What is 0.41212... as a Fraction?

0.412 equals 68165 as a fraction.

How do you turn 0.412 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.412 repeating into a fraction, begin writing this simple equation:

n = 0.412 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (12) and 1 digit in the non-repeating part (4). Multiply both sides by 10³ = 1000.

1000 × n = 412.12 (equation 2)

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

1000 × n = 412.12
  10 × n = 0.412
 990 × n = 408

408990 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 6 (the GCF - greatest common factor).

n = 408990 = 408 ÷ 6990 ÷ 6 = 68165. So,

0.412 = 68165 as the lowest possible fraction.

The repeating decimal 0.412 (vinculum notation) has a repeated block length of 2. It is also represented as 0.4121212... (ellipsis notation) which equals approximately 0.41212121212 (decimal approximation)(*).

The recurring decimal 0.412 can be written as a ratio of two integers having 68 as the numerator and 165 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.