What is 13.33461461461... as a Fraction?

13.33461 equals 33303224975 as a fraction.

How do you turn 13.33461 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.33461 repeating into a fraction, begin writing this simple equation:

n = 13.33461 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (461) and 2 digits in the non-repeating part (33). Multiply both sides by 10⁵ = 100000.

100000 × n = 1333461.461 (equation 2)

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

100000 × n = 1333461.461
   100 × n = 13.33461
 99900 × n = 1332128

133212899900 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 4 (the GCF - greatest common factor).

n = 133212899900 = 1332128 ÷ 499900 ÷ 4 = 33303224975. So,

13.33461 = 33303224975 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 33303224975 is also equal to 13835724975 when expressed as a mixed number.

The repeating decimal 13.33461 (vinculum notation) has a repeated block length of 3. It is also represented as 13.33461461461... (ellipsis notation) which equals approximately 13.33461461461461461 (decimal approximation)(*).

The recurring decimal 13.33461 can be written as a ratio of two integers having 333032 as the numerator and 24975 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.