What is 20.13142142142... as a Fraction?

20.13142 equals 201112999900 as a fraction.

How do you turn 20.13142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.13142 repeating into a fraction, begin writing this simple equation:

n = 20.13142 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (142) and 2 digits in the non-repeating part (13). Multiply both sides by 10⁵ = 100000.

100000 × n = 2013142.142 (equation 2)

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

100000 × n = 2013142.142
   100 × n = 20.13142
 99900 × n = 2011129

n = 201112999900 (answer)

20.13142 = 201112999900 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 201112999900 is also equal to 201312999900 when expressed as a mixed number.

The repeating decimal 20.13142 (vinculum notation) has a repeated block length of 3. It is also represented as 20.13142142142... (ellipsis notation) which equals approximately 20.13142142142142142 (decimal approximation)(*).

The recurring decimal 20.13142 can be written as a ratio of two integers having 2011129 as the numerator and 99900 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.