What is 13.22076076076... as a Fraction?

13.22076 equals 66037749950 as a fraction.

How do you turn 13.22076 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.22076 repeating into a fraction, begin writing this simple equation:

n = 13.22076 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (076) and 2 digits in the non-repeating part (22). Multiply both sides by 10⁵ = 100000.

100000 × n = 1322076.076 (equation 2)

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

100000 × n = 1322076.076
   100 × n = 13.22076
 99900 × n = 1320754

132075499900 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 2 (the GCF - greatest common factor).

n = 132075499900 = 1320754 ÷ 299900 ÷ 2 = 66037749950. So,

13.22076 = 66037749950 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 66037749950 is also equal to 131102749950 when expressed as a mixed number.

The repeating decimal 13.22076 (vinculum notation) has a repeated block length of 3. It is also represented as 13.22076076076... (ellipsis notation) which equals approximately 13.22076076076076076 (decimal approximation)(*).

The recurring decimal 13.22076 can be written as a ratio of two integers having 660377 as the numerator and 49950 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.