What is 12.106666... as a Fraction?

12.1066 equals 90875 as a fraction.

How do you turn 12.1066 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.1066 repeating into a fraction, begin writing this simple equation:

n = 12.1066 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 3 digits in the non-repeating part (106). Multiply both sides by 10⁴ = 10000.

10000 × n = 121066.6 (equation 2)

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

10000 × n = 121066.6
 1000 × n = 12.1066
 9000 × n = 108960

1089609000 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 120 (the GCF - greatest common factor).

n = 1089609000 = 108960 ÷ 1209000 ÷ 120 = 90875. So,

12.1066 = 90875 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 90875 is also equal to 12875 when expressed as a mixed number.

The repeating decimal 12.1066 (vinculum notation) has a repeated block length of 1. It is also represented as 12.106666... (ellipsis notation) which equals approximately 12.10666666 (decimal approximation)(*).

The recurring decimal 12.1066 can be written as a ratio of two integers having 908 as the numerator and 75 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.