What is 16.68571857185718571... as a Fraction?

16.68571 equals 1112276666 as a fraction.

How do you turn 16.68571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 16.68571 repeating into a fraction, begin writing this simple equation:

n = 16.68571 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (8571) and 1 digit in the non-repeating part (6). Multiply both sides by 10⁵ = 100000.

100000 × n = 1668571.8571 (equation 2)

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

100000 × n = 1668571.8571
    10 × n = 16.68571
 99990 × n = 1668405

166840599990 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 15 (the GCF - greatest common factor).

n = 166840599990 = 1668405 ÷ 1599990 ÷ 15 = 1112276666. So,

16.68571 = 1112276666 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 1112276666 is also equal to 1645716666 when expressed as a mixed number.

The repeating decimal 16.68571 (vinculum notation) has a repeated block length of 4. It is also represented as 16.6857185718571... (ellipsis notation) which equals approximately 16.685718571857185718571 (decimal approximation)(*).

The recurring decimal 16.68571 can be written as a ratio of two integers having 111227 as the numerator and 6666 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.