What is 20.11461461461... as a Fraction?

20.11461 equals 401891998 as a fraction.

How do you turn 20.11461 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.11461 repeating into a fraction, begin writing this simple equation:

n = 20.11461 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (461) and 2 digits in the non-repeating part (11). Multiply both sides by 10⁵ = 100000.

100000 × n = 2011461.461 (equation 2)

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

100000 × n = 2011461.461
   100 × n = 20.11461
 99900 × n = 2009450

200945099900 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 50 (the GCF - greatest common factor).

n = 200945099900 = 2009450 ÷ 5099900 ÷ 50 = 401891998. So,

20.11461 = 401891998 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 401891998 is also equal to 202291998 when expressed as a mixed number.

The repeating decimal 20.11461 (vinculum notation) has a repeated block length of 3. It is also represented as 20.11461461461... (ellipsis notation) which equals approximately 20.11461461461461461 (decimal approximation)(*).

The recurring decimal 20.11461 can be written as a ratio of two integers having 40189 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.