What is 12.22323... as a Fraction?

12.223 equals 12101990 as a fraction.

How do you turn 12.223 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.223 repeating into a fraction, begin writing this simple equation:

n = 12.223 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (23) and 1 digit in the non-repeating part (2). Multiply both sides by 10³ = 1000.

1000 × n = 12223.23 (equation 2)

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

1000 × n = 12223.23
  10 × n = 12.223
 990 × n = 12101

n = 12101990 (answer)

12.223 = 12101990 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 12101990 is also equal to 12221990 when expressed as a mixed number.

The repeating decimal 12.223 (vinculum notation) has a repeated block length of 2. It is also represented as 12.2232323... (ellipsis notation) which equals approximately 12.22323232323 (decimal approximation)(*).

The recurring decimal 12.223 can be written as a ratio of two integers having 12101 as the numerator and 990 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.