What is 13.0384384384... as a Fraction?

13.384 equals 217091665 as a fraction.

How do you turn 13.384 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.384 repeating into a fraction, begin writing this simple equation:

n = 13.384 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (384), so multiply both sides by 10³ = 1000.

1000 × n = 13384.384 (equation 2)

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

1000 × n = 13384.384
   1 × n = 13.384
 999 × n = 13371

13371999 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 = 13371999 = 13371 ÷ 3999 ÷ 3 = 217091665. So,

13.384 = 217091665 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 217091665 is also equal to 13641665 when expressed as a mixed number.

The repeating decimal 13.384 (vinculum notation) has a repeated block length of 3. It is also represented as 13.384384384... (ellipsis notation) which equals approximately 13.384384384384384 (decimal approximation)(*).

The recurring decimal 13.384 can be written as a ratio of two integers having 21709 as the numerator and 1665 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.