What is 21.21333... as a Fraction?

21.213 equals 159175 as a fraction.

How do you turn 21.213 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.213 repeating into a fraction, begin writing this simple equation:

n = 21.213 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (3) and 2 digits in the non-repeating part (21). Multiply both sides by 10³ = 1000.

1000 × n = 21213.3 (equation 2)

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

1000 × n = 21213.3
 100 × n = 21.213
 900 × n = 19092

19092900 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 = 19092900 = 19092 ÷ 12900 ÷ 12 = 159175. So,

21.213 = 159175 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 159175 is also equal to 211675 when expressed as a mixed number.

The repeating decimal 21.213 (vinculum notation) has a repeated block length of 1. It is also represented as 21.21333... (ellipsis notation) which equals approximately 21.2133333 (decimal approximation)(*).

The recurring decimal 21.213 can be written as a ratio of two integers having 1591 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.