What is 3.131538153815381538... as a Fraction?

3.131538 equals 12524939996 as a fraction.

How do you turn 3.131538 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.131538 repeating into a fraction, begin writing this simple equation:

n = 3.131538 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (1538) and 2 digits in the non-repeating part (13). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 3131538.1538 (equation 2)

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

1000000 × n = 3131538.1538
    100 × n = 3.131538
 999900 × n = 3131225

3131225999900 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 25 (the GCF - greatest common factor).

n = 3131225999900 = 3131225 ÷ 25999900 ÷ 25 = 12524939996. So,

3.131538 = 12524939996 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 12524939996 is also equal to 3526139996 when expressed as a mixed number.

The repeating decimal 3.131538 (vinculum notation) has a repeated block length of 4. It is also represented as 3.13153815381538... (ellipsis notation) which equals approximately 3.1315381538153815381538 (decimal approximation)(*).

The recurring decimal 3.131538 can be written as a ratio of two integers having 125249 as the numerator and 39996 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.