What is 20.6461461461... as a Fraction?

20.6461 equals 412511998 as a fraction.

How do you turn 20.6461 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.6461 repeating into a fraction, begin writing this simple equation:

n = 20.6461 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (461) and 1 digit in the non-repeating part (6). Multiply both sides by 10⁴ = 10000.

10000 × n = 206461.461 (equation 2)

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

10000 × n = 206461.461
   10 × n = 20.6461
 9990 × n = 206255

2062559990 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 5 (the GCF - greatest common factor).

n = 2062559990 = 206255 ÷ 59990 ÷ 5 = 412511998. So,

20.6461 = 412511998 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 412511998 is also equal to 2012911998 when expressed as a mixed number.

The repeating decimal 20.6461 (vinculum notation) has a repeated block length of 3. It is also represented as 20.6461461461... (ellipsis notation) which equals approximately 20.6461461461461461 (decimal approximation)(*).

The recurring decimal 20.6461 can be written as a ratio of two integers having 41251 as the numerator and 1998 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.