.