What is 16.17142142142... as a Fraction?

16.17142 equals 646213996 as a fraction.

How do you turn 16.17142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 16.17142 repeating into a fraction, begin writing this simple equation:

n = 16.17142 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (142) and 2 digits in the non-repeating part (17). Multiply both sides by 10⁵ = 100000.

100000 × n = 1617142.142 (equation 2)

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

100000 × n = 1617142.142
   100 × n = 16.17142
 99900 × n = 1615525

161552599900 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 25 (the GCF - greatest common factor).

n = 161552599900 = 1615525 ÷ 2599900 ÷ 25 = 646213996. So,

16.17142 = 646213996 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 646213996 is also equal to 166853996 when expressed as a mixed number.

The repeating decimal 16.17142 (vinculum notation) has a repeated block length of 3. It is also represented as 16.17142142142... (ellipsis notation) which equals approximately 16.17142142142142142 (decimal approximation)(*).

The recurring decimal 16.17142 can be written as a ratio of two integers having 64621 as the numerator and 3996 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.