What is 6.82307230723072307... as a Fraction?

6.82307 equals 22741333330 as a fraction.

How do you turn 6.82307 repeating into a fraction?

Detailed Answer:

Step 1: To convert 6.82307 repeating into a fraction, begin writing this simple equation:

n = 6.82307 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (2307) and 1 digit in the non-repeating part (8). Multiply both sides by 10⁵ = 100000.

100000 × n = 682307.2307 (equation 2)

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

100000 × n = 682307.2307
    10 × n = 6.82307
 99990 × n = 682239

68223999990 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 3 (the GCF - greatest common factor).

n = 68223999990 = 682239 ÷ 399990 ÷ 3 = 22741333330. So,

6.82307 = 22741333330 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 22741333330 is also equal to 62743333330 when expressed as a mixed number.

The repeating decimal 6.82307 (vinculum notation) has a repeated block length of 4. It is also represented as 6.8230723072307... (ellipsis notation) which equals approximately 6.823072307230723072307 (decimal approximation)(*).

The recurring decimal 6.82307 can be written as a ratio of two integers having 227413 as the numerator and 33330 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.