What is 16.45714571457145714... as a Fraction?

16.45714 equals 1645559999 as a fraction.

How do you turn 16.45714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 16.45714 repeating into a fraction, begin writing this simple equation:

n = 16.45714 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (5714) and 1 digit in the non-repeating part (4). Multiply both sides by 10⁵ = 100000.

100000 × n = 1645714.5714 (equation 2)

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

100000 × n = 1645714.5714
    10 × n = 16.45714
 99990 × n = 1645550

164555099990 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 10 (the GCF - greatest common factor).

n = 164555099990 = 1645550 ÷ 1099990 ÷ 10 = 1645559999. So,

16.45714 = 1645559999 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 1645559999 is also equal to 1645719999 when expressed as a mixed number.

The repeating decimal 16.45714 (vinculum notation) has a repeated block length of 4. It is also represented as 16.4571457145714... (ellipsis notation) which equals approximately 16.457145714571457145714 (decimal approximation)(*).

The recurring decimal 16.45714 can be written as a ratio of two integers having 164555 as the numerator and 9999 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.