What is 22.2384384384... as a Fraction?

22.2384 equals 370271665 as a fraction.

How do you turn 22.2384 repeating into a fraction?

Detailed Answer:

Step 1: To convert 22.2384 repeating into a fraction, begin writing this simple equation:

n = 22.2384 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (384) and 1 digit in the non-repeating part (2). Multiply both sides by 10⁴ = 10000.

10000 × n = 222384.384 (equation 2)

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

10000 × n = 222384.384
   10 × n = 22.2384
 9990 × n = 222162

2221629990 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 6 (the GCF - greatest common factor).

n = 2221629990 = 222162 ÷ 69990 ÷ 6 = 370271665. So,

22.2384 = 370271665 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 370271665 is also equal to 223971665 when expressed as a mixed number.

The repeating decimal 22.2384 (vinculum notation) has a repeated block length of 3. It is also represented as 22.2384384384... (ellipsis notation) which equals approximately 22.2384384384384384 (decimal approximation)(*).

The recurring decimal 22.2384 can be written as a ratio of two integers having 37027 as the numerator and 1665 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.