What is 3.12142142142... as a Fraction?

3.12142 equals 311839990 as a fraction.

How do you turn 3.12142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.12142 repeating into a fraction, begin writing this simple equation:

n = 3.12142 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (142) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁵ = 100000.

100000 × n = 312142.142 (equation 2)

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

100000 × n = 312142.142
   100 × n = 3.12142
 99900 × n = 311830

31183099900 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 10 (the GCF - greatest common factor).

n = 31183099900 = 311830 ÷ 1099900 ÷ 10 = 311839990. So,

3.12142 = 311839990 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 311839990 is also equal to 312139990 when expressed as a mixed number.

The repeating decimal 3.12142 (vinculum notation) has a repeated block length of 3. It is also represented as 3.12142142142... (ellipsis notation) which equals approximately 3.12142142142142142 (decimal approximation)(*).

The recurring decimal 3.12142 can be written as a ratio of two integers having 31183 as the numerator and 9990 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.