What is 16.809090909... as a Fraction?

16.809 equals 1849110 as a fraction.

How do you turn 16.809 repeating into a fraction?

Detailed Answer:

Step 1: To convert 16.809 repeating into a fraction, begin writing this simple equation:

n = 16.809 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (09) and 1 digit in the non-repeating part (8). Multiply both sides by 10³ = 1000.

1000 × n = 16809.09 (equation 2)

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

1000 × n = 16809.09
  10 × n = 16.809
 990 × n = 16641

16641990 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 = 16641990 = 16641 ÷ 9990 ÷ 9 = 1849110. So,

16.809 = 1849110 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 1849110 is also equal to 1689110 when expressed as a mixed number.

The repeating decimal 16.809 (vinculum notation) has a repeated block length of 2. It is also represented as 16.8090909... (ellipsis notation) which equals approximately 16.80909090909 (decimal approximation)(*).

The recurring decimal 16.809 can be written as a ratio of two integers having 1849 as the numerator and 110 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.