What is 12.121333... as a Fraction?

12.1213 equals 9091750 as a fraction.

How do you turn 12.1213 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.1213 repeating into a fraction, begin writing this simple equation:

n = 12.1213 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (3) and 3 digits in the non-repeating part (121). Multiply both sides by 10⁴ = 10000.

10000 × n = 121213.3 (equation 2)

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

10000 × n = 121213.3
 1000 × n = 12.1213
 9000 × n = 109092

1090929000 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 12 (the GCF - greatest common factor).

n = 1090929000 = 109092 ÷ 129000 ÷ 12 = 9091750. So,

12.1213 = 9091750 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 9091750 is also equal to 1291750 when expressed as a mixed number.

The repeating decimal 12.1213 (vinculum notation) has a repeated block length of 1. It is also represented as 12.121333... (ellipsis notation) which equals approximately 12.12133333 (decimal approximation)(*).

The recurring decimal 12.1213 can be written as a ratio of two integers having 9091 as the numerator and 750 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.