What is 21.13111... as a Fraction?

21.131 equals 9509450 as a fraction.

How do you turn 21.131 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.131 repeating into a fraction, begin writing this simple equation:

n = 21.131 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (1) and 2 digits in the non-repeating part (13). Multiply both sides by 10³ = 1000.

1000 × n = 21131.1 (equation 2)

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

1000 × n = 21131.1
 100 × n = 21.131
 900 × n = 19018

19018900 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 2 (the GCF - greatest common factor).

n = 19018900 = 19018 ÷ 2900 ÷ 2 = 9509450. So,

21.131 = 9509450 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 9509450 is also equal to 2159450 when expressed as a mixed number.

The repeating decimal 21.131 (vinculum notation) has a repeated block length of 1. It is also represented as 21.13111... (ellipsis notation) which equals approximately 21.1311111 (decimal approximation)(*).

The recurring decimal 21.131 can be written as a ratio of two integers having 9509 as the numerator and 450 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.