What is 12.121515... as a Fraction?

12.1215 equals 400013300 as a fraction.

How do you turn 12.1215 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.1215 repeating into a fraction, begin writing this simple equation:

n = 12.1215 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (15) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁴ = 10000.

10000 × n = 121215.15 (equation 2)

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

10000 × n = 121215.15
  100 × n = 12.1215
 9900 × n = 120003

1200039900 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 3 (the GCF - greatest common factor).

n = 1200039900 = 120003 ÷ 39900 ÷ 3 = 400013300. So,

12.1215 = 400013300 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 400013300 is also equal to 124013300 when expressed as a mixed number.

The repeating decimal 12.1215 (vinculum notation) has a repeated block length of 2. It is also represented as 12.12151515... (ellipsis notation) which equals approximately 12.121515151515 (decimal approximation)(*).

The recurring decimal 12.1215 can be written as a ratio of two integers having 40001 as the numerator and 3300 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.