What is 5.91428142814281428... as a Fraction?

5.91428 equals 19712333330 as a fraction.

How do you turn 5.91428 repeating into a fraction?

Detailed Answer:

Step 1: To convert 5.91428 repeating into a fraction, begin writing this simple equation:

n = 5.91428 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (1428) and 1 digit in the non-repeating part (9). Multiply both sides by 10⁵ = 100000.

100000 × n = 591428.1428 (equation 2)

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

100000 × n = 591428.1428
    10 × n = 5.91428
 99990 × n = 591369

59136999990 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 3 (the GCF - greatest common factor).

n = 59136999990 = 591369 ÷ 399990 ÷ 3 = 19712333330. So,

5.91428 = 19712333330 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 19712333330 is also equal to 53047333330 when expressed as a mixed number.

The repeating decimal 5.91428 (vinculum notation) has a repeated block length of 4. It is also represented as 5.9142814281428... (ellipsis notation) which equals approximately 5.914281428142814281428 (decimal approximation)(*).

The recurring decimal 5.91428 can be written as a ratio of two integers having 197123 as the numerator and 33330 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.