What is 19.6454545454545... as a Fraction?

19.645 equals 2161110 as a fraction.

How do you turn 19.645 repeating into a fraction?

Detailed Answer:

Step 1: To convert 19.645 repeating into a fraction, begin writing this simple equation:

n = 19.645 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (45) and 1 digit in the non-repeating part (6). Multiply both sides by 10³ = 1000.

1000 × n = 19645.45 (equation 2)

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

1000 × n = 19645.45
  10 × n = 19.645
 990 × n = 19449

19449990 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 = 19449990 = 19449 ÷ 9990 ÷ 9 = 2161110. So,

19.645 = 2161110 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 2161110 is also equal to 1971110 when expressed as a mixed number.

The repeating decimal 19.645 (vinculum notation) has a repeated block length of 2. It is also represented as 19.6454545... (ellipsis notation) which equals approximately 19.64545454545 (decimal approximation)(*).

The recurring decimal 19.645 can be written as a ratio of two integers having 2161 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.