What is 5.127142871428714287142871428... as a Fraction?

5.1271428 equals 128177292499975 as a fraction.

How do you turn 5.1271428 repeating into a fraction?

Detailed Answer:

Step 1: To convert 5.1271428 repeating into a fraction, begin writing this simple equation:

n = 5.1271428 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (71428) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 51271428.71428 (equation 2)

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

10000000 × n = 51271428.71428
     100 × n = 5.1271428
 9999900 × n = 51270916

512709169999900 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 4 (the GCF - greatest common factor).

n = 512709169999900 = 51270916 ÷ 49999900 ÷ 4 = 128177292499975. So,

5.1271428 = 128177292499975 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 128177292499975 is also equal to 53178542499975 when expressed as a mixed number.

The repeating decimal 5.1271428 (vinculum notation) has a repeated block length of 5. It is also represented as 5.12714287142871428... (ellipsis notation) which equals approximately 5.127142871428714287142871428 (decimal approximation)(*).

The recurring decimal 5.1271428 can be written as a ratio of two integers having 12817729 as the numerator and 2499975 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.