What is 15.7142714271427142... as a Fraction?

15.7142 equals 1571279999 as a fraction.

How do you turn 15.7142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 15.7142 repeating into a fraction, begin writing this simple equation:

n = 15.7142 (equation 1)

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

10000 × n = 157142.7142 (equation 2)

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

10000 × n = 157142.7142
    1 × n = 15.7142
 9999 × n = 157127

n = 1571279999 (answer)

15.7142 = 1571279999 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 1571279999 is also equal to 1571429999 when expressed as a mixed number.

The repeating decimal 15.7142 (vinculum notation) has a repeated block length of 4. It is also represented as 15.714271427142... (ellipsis notation) which equals approximately 15.71427142714271427142 (decimal approximation)(*).

The recurring decimal 15.7142 can be written as a ratio of two integers having 157127 as the numerator and 9999 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.