What is 0.22121... as a Fraction?

0.221 equals 73330 as a fraction.

How do you turn 0.221 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.221 repeating into a fraction, begin writing this simple equation:

n = 0.221 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (21) and 1 digit in the non-repeating part (2). Multiply both sides by 10³ = 1000.

1000 × n = 221.21 (equation 2)

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

1000 × n = 221.21
  10 × n = 0.221
 990 × n = 219

219990 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 3 (the GCF - greatest common factor).

n = 219990 = 219 ÷ 3990 ÷ 3 = 73330. So,

0.221 = 73330 as the lowest possible fraction.

The repeating decimal 0.221 (vinculum notation) has a repeated block length of 2. It is also represented as 0.2212121... (ellipsis notation) which equals approximately 0.22121212121 (decimal approximation)(*).

The recurring decimal 0.221 can be written as a ratio of two integers having 73 as the numerator and 330 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.