What is 20.210101... as a Fraction?

20.2101 equals 10004495 as a fraction.

How do you turn 20.2101 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.2101 repeating into a fraction, begin writing this simple equation:

n = 20.2101 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (01) and 2 digits in the non-repeating part (21). Multiply both sides by 10⁴ = 10000.

10000 × n = 202101.01 (equation 2)

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

10000 × n = 202101.01
  100 × n = 20.2101
 9900 × n = 200080

2000809900 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 20 (the GCF - greatest common factor).

n = 2000809900 = 200080 ÷ 209900 ÷ 20 = 10004495. So,

20.2101 = 10004495 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 10004495 is also equal to 20104495 when expressed as a mixed number.

The repeating decimal 20.2101 (vinculum notation) has a repeated block length of 2. It is also represented as 20.21010101... (ellipsis notation) which equals approximately 20.210101010101 (decimal approximation)(*).

The recurring decimal 20.2101 can be written as a ratio of two integers having 10004 as the numerator and 495 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.