What is 20.41414... as a Fraction?

20.414 equals 202199 as a fraction.

How do you turn 20.414 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.414 repeating into a fraction, begin writing this simple equation:

n = 20.414 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (14) and 1 digit in the non-repeating part (4). Multiply both sides by 10³ = 1000.

1000 × n = 20414.14 (equation 2)

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

1000 × n = 20414.14
  10 × n = 20.414
 990 × n = 20210

20210990 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 10 (the GCF - greatest common factor).

n = 20210990 = 20210 ÷ 10990 ÷ 10 = 202199. So,

20.414 = 202199 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 202199 is also equal to 204199 when expressed as a mixed number.

The repeating decimal 20.414 (vinculum notation) has a repeated block length of 2. It is also represented as 20.4141414... (ellipsis notation) which equals approximately 20.41414141414 (decimal approximation)(*).

The recurring decimal 20.414 can be written as a ratio of two integers having 2021 as the numerator and 99 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.