What is 22.180606... as a Fraction?

22.1806 equals 18299825 as a fraction.

How do you turn 22.1806 repeating into a fraction?

Detailed Answer:

Step 1: To convert 22.1806 repeating into a fraction, begin writing this simple equation:

n = 22.1806 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (06) and 2 digits in the non-repeating part (18). Multiply both sides by 10⁴ = 10000.

10000 × n = 221806.06 (equation 2)

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

10000 × n = 221806.06
  100 × n = 22.1806
 9900 × n = 219588

2195889900 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 12 (the GCF - greatest common factor).

n = 2195889900 = 219588 ÷ 129900 ÷ 12 = 18299825. So,

22.1806 = 18299825 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 18299825 is also equal to 22149825 when expressed as a mixed number.

The repeating decimal 22.1806 (vinculum notation) has a repeated block length of 2. It is also represented as 22.18060606... (ellipsis notation) which equals approximately 22.180606060606 (decimal approximation)(*).

The recurring decimal 22.1806 can be written as a ratio of two integers having 18299 as the numerator and 825 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.