What is 20.109999... as a Fraction?

20.109 equals 2011100 as a fraction.

How do you turn 20.109 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.109 repeating into a fraction, begin writing this simple equation:

n = 20.109 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (9) and 2 digits in the non-repeating part (10). Multiply both sides by 10³ = 1000.

1000 × n = 20109.9 (equation 2)

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

1000 × n = 20109.9
 100 × n = 20.109
 900 × n = 18099

18099900 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 9 (the GCF - greatest common factor).

n = 18099900 = 18099 ÷ 9900 ÷ 9 = 2011100. So,

20.109 = 2011100 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 2011100 is also equal to 2011100 when expressed as a mixed number.

The repeating decimal 20.109 (vinculum notation) has a repeated block length of 1. It is also represented as 20.10999... (ellipsis notation) which equals approximately 20.1099999 (decimal approximation)(*).

The recurring decimal 20.109 can be written as a ratio of two integers having 2011 as the numerator and 100 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.