What is 21.31212... as a Fraction?

21.312 equals 7033330 as a fraction.

How do you turn 21.312 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.312 repeating into a fraction, begin writing this simple equation:

n = 21.312 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (12) and 1 digit in the non-repeating part (3). Multiply both sides by 10³ = 1000.

1000 × n = 21312.12 (equation 2)

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

1000 × n = 21312.12
  10 × n = 21.312
 990 × n = 21099

21099990 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 = 21099990 = 21099 ÷ 3990 ÷ 3 = 7033330. So,

21.312 = 7033330 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 7033330 is also equal to 21103330 when expressed as a mixed number.

The repeating decimal 21.312 (vinculum notation) has a repeated block length of 2. It is also represented as 21.3121212... (ellipsis notation) which equals approximately 21.31212121212 (decimal approximation)(*).

The recurring decimal 21.312 can be written as a ratio of two integers having 7033 as the numerator and 330 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.