What is 11.3428571428571428571428571428571428571... as a Fraction?

11.3428571 equals 39735 as a fraction.

How do you turn 11.3428571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 11.3428571 repeating into a fraction, begin writing this simple equation:

n = 11.3428571 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (428571) and 1 digit in the non-repeating part (3). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 113428571.428571 (equation 2)

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

10000000 × n = 113428571.428571
      10 × n = 11.3428571
 9999990 × n = 113428458

1134284589999990 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 = 1134284589999990 = 113428458 ÷ 2857149999990 ÷ 285714 = 39735. So,

11.3428571 = 39735 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 39735 is also equal to 111235 when expressed as a mixed number.

The repeating decimal 11.3428571 (vinculum notation) has a repeated block length of 6. It is also represented as 11.3428571428571428571... (ellipsis notation) which equals approximately 11.3428571428571428571428571428571 (decimal approximation)(*).

The recurring decimal 11.3428571 can be written as a ratio of two integers having 397 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.