What is 8.30230230230... as a Fraction?

8.30230 equals 8294999 as a fraction.

How do you turn 8.30230 repeating into a fraction?

Detailed Answer:

Step 1: To convert 8.30230 repeating into a fraction, begin writing this simple equation:

n = 8.30230 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (230) and 2 digits in the non-repeating part (30). Multiply both sides by 10⁵ = 100000.

100000 × n = 830230.230 (equation 2)

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

100000 × n = 830230.230
   100 × n = 8.30230
 99900 × n = 829400

82940099900 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 100 (the GCF - greatest common factor).

n = 82940099900 = 829400 ÷ 10099900 ÷ 100 = 8294999. So,

8.30230 = 8294999 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 8294999 is also equal to 8302999 when expressed as a mixed number.

The repeating decimal 8.30230 (vinculum notation) has a repeated block length of 3. It is also represented as 8.30230230230... (ellipsis notation) which equals approximately 8.30230230230230230 (decimal approximation)(*).

The recurring decimal 8.30230 can be written as a ratio of two integers having 8294 as the numerator and 999 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.