What is 13.71538153815381538... as a Fraction?

13.71538 equals 137140199990 as a fraction.

How do you turn 13.71538 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.71538 repeating into a fraction, begin writing this simple equation:

n = 13.71538 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (1538) and 1 digit in the non-repeating part (7). Multiply both sides by 10⁵ = 100000.

100000 × n = 1371538.1538 (equation 2)

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

100000 × n = 1371538.1538
    10 × n = 13.71538
 99990 × n = 1371401

n = 137140199990 (answer)

13.71538 = 137140199990 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 137140199990 is also equal to 137153199990 when expressed as a mixed number.

The repeating decimal 13.71538 (vinculum notation) has a repeated block length of 4. It is also represented as 13.7153815381538... (ellipsis notation) which equals approximately 13.715381538153815381538 (decimal approximation)(*).

The recurring decimal 13.71538 can be written as a ratio of two integers having 1371401 as the numerator and 99990 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.