What is 12.22142142142... as a Fraction?

12.22142 equals 610464995 as a fraction.

How do you turn 12.22142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.22142 repeating into a fraction, begin writing this simple equation:

n = 12.22142 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (142) and 2 digits in the non-repeating part (22). Multiply both sides by 10⁵ = 100000.

100000 × n = 1222142.142 (equation 2)

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

100000 × n = 1222142.142
   100 × n = 12.22142
 99900 × n = 1220920

122092099900 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 20 (the GCF - greatest common factor).

n = 122092099900 = 1220920 ÷ 2099900 ÷ 20 = 610464995. So,

12.22142 = 610464995 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 610464995 is also equal to 1211064995 when expressed as a mixed number.

The repeating decimal 12.22142 (vinculum notation) has a repeated block length of 3. It is also represented as 12.22142142142... (ellipsis notation) which equals approximately 12.22142142142142142 (decimal approximation)(*).

The recurring decimal 12.22142 can be written as a ratio of two integers having 61046 as the numerator and 4995 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.