What is 20.19999... as a Fraction?

20.199 equals 1015 as a fraction.

How do you turn 20.199 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.199 repeating into a fraction, begin writing this simple equation:

n = 20.199 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (9) and 2 digits in the non-repeating part (19). Multiply both sides by 10³ = 1000.

1000 × n = 20199.9 (equation 2)

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

1000 × n = 20199.9
 100 × n = 20.199
 900 × n = 18180

18180900 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 180 (the GCF - greatest common factor).

n = 18180900 = 18180 ÷ 180900 ÷ 180 = 1015. So,

20.199 = 1015 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 1015 is also equal to 2015 when expressed as a mixed number.

The repeating decimal 20.199 (vinculum notation) has a repeated block length of 1. It is also represented as 20.19999... (ellipsis notation) which equals approximately 20.1999999 (decimal approximation)(*).

The recurring decimal 20.199 can be written as a ratio of two integers having 101 as the numerator and 5 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.