What is 22.22111... as a Fraction?

22.221 equals 19999900 as a fraction.

How do you turn 22.221 repeating into a fraction?

Detailed Answer:

Step 1: To convert 22.221 repeating into a fraction, begin writing this simple equation:

n = 22.221 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (1) and 2 digits in the non-repeating part (22). Multiply both sides by 10³ = 1000.

1000 × n = 22221.1 (equation 2)

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

1000 × n = 22221.1
 100 × n = 22.221
 900 × n = 19999

n = 19999900 (answer)

22.221 = 19999900 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 19999900 is also equal to 22199900 when expressed as a mixed number.

The repeating decimal 22.221 (vinculum notation) has a repeated block length of 1. It is also represented as 22.22111... (ellipsis notation) which equals approximately 22.2211111 (decimal approximation)(*).

The recurring decimal 22.221 can be written as a ratio of two integers having 19999 as the numerator and 900 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.