What is 21.146... as a Fraction?

21.146 equals 158675 as a fraction.

How do you turn 21.146 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.146 repeating into a fraction, begin writing this simple equation:

n = 21.146 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 2 digits in the non-repeating part (14). Multiply both sides by 10³ = 1000.

1000 × n = 21146.6 (equation 2)

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

1000 × n = 21146.6
 100 × n = 21.146
 900 × n = 19032

19032900 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 = 19032900 = 19032 ÷ 12900 ÷ 12 = 158675. So,

21.146 = 158675 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 158675 is also equal to 211175 when expressed as a mixed number.

The repeating decimal 21.146 (vinculum notation) has a repeated block length of 1. It is also represented as 21.14666... (ellipsis notation) which equals approximately 21.1466666 (decimal approximation)(*).

The recurring decimal 21.146 can be written as a ratio of two integers having 1586 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.