What is 16.34285742857428574285742857... as a Fraction?

16.342857 equals 8171347499995 as a fraction.

How do you turn 16.342857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 16.342857 repeating into a fraction, begin writing this simple equation:

n = 16.342857 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (42857) and 1 digit in the non-repeating part (3). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 16342857.42857 (equation 2)

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

1000000 × n = 16342857.42857
     10 × n = 16.342857
 999990 × n = 16342694

16342694999990 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 2 (the GCF - greatest common factor).

n = 16342694999990 = 16342694 ÷ 2999990 ÷ 2 = 8171347499995. So,

16.342857 = 8171347499995 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 8171347499995 is also equal to 16171427499995 when expressed as a mixed number.

The repeating decimal 16.342857 (vinculum notation) has a repeated block length of 5. It is also represented as 16.3428574285742857... (ellipsis notation) which equals approximately 16.34285742857428574285742857 (decimal approximation)(*).

The recurring decimal 16.342857 can be written as a ratio of two integers having 8171347 as the numerator and 499995 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.