What is 20.07142714271427142... as a Fraction?

20.7142 equals 100347149995 as a fraction.

How do you turn 20.7142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.7142 repeating into a fraction, begin writing this simple equation:

n = 20.7142 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (7142), so multiply both sides by 10⁴ = 10000.

10000 × n = 207142.7142 (equation 2)

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

10000 × n = 207142.7142
    1 × n = 20.7142
 9999 × n = 207122

n = 100347149995 (answer)

20.7142 = 100347149995 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 100347149995 is also equal to 20357149995 when expressed as a mixed number.

The repeating decimal 20.7142 (vinculum notation) has a repeated block length of 4. It is also represented as 20.714271427142... (ellipsis notation) which equals approximately 20.71427142714271427142 (decimal approximation)(*).

The recurring decimal 20.7142 can be written as a ratio of two integers having 1003471 as the numerator and 49995 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.