What is 16.4285742857428574285742857... as a Fraction?

16.42857 equals 164284199999 as a fraction.

How do you turn 16.42857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 16.42857 repeating into a fraction, begin writing this simple equation:

n = 16.42857 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (42857), so multiply both sides by 10⁵ = 100000.

100000 × n = 1642857.42857 (equation 2)

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

100000 × n = 1642857.42857
     1 × n = 16.42857
 99999 × n = 1642841

n = 164284199999 (answer)

16.42857 = 164284199999 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 164284199999 is also equal to 164285799999 when expressed as a mixed number.

The repeating decimal 16.42857 (vinculum notation) has a repeated block length of 5. It is also represented as 16.428574285742857... (ellipsis notation) which equals approximately 16.4285742857428574285742857 (decimal approximation)(*).

The recurring decimal 16.42857 can be written as a ratio of two integers having 1642841 as the numerator and 99999 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.