What is 12.1222... as a Fraction?

12.122 equals 109190 as a fraction.

How do you turn 12.122 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.122 repeating into a fraction, begin writing this simple equation:

n = 12.122 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (2) and 2 digits in the non-repeating part (12). Multiply both sides by 10³ = 1000.

1000 × n = 12122.2 (equation 2)

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

1000 × n = 12122.2
 100 × n = 12.122
 900 × n = 10910

10910900 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 = 10910900 = 10910 ÷ 10900 ÷ 10 = 109190. So,

12.122 = 109190 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 109190 is also equal to 121190 when expressed as a mixed number.

The repeating decimal 12.122 (vinculum notation) has a repeated block length of 1. It is also represented as 12.12222... (ellipsis notation) which equals approximately 12.1222222 (decimal approximation)(*).

The recurring decimal 12.122 can be written as a ratio of two integers having 1091 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.