What is 13.151428514285142851428514285... as a Fraction?

13.1514285 equals 13151297999990 as a fraction.

How do you turn 13.1514285 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.1514285 repeating into a fraction, begin writing this simple equation:

n = 13.1514285 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (14285) and 2 digits in the non-repeating part (15). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 131514285.14285 (equation 2)

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

10000000 × n = 131514285.14285
     100 × n = 13.1514285
 9999900 × n = 131512970

1315129709999900 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 = 1315129709999900 = 131512970 ÷ 109999900 ÷ 10 = 13151297999990. So,

13.1514285 = 13151297999990 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 13151297999990 is also equal to 13151427999990 when expressed as a mixed number.

The repeating decimal 13.1514285 (vinculum notation) has a repeated block length of 5. It is also represented as 13.15142851428514285... (ellipsis notation) which equals approximately 13.151428514285142851428514285 (decimal approximation)(*).

The recurring decimal 13.1514285 can be written as a ratio of two integers having 13151297 as the numerator and 999990 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.