What is 0.21111... as a Fraction?

0.211 equals 1990 as a fraction.

How do you turn 0.211 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.211 repeating into a fraction, begin writing this simple equation:

n = 0.211 (equation 1)

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

1000 × n = 211.1 (equation 2)

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

1000 × n = 211.1
 100 × n = 0.211
 900 × n = 190

190900 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 = 190900 = 190 ÷ 10900 ÷ 10 = 1990. So,

0.211 = 1990 as the lowest possible fraction.

The repeating decimal 0.211 (vinculum notation) has a repeated block length of 1. It is also represented as 0.21111... (ellipsis notation) which equals approximately 0.2111111 (decimal approximation)(*).

The recurring decimal 0.211 can be written as a ratio of two integers having 19 as the numerator and 90 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.