What is 19.999... as a Fraction?

19.9 equals 201 as a fraction.

How do you turn 19.9 repeating into a fraction?

Detailed Answer:

Step 1: To convert 19.9 repeating into a fraction, begin writing this simple equation:

n = 19.9 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (9), so multiply both sides by 10¹ = 10.

10 × n = 199.9 (equation 2)

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

10 × n = 199.9
 1 × n = 19.9
 9 × n = 180

1809 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 = 1809 = 180 ÷ 99 ÷ 9 = 201. So,

19.9 = 201 as the lowest possible fraction.

The repeating decimal 19.9 (vinculum notation) has a repeated block length of 1. It is also represented as 19.999... (ellipsis notation) which equals approximately 19.99999 (decimal approximation)(*).

The recurring decimal 19.9 can be written as a ratio of two integers having 20 as the numerator and 1 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.